# Transform orders and stochastic monotonicity of statistical functionals

Tommaso Lando Tommaso Lando was supported by the Italian funds ex MURST 60%
2021, by the Czech Science Foundation (GACR) under project 20-16764S and
moreover by SP2022/4, an SGS research project of VŠB-TU Ostrava. The support
is greatly acknowledged.
E-mail<EMAIL_ADDRESS>Corresponding author. Department of Economics,
University of Bergamo, Italy; Department of Finance, VŠB-TU Ostrava, Czech
Republic  Idir Arab Idir Arab and Paulo Eduardo Oliveira were partially
supported by the Centre for Mathematics of the University of Coimbra -
UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES.
E-mail<EMAIL_ADDRESS>CMUC, Department of Mathematics, University of
Coimbra, Portugal  Paulo Eduardo Oliveira E-mail<EMAIL_ADDRESS>CMUC,
Department of Mathematics, University of Coimbra, Portugal

###### Abstract

In some inferential statistical methods, such as tests and confidence
intervals, it is important to describe the stochastic behavior of statistical
functionals, aside from their large sample properties. We study such a
behavior in terms of the usual stochastic order. For this purpose, we
introduce a generalized family of stochastic orders, which is referred to as
transform orders, showing that it provides a flexible framework for deriving
stochastic monotonicity results. Given that our general definition makes it
possible to obtain some well known ordering relations as particular cases, we
can easily apply our method to different families of functionals. These
include some prominent inequality measures, such as the generalized entropy,
the Gini index, and its generalizations. We also illustrate the applicability
of our approach by determining the least favorable distribution, and the
behavior of some bootstrap statistics, in some goodness-of-fit testing
procedures.

Keywords: Gini index, hazard function, inequality, nonparametric test,
stochastic dominance.

##  1 Introduction

In statistics, one is often interested in estimating an unknown characteristic
of a given distribution $F$, rather than the distribution itself. In many such
cases, these characteristics may be represented by some probability functional
$T(F):\mathcal{F}\rightarrow\mathbb{R}$, where $\mathcal{F}$ is the space of
cumulative distribution functions (CDFs). The most intuitive way of estimating
$T(F)$ is the plug-in method, which simply consists in replacing the unknown
$F$ with its natural estimator, namely, the empirical CDF $\mathbb{F}_{n}$.
Correspondingly, we shall refer to $T(\mathbb{F}_{n})$ as a statistical
functional. However, since the empirical distribution is a random process,
that is, a function of the random sample, then statistical functionals are
random variables, and therefore it is important to study their behavior from a
stochastic point of view. While most of the work in the literature focuses on
large sample properties of statistical functionals, because of their
applications to estimation theory, we are concerned with a different problem,
namely, we are interested in establishing relationships between their
distribution functions that hold for any given sample, regardless of its size.

Let $T$ be some functional, and let $\mathbb{F}_{n}$ and $\mathbb{G}_{n}$ be
the empirical CDFs that correspond to random samples from $F$ and $G$,
respectively. We are interested in determining the conditions on $F$, $G$ and
$T$, such that $T(\mathbb{F}_{n})$ stochastically dominates
$T(\mathbb{G}_{n})$ with respect to the usual stochastic order (recalled below
in Definition 2). This is a relevant information as it may be used to help
characterizing finite sample properties of testing procedures, as we will be
describing later. Results of this kind may be rather complicated to derive,
depending on the mathematical form of $T$, and on the initial conditions on
the baseline distributions $F$ and $G$. In this regard, Arendarczyk et al.
(2021) recently studied the stochastic dominance behavior of the Greenwood
statistic (Greenwood, 1946) when the baseline distributions are comparable in
terms of the star order. We will show that this result is a particular case of
a much more general one, making it possible to establish dominance relations
within a wider family of statistical functionals, specifically, functionals
that are compatible with a general class of transform orders, generated by
some family of reference functions (see Definition 1 below).

Our main result, Theorem 1, presented in Section 2, shows that, if the
baseline distributions are ordered with respect to some transform order, the
statistical functionals are stochastically ordered, provided that the
corresponding probability functional is order-preserving with respect to the
same transform order. This result has a wide range of applicability, as our
definition of transform order includes the usual stochastic order, the convex
transform order, the star order, the superadditive and the dispersive orders
(Shaked and Shanthikumar, 2007). Therefore, we are able to derive stochastic
properties of several important families of functionals.

As an immediate application of Theorem 1, in Section 3 we obtain the
stochastic monotonicity of many important measures of inequality, including
the generalized entropy class (Shorrocks, 1980; Shorrocks and Slottje, 2002),
the well known Gini index (Gini, 1912) and some of its generalizations
(Mehran, 1976; Donaldson and Weymark, 1980). Differently, in Section 4, we use
Theorem 1 to determine, in a simple way, the least favorable distribution, as
well as the behavior of some bootstrap statistics, for some goodness-of-fit
testing problems, related to the convex order or to the star order, which are
relevant in areas such as survival analysis, reliability, and shape-
constrained inference (Barlow et al., 1971).

##  2 Main results

We begin by establishing some notations. The random variables $X$ and $Y$ have
CDFs $F$ and $G$ and supports $S_{F}$ and $S_{G}$, respectively. Let us denote
with $F^{-1}$ the left continuous generalized inverse, namely, the quantile
function, of $F$. We recall that a stochastic order is a binary relation
$\succ$ over $\mathcal{F}$ that is reflexive and transitive. In particular,
observe that $\succ$ does not generally satisfy the antisymmetry property,
that is, $F\succ G$ and $G\succ F$ do not necessarily imply $X=_{d}Y$, and it
is generally not total. A functional $T$ is said to be isotonic, or order-
preserving, with respect to $\succ$, whenever, for every pair
$F,G\in\mathcal{F}$ such that $F\succ G$, it holds that $T(F)\geq T(G)$. Let
us denote by $\mathcal{T}$ the class of functionals defined on $\mathcal{F}$.
Then, we may represent by $\mathcal{I}(\succ)$ the class of functionals that
are isotonic with respect to $\succ$, that is,

$\mathcal{I}(\succ)=\\{T\in\mathcal{T}:F\succ G\implies T(F)\geq
T(G),\;\forall F,G\in\mathcal{F}\\}.$

Likewise, the class of functionals that are antitonic, or order-reversing,
with respect to $\succ$, is denoted as
$\mathcal{A}(\succ)=\\{T\in\mathcal{T}:F\succ G\implies T(F)\leq
T(G),\;\forall F,G\in\mathcal{F}\\}$. It is worth noticing that, given a pair
of stochastic orders, say $\succ_{0}$ and $\succ_{1}$, if
$F\succ_{0}{G}\implies{F}\succ_{1}{G}$, then
$\mathcal{I}(\succ_{1})\subset\mathcal{I}(\succ_{0})$.

Given that stochastic orders and probability functionals depend only on the
distribution functions of the random variables, we may write $F\succ G$ or
$X\succ Y$, and $T(F)$ or $T(X)$, interchangeably.

###  2.1 A general family of orders

We now introduce the following definition of a transform order, generated by a
family of functions $\mathcal{C}$, whose choice enables the generalization of
some well known stochastic orders. As usual, $f|_{E}$ denotes the restriction
of a function $f$ to the set $E$.

######  Definition 1.

Let $\mathcal{C}$ be some family of non-decreasing functions. We say that $X$
dominates $Y$ with respect to the $\mathcal{C}$-transform order, denoted by
$X\geq_{\mathcal{C}}Y$, if $F^{-1}\circ G|_{S_{G}}\in\mathcal{C}$.

Note that $X=_{d}F^{-1}\circ G(Y)$, so, basically, a transform order holds
when the function that transforms (distributionally) one random variable into
the other satisfies some properties of interest. Such properties fully
characterize the dominance relations. It is easy to see that
$\geq_{\mathcal{C}}$ fulfills the basic properties of stochastic orders, and,
moreover, $\mathcal{C}_{0}\subset\mathcal{C}_{1}$ means that
$F\geq_{\mathcal{C}_{0}}G\implies F\geq_{\mathcal{C}_{1}}G$. As limiting
cases, if $\mathcal{C}$ is the class of non-decreasing functions, then
$\geq_{\mathcal{C}}$ is always verified, whereas if $\mathcal{C}$ contains
just the identity function then $\geq_{\mathcal{C}}$ coincides with equality
in distribution, $=_{d}$. The usual stochastic order, generally defined by the
relation $F(x)\leq G(x)$, for every $x\in\mathbb{R}$, (Shaked and
Shanthikumar, 2007), is a transform order, obtained by choosing $\mathcal{C}$
as the class of functions $\psi$ such that $\psi(x)\geq x$. Other well known
transform orders my be obtained by appropriately choosing $\mathcal{C}$: the
classes of convex, star-shaped, or superadditive functions, yield the convex
transform order, the star order, or the superadditive order, respectively.
Also the dispersive order can be obtained by choosing $\mathcal{C}$ as the
class of functions $\psi$ such that $\psi(x)-x$ is nondecreasing (Shaked and
Shanthikumar, 2007). Since these orders are particularly important, we recall
their definitions below (remember that $F^{-1}\circ G$ is always non-
decreasing by construction). However, Definition 1 above is quite general and
enables the possibility of defining new stochastic orders.

######  Definition 2.

We say that $X$ dominates $Y$ with respect to

  1. 1.

the usual stochastic order, denoted by $X\geq_{st}Y$, if $F^{-1}\circ
G|_{S_{G}}(x)\geq x$, or equivalently, $F(x)\leq G(x)$, for every
$x\in\mathbb{R}$;

  2. 2.

the convex transform order, denoted by $X\geq_{c}Y$, if $F^{-1}\circ
G|_{S_{G}}$ is convex;

  3. 3.

the star order, denoted by $X\geq_{*}Y$, if $F^{-1}\circ G|_{S_{G}}$ is star-
shaped;

  4. 4.

the superadditive order, denoted by $X\geq_{su}Y$, if $F^{-1}\circ G|_{S_{G}}$
is superadditive;

  5. 5.

the dispersive order, denoted by $X\geq_{disp}Y$, if $(F^{-1}\circ
G(x)-x)|_{S_{G}}$ is increasing.

For non-negative random variables, the following relations hold:

$X\geq_{c}Y\implies X\geq_{*}Y\iff\log X\geq_{disp}\log Y\implies
X\geq_{su}Y.$

In the sequel, we will further extend Definition 2 to enable comparisons of
(discrete) empirical distributions, which is necessary to obtain the desired
stochastic monotonicity between statistical functionals.

###  2.2 Stochastic monotonicity results

We are concerned with the problem of ranking statistical functionals in terms
of the usual stochastic order. Therefore, we are interested in determining the
conditions on $F$, $G$ and $T$ such that $P(T(\mathbb{F}_{n})\leq x)\leq
P(T(\mathbb{G}_{n})\leq x)$ holds for every $x\in\mathbb{R}$. Results of this
kind are not predictable in general. In this regard, intuitively, one may
wonder whether $T(F)\geq T(G)$ is sufficient for
$T(\mathbb{F}_{n})\geq_{st}T(\mathbb{G}_{n})$, or whether $X\succ Y$ and
$T\in\mathcal{I}(\succ)$ imply $T(\mathbb{F}_{n})\geq_{st}T(\mathbb{G}_{n})$,
for a general order $\succ$. It is important to remark that, in general, this
is not true (see Subsection 3.3 below).

Our main result, Theorem 1, shows that transform orders provide a general
framework for deriving stochastic monotonicity of statistical functionals of
the form $T(\mathbb{F}_{n})$, conditionally on the isotonicity properties of
$T$. However, before proceeding, we need some further preliminary definitions
and notations. Indeed, some transform orders may be not defined for
realizations, $F_{n}$ and $G_{n}$, of the empirical distribution functions
$\mathbb{F}_{n}$ and $\mathbb{G}_{n}$, as these realizations are discrete. For
instance, in the empirical case, the composition $F^{-1}_{n}\circ G_{n}$ is a
step function, hence neither convex nor star-shaped, so these CDFs are not
comparable in terms of $\geq_{c}$ or $\geq_{*}$. Therefore, to prove our main
results, we must extend, when needed, any transform order to the class of
observed empirical distributions, according to the following general
definition.

######  Definition 3.

Let $\mathcal{C}$ be some family of non-decreasing functions. We say that $X$
dominates $Y$ with respect to the extended $\mathcal{C}$-transform order,
denoted as $X\geq^{e}_{\mathcal{C}}Y$, if there exists a function
$\phi\in\mathcal{C}$ such that $\phi|_{S_{G}}=F^{-1}\circ G|_{S_{G}}$.

Note the distinction between Definitions 1 and 3: the former requires that
$F^{-1}\circ G$ satisfies some property, described by $\mathcal{C}$, on the
support of $G$, while the latter assumes the existence of some function
$\phi\in\mathcal{C}$, which coincides with the composition $F^{-1}\circ G$ on
the support of $G$, formally, $\phi|_{S_{G}}=F^{-1}\circ G|_{S_{G}}$. This is
useful because, for instance, notions such as convexity and star-shapedness
are generally not defined for functions whose support is a discrete set of
points. In such cases, Definition 3 makes it possible to consider
$\mathcal{C}$ as the class of convex or star-shaped functions, even when $F$
and $G$ are empirical distributions, which is out of the scope of Definition
1. Indeed, with regard to the pair $F_{n}$, $G_{n}$, where
$S_{G_{n}}=\\{y_{1},\ldots,y_{n}\\}$, the ordering
$F_{n}{\geq^{e}_{\mathcal{C}}}G_{n}$ holds if there exists a function
$\phi\in\mathcal{C}$ such that $\phi(y_{i})=F^{-1}_{n}\circ G_{n}(y_{i})$,
$i=1,\ldots,n$. Moreover, note that $\geq_{\mathcal{C}}^{e}$ is a weaker
version of $\geq_{\mathcal{C}}$, that is, $X\geq_{\mathcal{C}}Y$ implies
$X\geq_{\mathcal{C}}^{e}Y$.

We may now establish our main result. Consider a sample $y_{1},\ldots,y_{n}$
from $Y$. As $X=_{d}F^{-1}\circ G(Y)$, the values ${x_{i}^{\ast}=}F^{-1}\circ
G(y_{i})$, $i=1,\ldots,n$, may be seen as observations from $X$. Now, let us
denote with $G_{n}$ the realization of $\mathbb{G}_{n}$ corresponding to the
sample $y_{1},\ldots,y_{n}$, and with $F^{\ast}_{n}$ the particular
realization of $\mathbb{F}_{n}$ corresponding to the sample
$x_{1}^{\ast},\ldots,x_{n}^{\ast}$. Then, we obtain the following sufficient
conditions for $T(\mathbb{F}_{n})\geq_{st}T(\mathbb{G}_{n})$.

######  Theorem 1.

  1. 1.

If $X\geq_{\mathcal{C}}^{e}Y$ and $T\in\mathcal{I}(\geq^{e}_{\mathcal{C}})$,
then $T(\mathbb{F}_{n})\geq_{st}T(\mathbb{G}_{n})$.

  2. 2.

If $X\geq_{\mathcal{C}}^{e}Y$ and $T(F_{n}^{*})\geq T(G_{n})$, then
$T(\mathbb{F}_{n})\geq_{st}T(\mathbb{G}_{n})$.

###### Proof.

Let $\tilde{X}_{n}$ and $\tilde{Y}_{n}$ be two random variables whose
distributions are $F^{\ast}_{n}$ and $G_{n}$, respectively. Obviously,
$F^{-1}\circ G(\tilde{Y}_{n})=_{d}\tilde{X}_{n}$. If
$X\geq_{\mathcal{C}}^{e}Y$, then there exists some $\phi\in\mathcal{C}$
satisfying Definition 3. Let $Z$ be a random variable with distribution $H$
and support $S_{H}$ included in $S_{G}$. $X\geq_{\mathcal{C}}^{e}Y$ implies

$\phi|_{S_{H}}=(\phi|_{S_{G}})|_{S_{H}}=(F^{-1}\circ
G|_{S_{G}})|_{S_{H}}=F^{-1}\circ G\circ H^{-1}\circ H|_{S_{H}}.$

Note that the quantile function of $F^{-1}\circ G(Z)$ is $F^{-1}\circ G\circ
H^{-1}$, hence the latter relation is equivalent to $F^{-1}\circ
G(Z)\geq_{\mathcal{C}}^{e}Z$, for every random variable $Z$ with support
included in that of $Y$. Given that the support of $\tilde{Y}_{n}$ is always
included in the support of $Y$, this yields $\tilde{X}_{n}=_{d}F^{-1}\circ
G(\tilde{Y}_{n})\geq^{e}_{\mathcal{C}}\tilde{Y}_{n}$. Now, in case 1), as we
are assuming that $T\in\mathcal{I}(\geq^{e}_{\mathcal{C}})$, it follows that
$T(F^{\ast}_{n})\geq T(G_{n})$, while, in case 2), this holds by assumption.
Note that we are not interested in comparing $G_{n}$ and $F^{\ast}_{n}$,
however, this result contains information about the distributional behavior of
$T(\mathbb{F}_{n})$ and $T(\mathbb{G}_{n})$. In fact, since
$T(F^{\ast}_{n})\geq T(G_{n})$ holds for every possible pair of realizations
$x_{1}^{\ast},\ldots,x_{n}^{\ast}$ and $y_{1},\ldots,y_{n}$, obtained as
above, the latter relation can be equivalently expressed as
$T(\mathbb{F}_{n})\geq_{st}T(\mathbb{G}_{n})$ (again, this characterization of
the usual stochastic order is due to the probability integral transform, see
Theorem 1.A.1 of Shaked and Shanthikumar (2007)). ∎

Theorem 1 demonstrates the practical usefulness of our Definition 3. In fact,
transform orders may sometimes not be meaningful, especially when choosing an
unusual reference class $\mathcal{C}$. However, they may be employed merely as
tools for deriving stochastic properties of statistical functionals. For
instance, to apply part 1 of Theorem 1, it is enough to show that the
functional of interest $T$ belongs to some isotonic family,
$\mathcal{I}(\geq^{e}_{\mathcal{C}})$, and to do so, one may use existing
results which ensure that $T\in\mathcal{I}(\geq^{e}_{\mathcal{C}})$. In
particular, as we discuss in the next section, this approach works when
$\mathcal{C}$ is the class of functions that defines the star order or the
convex transform order, enabling the determination of the stochastic
monotonicity of the main measures of inequality and skewness. Obviously, if
$T$ is isotonic with more than one of the orders in Definition 3, we should
use the weakest one, in order to enlarge the range of applicability of the
result. On the other hand, it might happen that our functional $T$ is not
isotonic with known orders. In this case, we may try to define a new ad hoc
transform order such that $T\in\mathcal{I}(\geq^{e}_{\mathcal{C}})$, for
instance by choosing a wider class $\mathcal{C}$ or by defining $\mathcal{C}$
according to the properties of $T$. Alternatively, we may apply part 2 of
Theorem 1, replacing the isotonicity assumption,
$T\in\mathcal{I}(\geq_{\mathcal{C}}^{e})$, with the weaker order-preserving
property $T(F_{n}^{*})\geq T(G_{n})$.

Both approaches may be useful, depending on the situation. Several results
available in literature assert the isotonicity of many popular probability
functionals, with respect to some of the transform orders mentioned above,
hence part 1 immediately provides the ordering between the corresponding
statistical functionals, as discussed in the following section. When such
isotonicity results are not available, part 2 offers a weaker order-preserving
assumption, allowing for the same conclusion. This latter approach will be
explored in Section 4.

##  3 Stochastic behavior of inequality measures

We now consider some important families of statistical functionals that are
commonly employed in several fields, including statistics, economics and
finance. It should be noted that functionals that are isotonic with the usual
stochastic order are generally seen as location functionals. Since every
function of the random sample, which is nondecreasing in each argument, has a
stochastically increasing behavior (Shaked and Shanthikumar, 2007, Theorem
1.A.3), then the stochastic monotonicity of location estimators may be easily
derived. Therefore, we focus on some other families of functionals, that are
generally not isotonic with the usual stochastic order, namely inequality
measures. As discussed later in this section, inequality measures must be
isotonic with the star order, so they fit perfectly to our framework.
Similarly, notice that isotonicity with the convex transform order is a basic
condition for a skewness measure (van Zwet, 1964; Oja, 1981; Eberl and Klar,
2021), therefore our method may be also applied to skewness measures, but this
is beyond the purpose of this paper.

Hereafter, we focus on non-negative random variables with finite mean. Let
$L_{F}(p)=\int_{0}^{p}F^{-1}(t)\,dt/{\mu_{X}}$, $p\in[0,1]$ be the Lorenz
curve of $F$ (as usual, $\mu_{X}$ represents the mean of $X$). The Lorenz
curve is a primary tool for representation of inequality (e.g. income
inequality), as it is typically understood that the higher of two non-
intersecting Lorenz curves shows less inequality. This gives rise to the
definition of the Lorenz order.

######  Definition 4.

We say that $X$ dominates $Y$ with respect to the Lorenz order, denoted by
$X\geq_{L}Y$, if $L_{F}(p)\leq L_{G}(p)$ for every $p\in[0,1]$.

Note that the reverse relation ($L_{F}(p)\geq L_{G}(p)$) is sometimes used to
define the Lorenz order. We use the same notation as in Marshall et al.
(2011), that is, $X$ dominates $Y$ if it exhibits higher inequality, as
measured by the Lorenz curve. Using standard arguments in the ordering theory,
it is possible to derive several classes of probability functionals that are
isotonic with $\geq_{L}$. In particular, any functional $I(X)$ satisfying the
following basic properties (see Shorrocks (1980)), may be seen as an
inequality measure:

  1. 1.

$I\in\mathcal{I}(\geq_{L})$;

  2. 2.

$I(X)\geq I(\mu_{X})$;

  3. 3.

$I(aX)=I(X),\;a>0$;

  4. 4.

$I(X+b)\leq I(X),\;b>0$.

Actually, properties 2–4 are redundant, as they are implied by 1, that is, the
Lorenz isotonicity of the inequality measure $I$ is the crucial property. It
is well known that the star order implies the Lorenz order (Shaked and
Shanthikumar, 2007), so that
$\mathcal{I}({\geq_{L}})\subset\mathcal{I}(\geq_{*})$. However, since the star
order is usually defined only in the continuous case, we need the following
extension.

######  Lemma 1.

$\mathcal{I}({\geq_{L}})\subset\mathcal{I}(\geq_{*}^{e})$.

###### Proof.

Choose $T\in\mathcal{I}({\geq_{L}})$. The inclusion follows if we prove that
$F_{n}\geq_{*}^{e}G_{n}\implies F_{n}\geq_{L}G_{n}$, as this latter implies
that $T(F_{n})\geq T(G_{n})$. Let $\tilde{F}_{n}$ and $\tilde{G}_{n}$ be the
linear interpolators of the jump points of $F_{n}$ and $G_{n}$, respectively.
If $F_{n}\geq_{*}^{e}G_{n}$, then we can take
$\phi=\tilde{F}_{n}^{-1}\circ\tilde{G}_{n}$ in Definition 3. The function
$\phi$ coincides with the linear interpolator of ${F}_{n}^{-1}\circ{G}_{n}$,
which is star-shaped by construction. Then, since $\tilde{F}_{n}$,
$\tilde{G}_{n}$ are continuous, $\tilde{F}_{n}\geq_{\ast}\tilde{G}_{n}$ and
the ratio ${\tilde{F}_{n}^{-1}}/{\tilde{G}_{n}^{-1}}$ is nondecreasing (Shaked
and Shanthikumar, 2007, p. 214), the sequence
${{F}_{n}^{-1}(i/n)}/{{G}_{n}^{-1}(i/n)}$ is nondecreasing for $i=1,\ldots,n$,
whereas ${{F}_{n}^{-1}(p)}/{{G}_{n}^{-1}(p)}$ is constant between ${(i-1)}/n$
and ${i}/n$, for $i=2,\ldots,n$. Hence, the function
$R={{F}_{n}^{-1}}/{{G}_{n}^{-1}}$ is a nondecreasing step function. Without
loss of generality, let $\mu_{X}=\mu_{Y}=1$. Then
$L_{F_{n}}(p)-L_{G_{n}}(p)=\int_{0}^{p}({F}_{n}^{-1}(u)-{G}_{n}^{-1}(u))\,du$.
If $R$ is nondecreasing, and the means are equal, the quantile functions must
cross, and $R(\cdot)-1$ must have one sign change. However, $R(\cdot)-1$ has
the same sign as ${F}_{n}^{-1}-{G}_{n}^{-1}$, therefore the argument of
Theorem 4.B.4 of Shaked and Shanthikumar (2007) implies the result. ∎

Now, the following result is an immediate consequence of Theorem 1 and Lemma
1.

######  Corollary 1.

If $I\in\mathcal{I}(\geq_{L})$ and $X\geq_{*}Y$, then
$I(\mathbb{F}_{n})\geq_{st}I(\mathbb{G}_{n})$.

In the next subsections, we will focus on some particularly relevant families
of inequality measures.

###  3.1 Expectation-type transformations

It is well known that $X\geq_{L}Y$ if and only if
$\mathbb{E}(\phi(X/{\mu_{X}}))\geq\mathbb{E}(\phi(Y/{\mu_{Y}}))$, for every
convex function $\phi$ (Marshall et al., 2011). Therefore, any functional of
the form

$T_{\phi}(F)=\mathbb{E}\left(\phi\left(\tfrac{X}{\mu_{X}}\right)\right)=\int_{0}^{\infty}\phi\left(\tfrac{x}{\mu_{X}}\right)\,dF(x)=\int_{0}^{1}\phi\left(\tfrac{F^{-1}(p)}{\mu_{X}}\right)\,dp,$

where $\phi$ is convex, is isotonic with $\geq_{L}$, as it is easily seen to
satisfy the properties of inequality measures (Lando and Bertoli-Barsotti,
2016). Now, the corresponding statistical functional is
$T_{\phi}(\mathbb{F}_{n})=\sum\phi({X_{i}}/{\overline{X}_{n}})/n$, where
$X_{1},\ldots,X_{n}$ is a random sample from $X$ and $\overline{X}_{n}$ is the
sample mean. Several well-known indices belong to this general family, among
which we may note the class of generalized entropy, or additively decomposable
measures of inequality (Shorrocks, 1980; Shorrocks and Slottje, 2002),
obtained by setting $\phi_{r}(x)={x^{r}}/({r(r-1)})$, $r\neq 0,1$,
$\phi_{0}(x)=-\log x$, or $\phi_{1}(x)=x\log x$, respectively, which yield

$I_{r}(\mathbb{F}_{n})=\begin{cases}\frac{1}{r(r-1)n}\sum(\frac{X_{i}}{\overline{X}_{n}})^{r}&r\neq
0,1\\\ \frac{1}{n}\sum\log(\frac{\overline{X}_{n}}{X_{i}})&r=0\\\
\frac{1}{n}\sum\frac{X_{i}}{\overline{X}_{n}}\log(\frac{X_{i}}{\overline{X}_{n}})&r=1.\end{cases}$

This class gives, for $r=1$, the Theil index, that is, a shifted version of
the Shannon’s entropy measure, applied to the quantities
${X_{i}}/({n\overline{X}_{n}})$ instead of probabilities; for $r\in(0,1]$, a
monotonic transformation of the Atkinson’s class (Atkinson, 1970); and, for
$r=2$, a simple transformation of the coefficient of variation $CV_{n}$, that
is, $(2I_{2})^{1/2}=CV_{n}$. Moreover, when $\phi(x)=|x-1|$, we obtain the
relative mean absolute deviation
$T_{\phi}(\mathbb{F}_{n})=\sum|X_{i}-\overline{X}_{n}|/({n\overline{X}_{n}})$.
Then, the following result is an immediate consequence of Corollary 1.

######  Corollary 2.

If $X\geq_{*}Y$, then
$\sum\phi({X_{i}}/{\overline{X}_{n}})/n\geq_{st}\sum\phi({Y_{i}}/{\overline{Y}_{n}})/n$,
for every convex function $\phi$.

Notice that Arendarczyk et al. (2021) proved that the Greenwood statistic has
a stochastic increasing behavior with respect to the star order. However,
since the Greenwood statistic is a transformation of the coefficient of
variation, namely ${(1+CV_{n}^{2})}/n$, Theorem 1 of Arendarczyk et al. (2021)
follows as a consequence of Corollary 2.

###  3.2 Distorted expectations

Let $H$ be a distortion function, that is a non-decreasing function on the
unit interval, such that $H(0)=0$ and $H(1)=1$, and let
$\tilde{H}(p)=1-H(1-p)$ be the corresponding dual distortion function.
Probability functionals of the form

$\mathbb{E}_{H}(F)=\frac{1}{\mu_{X}}\int_{0}^{\infty}x\,dH\circ
F(x)=\frac{1}{\mu_{X}}\int_{0}^{1}F^{-1}(p)\,dH(p)=\frac{1}{\mu_{X}}\int_{0}^{\infty}\tilde{H}(1-F(x))\,dx,$

are generally referred to as distorted expectations, distortion risk measures,
or Gini-type functionals (Wang and Young, 1998; Muliere and Scarsini, 1989;
Lando and Bertoli-Barsotti, 2020). It can be seen that $X\geq_{L}Y$ if and
only if $\mathbb{E}_{H}(F)\leq(\geq)\mathbb{E}_{H}(G)$, for every concave
(convex) distortion $H$. Then, $\mathbb{E}_{H}$ is antitonic with respect to
$\geq_{L}$, provided that $H$ is concave. If we denote by
$X_{(1)},\ldots,X_{(n)}$ the order statistics of a random sample from $X$, the
corresponding statistical functional is

$\mathbb{E}_{H}(\mathbb{F}_{n})=\frac{1}{\overline{X}_{n}}\sum{X_{(i)}}\int^{\frac{i}{n}}_{\frac{i-1}{n}}\,dH(p)=\frac{1}{{\overline{X}_{n}}}\sum{X_{(i)}}\left[H\left(\tfrac{i}{n}\right)-H\left(\tfrac{i-1}{n}\right)\right].$

Such linear combinations of order statistics are generally referred to as
L-statistics (Serfling, 1984). Several important indices belong to this
family. For instance, by choosing $H(p)=1-(1-p)^{k}$, $k\geq 1$, we obtain the
class of generalized Gini indices $\Gamma_{H}=1-\mathbb{E}_{H}$, introduced by
Donaldson and Weymark (1980). In particular, the classic Gini coefficient of
inequality (Gini, 1912) is given by $\Gamma=\Gamma_{H}$, with
${H}(p)=1-(1-p)^{2}$, that is

$\Gamma(\mathbb{F}_{n})=1-\frac{1}{n^{2}{\overline{X}_{n}}}\sum{X_{(i)}}(2n-2i+1)=1-2\int_{0}^{1}L_{\mathbb{F}_{n}}(p)\,dp=\frac{\sum_{i=1}^{n}\sum_{j=1}^{n}|X_{(i)}-X_{(j)}|}{2n^{2}{\overline{X}_{n}}}.$

As for the expressions above, we recall that there are several alternative
ways to represent $\Gamma$ (see Chapter 1 of Yitzhaki and Schechtman (2013)).
For example, note that, if $k$ is a positive integer,
$\int_{0}^{\infty}(1-F(x))^{k}\,dx=\mathbb{E}(\min(X_{1},\ldots,X_{k}))$, so
that $\Gamma(F)=1-{\mathbb{E}(\min(X_{1},X_{2}))}/{\mu_{X}}$.

Similarly, the functional $\int_{0}^{\infty}F^{-1}(p)h(p)\,dp/{\mu_{X}}$,
where $h$ is non-decreasing and such that $H(p)=\int_{0}^{p}h(t)\,dt$,
$p\in[0,1]$, is isotonic with respect to $\geq_{L}$. If we set $w(p)=h(p)-1$,
$p\in[0,1]$, without loss of generality in terms of isotonicity, we obtain the
family of linear inequality measures

$\tilde{\Gamma}_{w}(F)=\frac{1}{\mu_{X}}\int_{0}^{1}F^{-1}(p)w(p)\,dp,$

studied by Mehran (1976). In particular, it can be shown that the weight
function $w(p)=2p-1$ yields again the Gini coefficient, whereas $w(p)=0$,
$p\in(0,1)$, $w(0)=-1$, $w(1)=1$ gives the relative range
$\tilde{\Gamma}_{w}(\mathbb{F}_{n})=(X_{(n)}-X_{(1)})/{\overline{X}_{n}}$. It
is easy to see that $\Gamma_{H}$ and $\tilde{\Gamma}_{w}$, as probability
functionals, satisfy the four properties of inequality measures discussed
earlier. Again, Corollary 1 gives the following result.

######  Corollary 3.

If $X\geq_{*}Y$, then
$\Gamma_{H}(\mathbb{F}_{n})\geq_{st}\Gamma_{H}(\mathbb{G}_{n})$, for every
concave distortion function $H$, and
$\tilde{\Gamma}_{w}(\mathbb{F}_{n}){\geq_{st}}\tilde{\Gamma}_{w}(\mathbb{G}_{n})$,
for every non-decreasing weight function $w$ on $[0,1]$ such that
$\int_{0}^{1}w(p)\,dp=0$.

###  3.3 Some useful remarks

One may wonder if, in Theorem 1, a transform order may be replaced by a
different kind of order $\succ$, that is, if, in general, $F\succ G$ and
$T\in\mathcal{I}(\succ)$ imply $T(\mathbb{F}_{n})\geq_{st}T(\mathbb{G}_{n})$,
where $\succ$ is not a transform order. For instance, can one replace
$\geq_{*}$ by $\geq_{L}$ in Corollary 1? More generally, one may even wonder
if the stochastic ordering assumption between the baseline distributions could
be relaxed: for instance, would $T(F)\geq T(G)$ suffices to obtain
$T(\mathbb{F}_{n})\geq_{st}T(\mathbb{G}_{n})$? (This would also mean that
$T(F)=T(G)$ implies $T(\mathbb{F}_{n})=_{d}T(\mathbb{G}_{n})$). The following
counterexample provides a negative answer to these conjectures.

Let $X$ and $Y$ be discrete random variables with uniform distributions on the
supports $\\{1,3.5,6,6.5,9,11\\}$ and $\\{2,3,5,7,7.5,10\\}$, respectively. It
is easily seen that $F\geq_{L}G$ and the Gini indices are such that
$\Gamma(F)>\Gamma(G)$, but $F\not\geq_{*}^{e}G$. Note that we focus on
discrete distributions, because, using continuous parametric models, it is
difficult to find instances such that $F\geq_{L}G$ and $F\not\geq_{*}^{e}G$.
In the discrete case, the distributions of $\Gamma(\mathbb{F}_{n})$ and
$\Gamma(\mathbb{G}_{n})$ are also discrete, with finite support included in
the unit interval. Therefore, we may obtain a precise approximation of these
distributions, for small sample sizes, by using a large number of simulation
runs. An approximate representation of the CDFs of $\Gamma(\mathbb{F}_{n})$
and $\Gamma(\mathbb{G}_{n})$, based on one million random samples of size
$n=3$ from $F$ and $G$, is shown in Figure 1. The functions are clearly
crossing, hence $\Gamma(\mathbb{F}_{n})\not\geq_{st}\Gamma(\mathbb{G}_{n})$.
In particular, we observe that these functions exhibit some “bumps” within
different intervals, and this represents an obstruction for the dominance
relation. When the distributions are star ordered ($F\geq_{\ast}^{e}G$), such
bumps occur within some overlapping intervals, indeed the dominance relation
is guaranteed by Corollary 1. We obtained a similar behavior for other small
sample sizes, such as $n=4,5$. This result is quite unexpected, however, we
may conclude that both conjectures above are false. In particular, the
behavior of the functionals is not “under control” if the sample size is
small, whereas we know that, for large sample sizes, under some conditions
(which are satisfied for probability functionals of the form $\int
t(x)\,dF(x)$, such as transformed and distorted expectations), statistical
functionals converge to the constant value $T(F)$. Therefore, transform orders
provide suitable tools for controlling the stochastic behavior of statistical
functionals.

Figure 1: CDFs of the Gini coefficient, generated by $10^{6}$ samples from $F$
(thick) and $G$ (thin).

##  4 Application to goodness-of-fit tests

In the statistical literature, tests of the null hypothesis
$\mathcal{H}^{R}_{0}:G^{-1}\circ F|_{R}\in\mathcal{C}$, versus the alternative
$\mathcal{H}^{R}_{1}:G^{-1}\circ F|_{R}\notin\mathcal{C}$, where $R\subset
S_{F}$ is some interval, $F$ is the distribution of interest and $G$ is known,
may be particularly interesting. Hereafter, we will focus on the case in which
$G(0)=0$, to avoid some technical issues. Given $G$, $G^{-1}\circ F$ is called
the generalized hazard function of $F$ (Barlow et al., 1971). Tests for
$G\geq_{c}F$ and $G\geq_{*}F$ have several applications: for example, by
choosing $G(x)=\mathcal{E}(x):=1-\exp(-x)$, $x\geq 0$, namely, the unit
exponential distribution, we obtain tests for the increasing hazard rate (IHR)
and the increasing hazard rate average (IHRA) properties, respectively, which
are fundamental tools for survival analysis and reliability theory (Marshall
and Olkin, 2007). Non parametric tests of these properties have been studied
extensively in the literature, most of which deal with the null hypothesis of
exponentiality ($F=\mathcal{E}$) versus the alternative that $F$ is IHR
(Proschan and Pyke, 1967; Barlow and Proschan, 1969; Bickel, 1969; Bickel and
Doksum, 1969) or IHRA (Deshpande, 1983; Klefsjö, 1983; Kochar, 1985; Link,
1989; Wells and Tiwari, 1991; Ahmad, 1994), but not exponential. In such
cases, it is clear that the distribution of the test statistic can be obtained
using $F=\mathcal{E}$. Differently, we study goodness-of-fit tests where the
determination of the distribution of the test statistic under the (composite)
null hypothesis is not obvious and it represents an important issue (Tenga and
Santner, 1984; Hall and van Keilegom, 2005; Durot, 2007; Groeneboom and
Jongbloed, 2012; Gijbels and Heckman, 2004; Lando et al., 2021; Lando, 2022).
As shown in the sequel, part 2 of Theorem 1 may be used to establish
stochastic ordering properties of test statistics under some circumstances of
interest. These results make it possible to control the behavior of the test
for a given sample size, which is an important aspect, comparable to
asymptotic properties.

In the next subsections, we shall remove the restriction $R$ and just consider
$\mathcal{H}_{0}:G^{-1}\circ F\in\mathcal{C}$ versus
$\mathcal{H}_{1}:G^{-1}\circ F\notin\mathcal{C}$, to simplify the notations.
It is easily seen that the results in Subsections 4.1 and 4.2 hold even in the
restricted case. Differently, a restricted null hypothesis is considered in
Subsection 4.2.1.

###  4.1 Testing convexity

Define the greatest convex minorant of a function $\phi$ as the largest convex
function $\phi_{c}$ that does not exceed $\phi$ (see Tenga and Santner (1984)
for a formal definition of this function). An intuitive way of testing
$\mathcal{H}_{0}:G\geq_{c}F$ versus $\mathcal{H}_{1}:G\not\geq_{c}F$ consists
in measuring the distance between $G^{-1}\\!\circ\\!\mathbb{F}_{n}$ and its
greatest convex minorant $(G^{-1}\\!\circ\\!\mathbb{F}_{n})_{c}$, or, more
generally,

$T^{{K}}_{c}(\mathbb{F}_{n})=\sup_{i=2,\ldots,n-1}\left|K\circ
G^{-1}\\!\circ\\!\mathbb{F}_{n}(X_{(i-1)})-K\circ(G^{-1}\\!\circ\\!\mathbb{F}_{n})_{c}(X_{(i)})\right|\\\
=\sup_{i=2,\ldots,n-1}\left(K\circ
G^{-1}(\tfrac{i-1}{n})-K\circ(G^{-1}\\!\circ\\!\mathbb{F}_{n})_{c}(X_{(i)})\right),$

where $K$ is some strictly increasing transformation defined on $S_{G}$. This
family of tests is quite general. Some interesting examples are obtained by
choosing $K$ as the identity function (Tenga and Santner, 1984; Durot, 2007;
Lando et al., 2021); or setting $K=G$ (Groeneboom and Jongbloed, 2012; Lando,
2022), in this latter case $T^{K}_{c}(\mathbb{F}_{n})$ boils down to a
Kolmogorov-Smirnov type test statistic. It can be seen that $T_{c}^{K}$ is
scale invariant, which is a basic condition for tests of this kind. Note that
the difference is evaluated among the shifted points $X_{(i)}$ and $X_{(i-1)}$
because, by construction,
$G^{-1}\\!\circ\\!\mathbb{F}_{n}(X_{(i-1)})\geq(G^{-1}\\!\circ\\!\mathbb{F}_{n})_{c}(X_{(i)})$.
A weighted version of $T^{K}_{c}(\mathbb{F}_{n})$ may also be considered
without loss of generality. Of course, since $\mathcal{H}_{0}$ is a composite
hypothesis, the determination of the least favorable distribution of
$T^{K}_{c}(\mathbb{F}_{n})$ under $\mathcal{H}_{0}$ is especially critical.
With regard to the IHR hypothesis, this distribution is indeed obtained by
simulating from the unit exponential, as proved by Tenga and Santner (1984).
We remark that a more general result may be achieved as an application of
Theorem 1. As expected, the least favorable distribution of
$T^{K}_{c}(\mathbb{F}_{n})$ under $\mathcal{H}_{0}:G\geq_{c}F$, is determined
just by simulating from $G$. Let $\Phi^{G}_{p}(F)=(G^{-1}\circ
F)_{c}(F^{-1}(p))$. The following lemma is a consequence of Theorem 2.2 of
Tenga and Santner (1984), and it establishes the order-reversing behavior of
$\Phi^{G}_{p}$.

######  Lemma 2.

Assume that $G\geq_{c}F$. Then
$\Phi^{G}_{i/n}(G_{n})\leq\Phi^{G}_{i/n}(F^{*}_{n})$, for $i=1,\ldots,n$.

Theorem 1 and Lemma 2 imply the following result. Note that this is a slight
generalization of existing results (Tenga and Santner, 1984; Lando et al.,
2021), however, here we show it to illustrate our method, as an immediate
application of part 2 of Theorem 1.

######  Proposition 1.

Under $\mathcal{H}_{0}:G\geq_{c}F$,
$T^{K}_{c}(\mathbb{G}_{n})\geq_{st}T^{K}_{c}(\mathbb{F}_{n})$.

###### Proof.

Clearly, by Lemma 2, $\Phi^{G}_{i/n}(G_{n})\leq\Phi^{G}_{i/n}(F^{*}_{n})$.
Note that

$T^{K}_{c}(\mathbb{F}_{n})=\sup_{i=2,\ldots,n-1}\left({K\circ}G^{-1}(\tfrac{i-1}{n})-{K\circ}\Phi^{G}_{\frac{i}{n}}(\mathbb{F}_{n})\right),$

then Theorem 1 implies the result, since $K$ is increasing.∎

Therefore, we reject $\mathcal{H}_{0}$ when $T^{K}_{c}(F_{n})\geq
c_{\alpha,n}$, where $c_{\alpha,n}$ is the solution of
$P(T^{K}_{c}(\mathbb{G}_{n})\geq c_{\alpha,n})=\alpha$. Proposition 1 ensures
that the size of the test is $\alpha$, that is, under $\mathcal{H}_{0}$,
$P(T^{K}_{c}(\mathbb{F}_{n})\geq c_{\alpha,n})\leq\alpha$. Moreover, under the
alternative “$G^{-1}\circ F$ is concave”, reversing Proposition 1, we can
similarly obtain $P(T^{K}_{c}(\mathbb{F}_{n})\geq c_{\alpha,n})\geq
P(T^{K}_{c}(\mathbb{G}_{n})\geq c_{\alpha,n})=\alpha$, that is, the test is
unbiased under concave alternatives. The $p$-value of the test is
$p=P(T^{K}_{c}(\mathbb{G}_{n})\geq T^{K}_{c}(F_{n}))$.

As well known, the determination of the least favorable distribution of
$T^{K}_{c}$ yields conservative tests. In order to improve the power of the
test against non-convex alternatives ($\mathcal{H}_{1}:G\not\geq_{c}F$), a
more modern approach consists in performing bootstrap resampling from the
distribution $F^{G}_{n}=G\circ(G^{-1}\circ F_{n})_{c}$, that is an isotonic
estimate of $F$, instead of sampling from $G$ (Groeneboom and Jongbloed,
2012). In particular, denote with $\hat{\mathbb{F}}^{G}_{n}$ the (bootstrap)
empirical distribution obtained by sampling from $F^{G}_{n}$. The following
result holds.

######  Proposition 2.

Under $\mathcal{H}_{0}:G\geq_{c}F$,
$T^{K}_{c}(\mathbb{G}_{n})\geq_{st}T^{K}_{c}(\hat{\mathbb{F}}^{G}_{n})$.

###### Proof.

Let $F^{G}_{n}$ be the estimate of $F$ which corresponds to the empirical
distribution $F_{n}^{*}$. $F_{n}^{G}$ is a realization of
$\hat{\mathbb{F}}^{G}_{n}$. Lemma 2 yields
$\Phi^{G}_{i/n}(G_{n})\leq\Phi^{G}_{i/n}(F^{*}_{n})$, for $i=1,\ldots,n$. Note
that $F^{G}_{n}$ is a minorant of $F^{*}_{n}$ (equivalently,
$(F^{*}_{n})^{-1}$ is a minorant of $(F^{G}_{n})^{-1}$). Then, since
$G^{-1}\circ F_{n}^{G}=(G^{-1}\circ F^{*}_{n})_{c}$, we obtain

$\Phi^{G}_{\frac{i}{n}}(G_{n})\leq\Phi^{G}_{\frac{i}{n}}(F^{*}_{n})=(G^{-1}\circ
F_{n}^{*})_{c}((F^{*}_{n})^{-1}(\tfrac{i}{n}))\\\ \leq(G^{-1}\circ
F_{n}^{*})_{c}((F^{G}_{n})^{-1}(\tfrac{i}{n}))=G^{-1}\circ
F_{n}^{G}\circ(F^{G}_{n})^{-1}(\tfrac{i}{n})=\Phi^{G}_{\frac{i}{n}}(F^{G}_{n}).$

Hence, as shown in the proof of Proposition 1, we obtain that
$T^{K}_{c}(\mathbb{G}_{n})\geq_{st}T^{K}_{c}(\hat{\mathbb{F}}^{G}_{n})$. ∎

Since quantiles are isotonic with $\geq_{st}$, Proposition 2 ensures that,
with this method, we obtain a smaller critical value and, correspondingly, a
higher value of the power function under $\mathcal{H}_{1}$. On the other hand,
a test obtained by this method does not satisfy anymore the properties implied
by Proposition 1 (size and unbiasedness).

###  4.2 Testing star-shapedness

Now, let us consider tests for $\mathcal{H}_{0}:G\geq_{*}F$ versus
$\mathcal{H}_{1}:G\not\geq_{*}F$. Note that $G\geq_{*}F$ requires that
$F(0)=0$. Define the greatest star-shaped minorant of a function $\phi$, as
the largest star-shaped function $\phi_{*}$ that does not exceed $\phi$ (see
Sect. 2 of Wang (1987) for a formal definition). The same arguments used
earlier may determine the least favorable distribution, under
$\mathcal{H}_{0}$, of the test statistic

${T^{K}_{*}(\mathbb{F}_{n})=\sup_{i=2,\ldots,n-1}\left(K\circ
G^{-1}(\tfrac{i-1}{n})-K\circ(G^{-1}\\!\circ\\!\mathbb{F}_{n})_{*}(X_{(i)})\right).}$

Let $\Psi^{G}_{p}(F)=(G^{-1}\circ F)_{*}(F^{-1}(p))$. The following lemma
establishes the order-reversing behavior of $\Psi_{p}^{G}$, corresponding to
Lemma 2.

######  Lemma 3.

Assume that $G\geq_{\ast}F$. Then
$\Psi^{G}_{i/n}(G_{n})\leq\Psi^{G}_{i/n}(F^{*}_{n})$, for $i=1,\ldots,n$.

###### Proof.

To simplify notations, let $y_{1},\ldots,y_{n}$ be ordered realizations from
$G$. The function $u=F^{-1}\circ G$ is, by assumption, anti-star-shaped, and
recall that $x_{i}^{*}=u(y_{i})$ are the observations which determine
$F_{n}^{*}$. Note that $G^{-1}\circ G_{n}(y_{i})=G^{-1}(i/n)$. We now describe
explicitly $(G^{-1}\circ G_{n})_{*}$ (see Wang (1988)). Let

$a_{i}=\frac{1}{y_{i}}G^{-1}\left(\frac{i-1}{n}\right),\quad i=1,\ldots,n,$

be the slope of the line connecting the origin to the point
$(y_{i},G^{-1}(({i-1})/n))$. The analytical expression of the greatest star-
shaped minorant is given by $(G^{-1}\circ G_{n})_{*}(x)=\alpha_{i}x$, for
$x\in[y_{i-1},y_{i})$, where $y_{0}=0$ and
$\alpha_{i}=\min\\{a_{j},\,j=i,\ldots,n\\}$ (note that $\alpha_{1}=0$, and,
moreover, star-shapedness requires that the slopes are increasing).
$(G^{-1}\circ F^{*}_{n})_{*}$ is defined similarly. Note that the two
functions have different domains, but coincide at corresponding points:
$G^{-1}\circ F^{*}_{n}(x_{i}^{*})=G^{-1}\circ G_{n}(y_{i})=G^{-1}(({i-1})/n)$.
We want to prove that $(G^{-1}\circ F^{*}_{n})_{*}(u(y_{i}))\geq(G^{-1}\circ
G_{n})_{*}(y_{i})$, for $i=1,\ldots n$. It is sufficient to prove

$\frac{G^{-1}(\frac{k-1}{n})}{u(y_{k})}u(y_{i})\geq\frac{G^{-1}(\frac{k-1}{n})}{y_{k}}y_{i}\iff\frac{u(y_{i})}{y_{i}}\geq\frac{u(y_{k})}{y_{k}},$

for $k=i,\ldots,n$, which follows form the anti-star-shapedness of the
function $u$, that is, $u(x)/x$ is decreasing. ∎

Using Lemma 3 and similar arguments as those used in the previous subsection,
we can ensure that the least favorable distribution of $T^{K}_{*}$, under
$\mathcal{H}_{0}$, is determined by simulating from $G$. Accordingly, the test
determined by $P(T^{K}_{*}(\mathbb{G}_{n})\geq c_{\alpha,n})=\alpha$ has size
$\alpha$ and it is unbiased against anti-star-shaped alternatives. Moreover,
the bootstrap estimator which arises from the isotonic estimate
$G\circ(G^{-1}\circ F_{n})_{*}$, denoted as $\tilde{\mathbb{F}}^{G}_{n}$,
satisfies an equivalent property to that established by Proposition 2. We
summarize these results as follows (the proof is omitted).

######  Proposition 3.

Under $\mathcal{H}_{0}:G\geq_{*}F$,

  1. 1.

$T^{K}_{*}(\mathbb{G}_{n})\geq_{st}T^{K}_{*}(\mathbb{F}_{n})$;

  2. 2.

$T^{K}_{*}(\mathbb{G}_{n})\geq_{st}T^{K}_{*}(\tilde{\mathbb{F}}^{G}_{n})$.

Differently from the tests discussed in the Section 4.1, which have been
studied extensively in the literature, tests of the null hypothesis
$G\geq_{*}F$ are relatively unexplored, so it is worth discussing them
further. In fact, our results can be useful for testing goodness-of-fit to the
IHRA family, which is particularly relevant in reliability and survival
analysis, or some generalizations of this property, as described by Barlow et
al. (1971).

####  4.2.1 An example: testing the IHRA property

A CDF $F$ is said to be IHRA if the hazard function $H=-\ln(1-F)$ is star-
shaped, or equivalently, if the hazard rate average (HRA) function ${H(x)}/x$
is increasing (Marshall and Olkin, 2007). Tests of the IHRA property have
already been considered in the literature, however, these are based on the
null hypothesis that $F$ is exponential, versus IHRA alternatives (Deshpande,
1983; Klefsjö, 1983; Kochar, 1985; Link, 1989; Wells and Tiwari, 1991; Ahmad,
1994), which is clearly different from our approach.

Plugging $G=K=\mathcal{E}$ into the expression of $T^{K}_{*}$, we obtain a
Kolmogorov-Smirnov type test statistic, which can be used to test goodness-of-
fit to the IHRA family. More generally, we may consider the following
restricted hypotheses $\mathcal{H}^{\nu}_{0}:H|_{S_{\nu}}$ is star-shaped
versus $\mathcal{H}^{\nu}_{1}:H|_{S_{\nu}}$ is not star-shaped, where
$S_{\nu}=\\{x:x\leq F^{-1}(1-\nu)\\}$, $\nu\in[0,1]$ (clearly
$\mathcal{H}_{0}$ coincides with $\mathcal{H}_{0}^{\nu}$). Let us define
$n_{\nu}$ by $X_{(n_{\nu})}=\mathbb{F}_{n}^{-1}(1-\nu)$, and
$S_{n,\nu}=\\{x\leq X_{(n_{\nu})}\\}$. Accordingly, we have the following
restricted test statistic:

$T^{\nu}_{*}(\mathbb{F}_{n})=\sup_{i=2,\ldots,n_{\nu}-1}\left(\tfrac{i-1}{n}-\mathcal{E}\circ\left(\mathbb{H}_{n}|_{S_{n,\nu}}\right)_{*}(X_{(i)})\right),$

where $\mathbb{H}_{n}=-\ln(1-\mathbb{F}_{n})$ is the empirical hazard
function. It is easy to see that this restricted test still satisfies the
properties established by Proposition 3, for every $\nu\in[0,1]$. Therefore,
the distribution of $T^{\nu}_{*}$ is obtained by simulating from
$G=\mathcal{E}$. We reject $\mathcal{H}_{0}$ when $T_{*}^{\nu}(F_{n})\geq
c_{\alpha,n,\nu}$, where $c_{\alpha,n,\nu}$ is the solution of
$P(T_{*}^{\nu}(\mathbb{G}_{n})\geq c_{\alpha,n,\nu})=\alpha$, giving the size
of the test. The test is unbiased against alternatives that exhibit a
decreasing hazard rate average (DHRA), corresponding to an anti-star-shaped
behavior of $H$, in $S_{\nu}$. Moreover, for every $\nu\in(0,1]$, it is
possible to establish the following consistency property.

######  Theorem 2.

If $\mathcal{H}_{0}^{\nu}$ is false,
$\lim_{n\rightarrow\infty}P(T_{*}^{\nu}(\mathbb{F}_{n})\geq
c_{\alpha,n,\nu})=1$, for every $\nu>0$.

###### Proof.

Note that
$T^{\nu}_{*}(\mathbb{F}_{n})=\sup_{S_{n,\nu}}\left|\mathbb{F}_{n}(x)-\mathcal{E}\circ(\mathbb{H}_{n}|_{S_{n,\nu}})_{*}(x)\right|-1/n$,
so we may equivalently consider the “$\sup$” on the right-hand side, which is
easier to manipulate, instead of $T^{\nu}_{*}$. By the Glivenko-Cantelli
Theorem, $\mathbb{F}_{n}$ converges a.s. and uniformly to $F$, whereas
$\mathcal{E}^{-1}(p)=-\ln(1-p)$ is uniformly continuous in $[0,1-\nu]$ (for
$\nu>0$), then $\mathbb{H}_{n}=\mathcal{E}^{-1}\circ\mathbb{F}_{n}$ converges
a.s. and uniformly to $H=\mathcal{E}^{-1}\\!\circ\\!F$ in $S_{\nu}$. If
$\mathcal{H}_{0}^{\nu}$ is true, we can apply Lemma 3.1 of Wang (1988), which
yields

$\sup_{S_{n,\nu}}\left|\mathbb{H}_{n}(x)-H(x)\right|\geq\sup_{S_{n,\nu}}\left|(\mathbb{H}_{n}|_{S_{n,\nu}})_{*}(x)-H(x)\right|,$

implying strong uniform consistency of $(\mathbb{H}_{n}|_{S_{n,\nu}})_{*}$ in
${S_{\nu}}$ (clearly, if $\nu>0$, $\mathbb{F}_{n}^{-1}(1-\nu)$ converges a.s.
to $F^{-1}(1-\nu)$). Since $\mathcal{E}$ is absolutely continuous, then
$\mathbb{F}^{\mathcal{E}}_{n,\nu}:=\mathcal{E}\circ(\mathbb{H}_{n}|_{S_{n,\nu}})_{*}$
converges a.s. and uniformly to $F$ in $S_{\nu}$, for every $\nu>0$. Now, it
can be easily seen that, under $\mathcal{H}_{0}^{\nu}$,
$c_{\alpha,n,\nu}\rightarrow 0$ a.s. for $n\rightarrow\infty$.

Define $F^{\mathcal{E}}_{\nu}=\mathcal{E}\circ(\mathcal{E}^{-1}\circ
F|_{S_{\nu}})_{*}$. Suppose that $\mathcal{H}_{0}^{\nu}$ is false, that is,
$\mathcal{E}\circ F$ is not star-shaped in $S_{\nu}$, and let
$d=\sup_{S_{\nu}}|F-F^{\mathcal{E}}_{\nu}|=\sup_{S_{\nu}}(F-F^{\mathcal{E}}_{\nu})$.
Then, $F_{n}$ converges uniformly to $F$, whereas
$\mathbb{F}^{\mathcal{E}}_{n,\nu}$ converges uniformly to
$F^{\mathcal{E}}_{\nu}$ on $S_{\nu}$, with probability 1. Moreover, by
assumption, $d>0$. Therefore, given some $\epsilon\in(0,d)$, there exists some
$n_{0}$ such that, for $n>n_{0}$, $\sup_{S_{n,\nu}}|F_{n}-F|<\epsilon/2$ and
$\sup_{S_{n,\nu}}|\mathbb{F}^{\mathcal{E}}_{n,\nu}-F^{\mathcal{E}}_{\nu}|<\epsilon/2$,
with probability 1. Then, for $n>n_{0}$,

$\mathbb{F}_{n}(x)-\mathbb{F}^{\mathcal{E}}_{n,\nu}(x)>F(x)-\frac{\epsilon}{2}-(F^{\mathcal{E}}_{\nu}(x)+\frac{\epsilon}{2})=F(x)-F^{\mathcal{E}}_{\nu}(x)-\epsilon$

almost surely, for every $x\in S_{n,\nu}$, which implies

$\sup_{S_{n,\nu}}\left|\mathbb{F}_{n}(x)-\mathbb{F}^{\mathcal{E}}_{n,\nu}(x)\right|>\sup_{S_{n,\nu}}\left|F(x)-\mathbb{F}^{\mathcal{E}}_{n,\nu}(x)-\epsilon\right|=\sup_{S_{n,\nu}}\left(F(x)-\mathbb{F}^{\mathcal{E}}_{n,\nu}(x)\right)-\epsilon=d-\epsilon>0.$

Therefore, since $\epsilon$ can be arbitrarily small,
$P(\sup_{S_{n,\nu}}\left|\mathbb{F}_{n}(x)-\mathbb{F}^{\mathcal{E}}_{n,\nu}(x)\right|\geq
d)\rightarrow 1$, for $n\rightarrow\infty$. But since
$c_{\alpha,n,\nu}\rightarrow 0$, then $P(T_{*}^{\nu}(\mathbb{F}_{n})\geq
c_{\alpha,n,\nu})\rightarrow 1$.∎

The value of $\nu$ should be chosen as small as possible, e.g.
$\nu=0.01,0.05$. One might even choose $\nu$ as a function of $n$, for example
$\nu=n^{-1/2}$. Yet, smaller $\nu$’s yield slower convergence, as we will need
more observations in order to get $c_{\alpha,n,\nu}$ close enough to 0. In the
limit case $\nu=0$, we cannot ensure uniform a.s. consistency of
$\mathbb{F}^{\mathcal{E}}_{n,0}$ in $S_{0}$, accordingly, the consistency of
the test $T^{0}_{*}$ cannot be established.

####  4.2.2 Simulations

The numerical performance of the test $T^{\nu}_{*}$ can be depicted by a
simulation study. We focus on some critical alternatives, obtained using
popular parametric families. By scale invariance, the scale parameters are
always set to 1. We consider DHRA, bathtub-shaped (decreasing-increasing) HRA,
and bell-shaped (increasing-decreasing) HRA models. The Weibull distribution
with CDF $F(x)=1-e^{-x^{a}}$, $a,x>0$, is DHRA for $a<1$ and IHRA for $a\geq
1$; the beta distribution, with CDF
$F(x)=\int_{0}^{x}t^{a-1}(1-t)^{b-1}\,dt/{B(a,b)}$, $a,b>0$, $x\in[0,1]$, is
bathtub-shaped HRA for $a<1<b$ and IHRA for $a,b\geq 1$; the Burr distribution
with CDF $F(x)={1}-\left({1+x^{b}}\right)^{-a}$, $x,a,b>0$, is DHRA for $b\leq
1$ and bell-shaped HRA for $b>1$.

The results are reported in Figure 2, which contains the plots of the
rejection probabilities at level $\alpha=0.1$, estimated using 2000 simulation
runs. To cover a large number of alternatives, for each distribution
considered we fixed a grid of parameters within some intervals of interest. We
considered sample sizes $n=10,50,100,200,300$, setting $\nu=0.05$, so the
restriction has an effect just for $n\geq 50$. The figures show that the test
has a remarkable performance in terms of empirical power, coherently with the
behavior of the HRA. In fact, some alternatives considered are particularly
difficult to detect, as the HRA may exhibit a very slight decrease (Weibull
with $a\in[0.8,1)$) or a subtle non-monotonic behavior (beta with $a\geq 0.7$;
Burr with $b\geq 1.5$). In the first case, the power is always larger than
$\alpha$, independently of the sample size, confirming the unbiasedness
property implied by Proposition 3; in the latter case, the power may be
smaller than $\alpha$, for small values of $n$ (beta with $a\geq 0.7$),
confirming that unbiasedness of tests of this type holds just against DHRA
alternatives, but when the HRA has a non-monotonic behavior, this is not
guaranteed. Under the null hypothesis, the size is always bounded by $\alpha$,
even when the HRA is close to the boundary of DHRA (Weibull with $a>1)$.
Clearly, as the test is consistent, the empirical power increases with sample
size, when $\mathcal{H}_{0}^{0.05}$ is false, even under the most citical
alternatives (this should be evaluated just for $n\geq 50$, as remarked
earlier; the beta with $a\geq 0.7$ is the most critical case and the power may
be increased by increasing the sample size, or by choosing a larger value of
$\nu$).

Figure 2: Rejection rates of $T_{*}^{0.05}$ and $J_{0.5}$ for IHRA (Weibull
$a\geq 1$), DHRA (Weibull $a<1$), bathtub-shaped HRA (beta) and bell-shaped
HRA (Burr) distributions.

The test is compared with the one proposed by Deshpande (1983) and later
studied by Bandyopadhyay and Basu (1989), aimed at testing the null hypothesis
$\widetilde{\mathcal{H}}_{0}:F=\mathcal{E}$ versus
$\widetilde{\mathcal{H}}_{1}$: “$F$ is IHRA and not exponential”, as in
Klefsjö (1983), Kochar (1985), Link (1989), Wells and Tiwari (1991), Ahmad
(1994). This approach is quite different from ours, in fact, such tests are
supposed to reject the IHRA hypothesis more easily, when it is false, compared
to $T_{*}^{\nu}$. However, our simulation results reveal some possible issues
of these kinds of tests. Actually, Deshpande (1983) introduced a family of
rank-based test statistics, denoted as $J_{p}$, where the parameter $p$ ranges
in $(0,1)$. As proved by Bandyopadhyay and Basu (1989), the choice for $p$
does not have consequences on the properties of the test, therefore, in the
plots we show the results of $J_{0.5}$, which is the test recommended by
Deshpande (1983). This test is unbiased and consistent, for every $p$, under
IHRA alternatives. However, deviations from exponentiality should not
necessarily imply that $\widetilde{\mathcal{H}}_{1}$ is true, as tests of this
type are not intended to deal with bathtub or bell-shaped HRA models; indeed
$J_{p}$ may be misleading when $F$ has a non-monotone HRA, as it can be seen
in the case of the beta distribution ($a\geq 0.7$) or Burr distribution. In
such cases, by increasing the sample size we also increase the power, that is,
the probability of rejecting $\widetilde{\mathcal{H}}_{0}$, suggesting that
the (false) strict IHRA alternative may be true (we obtained similar results
for several values of $p$). In other words, in some critical bathtub and bell-
shaped HRA situations, $J_{p}$ might lead to the wrong decision. So, given
that non-monotone HRA distributions may be quite common, we recommend using
$T_{*}^{\nu}$. Looking at the plots, it is important to remember that
$T_{*}^{\nu}$ and $J_{p}$ cannot be compared in terms of power, since they
deal with reversed hypotheses, hence one may only compare their overall
performance. This example also shows why it is important to study and
understand the behavior of the test statistics using ordering constraints.

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# PixSet : An Opportunity for 3D Computer Vision to Go Beyond Point Clouds
With a Full-Waveform LiDAR Dataset

Jean-Luc Déziel  LeddarTech  Pierre Merriaux  LeddarTech  Francis Tremblay
LeddarTech  Dave Lessard  LeddarTech  Dominique Plourde  LeddarTech  Julien
Stanguennec  LeddarTech  Pierre Goulet  LeddarTech  Pierre Olivier  LeddarTech

###### Abstract

Leddar PixSet is a new publicly available dataset (dataset.leddartech.com) for
autonomous driving research and development. One key novelty of this dataset
is the presence of full-waveform data from the Leddar Pixell sensor, a solid-
state flash LiDAR. Full-waveform data has been shown to improve the
performance of perception algorithms in airborne applications but is yet to be
demonstrated for terrestrial applications such as autonomous driving. The
_PixSet_ dataset contains approximately 29k frames from 97 sequences recorded
in high-density urban areas, using a set of various sensors (cameras, LiDARs,
radar, IMU, etc.) Each frame has been manually annotated with 3D bounding
boxes.

††preprint: APS/123-QED

##  I Introduction

Autonomous vehicles (AVs) have the potential to transform how transportation
is done for people and merchandise, while improving both safety and
efficiency. In order to reach the highest levels of autonomy, one of the main
challenges that AVs are currently facing is to leverage the data from multiple
types of sensors, each of which has its own strengths and weaknesses. Sensor
fusion techniques are widely used to improve the performance and robustness of
computer vision algorithms.

Nowadays, the best performing computer vision algorithms are neural networks
that are optimized using a deep learning approach Lang  _et al._ (2019); He
_et al._ (2020); Shi  _et al._ (2020), which requires large amount of data.
Multiple datasets have been made publicly available in order to boost research
and development of such algorithms Geiger  _et al._ (2013); Pitropov  _et al._
(2020); Caesar  _et al._ (2020); Sun  _et al._ (2020); Houston  _et al._
(2020). In particular, most of these datasets include data acquired with a
LiDAR sensor (Light Detection and Ranging) which is considered to be an
essential component of highly autonomous vehicles Feng  _et al._ (2020); Li
_et al._ (2020).

In this paper, we present our contribution to this effort, the _PixSet_
dataset111Download link: dataset.leddartech.com. What makes this new dataset
unique is the use of a flash LiDAR and the inclusion of the full-waveform raw
data, in addition to the usual point cloud data. The use of full-waveform data
from a flash LiDAR has been shown to improve the performance of segmentation
and object detection algorithms in airborne applications Reitberger and
Krzystek (2009); Shinohara  _et al._ (2020), but is yet to be demonstrated for
terrestrial applications such as autonomous driving.

The _PixSet_ dataset contains 97 sequences, each averaging a few hundreds of
frames, for a total of roughly 29000 frames. Each frame has been manually
annotated with 3D bounding boxes. The sequences have been gathered in various
environments and climatic conditions with the instrumented vehicle shown in
Figure 1.

Figure 1: Instrumented vehicle (_RAV4_) used for data acquisition.

Our main contributions are summarized as follows:

  * •

Introduce to the community a new dual LiDAR-type dataset using solid-state and
mechanical LiDARs, with 3D bounding box annotation.

  * •

Provide full-waveform data for the solid-state LiDAR.

  * •

Trigger exteroceptive sensors to improve 3D bounding box annotation accuracy

  * •

Provide API and dataset viewer to facilitate algorithm research and
development.

The paper is organized as follows: sections II.1 to II.3 present the recording
setup and conditions. Section II.5 briefly describes how the data is stored
and how to read/viewed it with an optional open source library we have
developed for _PixSet_. The waveform data is described in section II.6.
Section III describes the dataset annotations (3D bounding boxes). Then
section IV provides baseline object detection results, and a brief conclusion
is presented in section V.

##  II Dataset

The dataset is available at the link dataset.leddartech.com. The format of the
data is discussed in section II.5.

###  II.1 Sensors

The sensors used to collect the dataset were mounted on a car (see Figure 1)
and are listed in Table 1. Most sensors (cameras, LiDARs and the radar) are
positioned close to each other at the front of the car in a configuration
shown in Figure 2. This proximity is deliberate in order to minimize the
parallax effect. The GPS antennas for the inertial measurement unit (IMU) are
located on the top of the vehicle.

_Sensor label_ | _Description_
---|---
pixell_bfc |  Leddar Pixell solid-state LiDAR with waveforms
ouster64_bfc |  Ouster OS1-64 mechanical LiDAR
flir_bfl/bfc/bfr |  3 FLIR BFS-PGE-16S2C-CS cameras + 90°optics
flir_bbfc |  FLIR BFS-PGE-16S2C-CS camera + Immervision panomorph 180°optic.
radarTI_bfc |  TI AWR1843 mmWave radar
sbgekinox_bcc |  SBG Ekinox IMU with RTK GPS dual antenna
peakcan_fcc |  Toyota RAV4 CAN bus
Table 1: List of sensors used for _PixSet_ dataset data acquisition. Figure 2:
Spatial configuration of the sensor stack on the front of the vehicle. Arrows
show the coordinate system used by each sensor. See Table 1 for a description
of the sensors.

Most sensors are fairly standard and well known by the community, except for
the Leddar Pixell. It is a solid-state (no moving parts) flash and full-
waveform LiDAR (see section II.6 for a description of the waveforms). Its
field of view is 180°horizontally and 16°vertically. While it has a relatively
low resolution (96 horizontal channels by 8 vertical channels), it should
provide more information per channel than a typical non-flash LiDAR sensor.
Testing this hypothesis is one of the motivations to create this dataset.

###  II.2 Time Synchronization

One of the objectives of _PixSet_ was to obtain the highest accuracy possible
in the positioning of the bounding boxes for data annotation. Since the
environment is dynamic, combining the data from multiple sensors will yield
the best result when the sensors are gathering data simultaneously. This was
achieved by triggering each frame acquisition from a single periodic signal
source generated by the OS1-64 mechanical LiDAR. The Pixell LiDAR and the 4
cameras are triggered as such, at a frequency of 10 Hz. The other sensors
(radar, IMU and the vehicle CAN bus) are not triggered, but are recording at
much higher frame rates which minimizes the timing errors. The sensor
triggering is used to minimize the acquisition time difference between
extrinsic sensors. In Figure 3, the average offset differences were measured
for every channel of the Pixell sensor, which is 40 ms at most.

A timestamp is associated with each frame of each sensor, or each point for
LiDAR. To make sure that all sensors are precisely synchronized, we run a
server that collects the time from the GPS antenna (PTP protocol) and connect
all sensors to that server to synchronize their internal clocks. The only
exceptions are the radar and the vehicle CAN bus for which the timestamps are
assigned at data reception in the computer (the computer’s clock is also
synchronized with the same time server). Moreover, the quality of the
synchronization is always monitored in real time by comparing the trigger
signal to a separate time measurement from the IMU. The typical measured
synchronization error was 6 $\mu$s for cameras and 350 $\mu$s for Pixell.

One more thing that was considered is the fact that LiDARs are not global
shutter sensors, in contrast with cameras. What is referred to as a ”frame”
for a LiDAR, or a single complete scan of the field of view, is not measured
all at once. For example, a typical mechanical LiDAR such as the OS1-64 is
simultaneously measuring a single vertical line, which is continuously
rotating during 100ms. Thanks to accurate timestamping LiDAR and IMU, the
motion of the car during the frame acquisition time can be compensated. The
API (section II.5) provides this functionality to algorithm developers.

Figure 3: Average difference between the timestamp of each channel of the
Pixell sensor and the timestamps of the projected overlapping OS1-64
detections.

###  II.3 Calibration

Sensor projection is mandatory for many algorithms like sensor fusion. To
obtain an accurate projection (Figure 4), a calibration must be performed,
which can be done by multiple methods. The following describes the methods
chosen to obtain the calibration matrices included in the dataset.

Figure 4: Projection of an OS1-64 point cloud in a camera image, using the
calibration matrices provided with the dataset.

To compensate for the camera lens distortion, a set of images of a chessboard
was gathered for each camera and the openCV library that calculates the
desired matrices from these images was used. For the Pixell sensor, the
direction of each channel must be known in order to position each detection in
3D space. Thankfully, these directions are calibrated at the factory with
specialized equipment and provided in the dataset.

Next, the extrinsic calibration matrices are also needed to change the
referential of the coordinate system from/to any pair of sensors. The
extrinsic calibration between pairs of cameras is obtained, again, by using
the openCV library to extract the 3D coordinates of the corners of a
chessboard in multiple images. A closed loop optimization method was closed to
minimize the transformation matrices. A similar method is used for the
extrinsic calibration between the OS1-64 LiDAR and the cameras, by first
extracting the 3D coordinates of the corners of a chessboard in each sensor
and then solving the transformation matrices with the perspective-n-point
method. The extrinsic calibration between both LiDARs (Pixell and OS1-64) is
obtained by averaging multiple matrices, each obtained by using the iterative
closest point (ICP) method with pairs of simultaneous scans from both sensors.
The calibration between the radar and the OS1-64 LiDAR is obtained similarly,
except that only the high-intensity detections from a specific metallic target
are accounted for. Finally, the calibration between the OS1-64 LiDAR and the
IMU is obtained by minimizing plane thickness of multiple sequential point
clouds in the world coordinates provided by the IMU. The pairs of sensors that
have not been mentioned are obtained by combining other matrices (i.e. the
Pixell to IMU transformation matrix is obtained by multiplying the matrices
for Pixell to OS1-64 and for OS1-64 to IMU).

###  II.4 Recording Conditions

Figure 5: Locations of the recorded data.

The _PixSet_ dataset has been recorded in Canada, in both Quebec City and
Montreal, during the summer of 2020. Locations are shown in Figure 5. A
summary of the recording conditions is presented in Figure 6. Most of the
dataset has been recorded in high-density urban environments, such as downtown
areas where pedestrians are almost always present and on boulevards where the
car density is high. Most of the dataset was recorded during day time with dry
weather, but a few thousand frames were recorded at night and/or while
raining, providing a great variety of situations with real-world data for
autonomous driving. Figure 7 showcases a few samples taken from the dataset.

Figure 6: Summary of recording conditions. Figure 7: _PixSet_ dataset overview
of the variety of scenes and environmental conditions with samples from the
cameras and the OS1-64 LiDAR with 3D boxes annotation.

###  II.5 Dataset Format and Access

Each of the 97 sequences of the dataset is contained in an independent
directory. Each of them contains the following elements: (i) a configuration
file named platform.yml that contains all the required parameters to read and
process the data from the optional library provided along with the dataset
(see below for details). (ii) An intrinsics directory that contains the
calibration matrices for each camera (lens distortion compensation, see
Section II.3). (iii) An extrinsics directory that contains the extrinsic
calibrations (4x4 affine transformation matrices) for the pairs of sensors
that have been calibrated. (iv) A _zip_ file for each source of data (i.e.
images from a camera).

A given _zip_ file is named after the label of the sensor (see Table 1, with
pixell_bfc for the Pixell for example) and the type of data, which can be
multiple for a single sensor. The raw waveforms from Pixell are stored in the
pixell_bfc_ftrr.zip file and the detections that have been processed from
these waveforms are stored in the pixell_bfc_ech.zip file. In each of these
files, a series of _pickle_ files (or a _.jpg_ for camera images) are stored,
each of them containing the data of a single frame (a full LiDAR scan or a
single camera image). Along the raw data, a timestamps.csv file is also
stored, containing all timestamps.

Along with the dataset, we also provide an API to a Python library that was
developed to read, process and view the
data222https://github.com/leddartech/pioneer.das.api
https://github.com/leddartech/pioneer.das.view. This tool can perform several
of the most complex and common operations such as referential transformations,
time synchronization of the different sources of data, motion compensation for
point clouds and much more.

###  II.6 Waveforms

Typically, a LiDAR provides data in the form of point clouds, a collection of
three-dimensional coordinates. Usually, an intensity value is also attached to
each point. Point clouds are not the raw data measured by the sensor but are
rather processed from the waveforms. A waveform is the measured intensity as a
function of time after the emission of the laser pulse from the LiDAR (See
Figure 8 for an example). A point is the result of a peak found in a waveform.
While the information about the positions and the amplitudes of the peaks is
preserved in the point cloud, all additional information about the shape of
the waveform is lost.

Figure 8: (a) 3D view of detections provided by the Pixell sensor. (b) 2D
front view of the amplitudes from the same detections as in (a). (c) Example
of raw waveforms obtained from a single Pixell channel. (See text for
details).

The Pixell sensor measures a set of two waveforms for each channel of each
frame. The first waveform is gathered after the emission of a high-intensity
laser pulse and the second one after the emission of a laser pulse with a
quarter of the intensity. This effectively increases the dynamic range of the
sensor, because the high-intensity waveforms have a tendency to saturate the
sensor when looking at close and/or reflective targets. This is analogous to
high-dynamic range imagery with cameras that can be achieved by combining
images with different exposure levels.

The waveforms are sampled at an 800 MHz rate. The high-intensity waveforms
contain 512 samples, while the low-intensity ones contain 256 samples. One
more important aspect to keep in mind is that waveforms from different
channels have slightly different time offsets with respect to each other.
These offsets are calibrated and compensated for in the provided point clouds,
but not for the raw waveforms. The offsets are provided with the dataset and
the waveforms can easily be adjusted (by linear interpolation). The API
provided (see end of section II.5) contains multiple functions to deal with
waveform processing, including time realignment.

##  III Annotations

Each frame ($\sim$29k) of the dataset has been manually annotated with 3D
bounding boxes (Figure 7). The annotators have been provided with the point
clouds from both the Pixell and OS1-64 LiDARs, merged into the same
referential (see section II.3). Motion compensation have been applied to both
point clouds (see end of section II.5). Images from the cameras _flir_bfl,
flir_bfc_ and _flir_bfr_ (see Figure 2) along with their calibration data have
also been provided to be able to project the point clouds into the images to
help identity objects.

Each bounding box is associated with one physical object and describes its
central position (in the Pixell sensor’s referential), its dimensions and
orientations with respect to all three axes, its category (car, pedestrian,
etc.), an inter-frame persistent identification number and a set of attributes
such as an occlusion level. The complete list of the 20 categories and the
number of annotated objects per category are shown in Figure 9(a). The
distributions or distances and orientations of the bounding boxes for a few
categories are shown in Figure 9(b). An overview of the attributes for some
categories is also presented in Figure 9(c).

(a) Total number of annotated objects per category. Interestingly, the
relative amounts seem to follow Zipf’s law Zipf (1949). (b) Distribution of
the distances and orientations of cars, pedestrians and cyclists.

(c) Relative amount of each attribute values found in the dataset for cars,
pedestrians and cyclists. Occlusion refers to a fraction of the object hidden
behind other things. Truncation refers to a fraction of the object that is
outside the LiDARs’ field of view. Take note that an occlusion/truncation
value of 0 means $0\%$, a value of 1 means $<50\%$ and a value of 2 means
$>50\%$.

Figure 9: Overview of data annotations.

A notable feature of the _PixSet_ dataset is the variable size of the bounding
boxes for pedestrians. It is common to use a fixed size for a given object
instance for the whole duration of a sequence. It makes sense for cars, or any
solid object that has a fixed size and shape, but leads to complications for
pedestrians that are deformable objects. For example, if a pedestrian extends
their arm for a few seconds, using a fixed bounding box size is problematic.
Should the arm be ignored or included in the box, and an oversized bounding
box maintained for the remainder of the sequence? Thus, bounding boxes for
pedestrians in the _PixSet_ dataset have variable sizes, adjusted frame per
frame, to avoid these issues.

##  IV Object Detection

This section presents results from an object detection algorithm in order to
provide a baseline for the perception performance of the Pixell sensor. It
describes the model and the metrics at a coarse level, while the source code
and all details can be found in obj .

#### Preprocessing.

The raw data is prepared and preprocessed by using the library that was made
publicly available (See section II.5). The steps performed by this tool are
the following: (i) match the synchronized data from Pixell, annotations and
the IMU for egomotion. (ii) Convert raw detection data of the Pixell in point
clouds. (iii) Apply motion compensation to the point cloud (See end of section
II.2).

Then, data augmentation is performed using the following methods: (i)
insertion of bounding boxes and contained points from other frames (up to 5
cars, 5 pedestrians and 5 cyclists); (ii) With a 50% probability, flip the
left/right axis. (iii) Random translations along all three axes within the
values $x\in[-3,0]$, $y\in[-3,3]$, $x\in[-0.5,0.5]$ (in meters, see Figure 2
for the coordinate system). (iv) Random rotation (_yaw_
$\in[-5\degree,5\degree]$). (v) Random scaling of the scene by a factor
$f\in[0.95,1.05]$. The translations, rotation and scaling are applied to the
entire frame, using the position of the sensor as the origin.

#### Neural network.

The neural network is composed of three parts: the encoder, the backbone and
the detection head. The encoder is similar to the one from PointPillars Lang
_et al._ (2019), except that the single encoding layer is replaced by a dense
block inspired by DenseNet Huang  _et al._ (2017) (4 layers, with a growth
parameter of 16). This results in a 2D bird’s eye view of a set of encoded
features. The features then go through the backbone of the network, also a
dense block, with 60 layers and a growth parameter of 16. The detection head
is a final dense block of 5 layers and a growth parameter of 7. The detection
head is inspired by CenterNet Duan  _et al._ (2019) and produces bird’s eye
view heat maps where local maxima are interpreted as objects. Non-maximum
suppression is not necessary with this method so it is not used.

The neural network has a total of 5.2M parameters. The model was trained for
100 epochs with the Adam optimizer Kingma and Ba (2015). The learning rate was
exponentially decaying from 0.001 down to 0.000174 between each epoch. A test
set ($\sim$3k frames) was removed from the training data. The sequences that
were kept for the test set are parts 1, 9, 26, 32, 38 and 39 (these part
numbers are in the name of each downloadable directory of the dataset).

Figure 10: Average precision (AP) as a function of distance for cars,
pedestrians and cyclists found in the test set (parts 1, 9, 26, 32, 38 and 39
of the dataset). To be considered a true positive, a prediction must have an
IoU (intersection over union) larger than the thresholds indicated in the
legends.

#### Metrics.

The most popular metric to evaluate the performance of an object detection
model is the average precision (AP). This metric is calculated as a function
of distance, as shown in Figure 10 (See table 2 for the global AP values).
This better shows how a sensor can perform quite well at short range and at
what distance the model’s predictions can be trusted. Admittedly, the Pixell
detection capabilities seem to rapidly drop beyond 10-15 meters, but it is not
meant to be used as the only sensor on an AV. These results are mostly
presented as a reference baseline. Future work will focus on the performance
of a fused sensor stack and the potential improvements of including full-
waveform data from the Pixell.

IoU threshold | Car AP | Pedestrian AP | Cyclist AP
---|---|---|---
25% | 68.33% | 37.15% | 35.87%
50% | 52.04% | 14.13% | 19.69%
Table 2: Average precision (AP) results. The evaluation only account for
bounding boxes found within 32 meters from the Pixell sensor.

A few notes on the metric calculation: a predicted bounding box is considered
a true positive if its three-dimensional intersection over union (IoU) is
above a certain threshold (indicated in the figures) and if the classification
of the bounding box is correct. Moreover, there can be only one true positive
prediction per ground truth bounding box (duplicates are false positives).

##  V Conclusion

This work presented our contribution, the _PixSet_ dataset, to the collective
effort of developing safe autonomous vehicles. We believe that there is a
great potential to improve further the perception algorithms by leveraging the
raw data from the full-waveforms provided by the Pixell flash LiDAR. We have
also provided a baseline for 3D object detection performance from the Pixell
point clouds.

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