$M_{2,n}=M_{2,n-1}+(\text{{newdata}}-\bar{x}_{n-1})\odot(\text{{newdata}}-\bar{x}_{n})$;

10: $s_{n}^{2}=\frac{1}{n-1}M_{2,n}$;

11: return $n$, $\bar{x}_{n}$, $M_{2,n}$, $s_{n}^{2}$

The initial dark and illuminated sample sizes for the data collection
algorithm are determined from looking back at the plots for
$n_{1}^{\text{opt}}$ and $n_{2}^{\text{opt}}$ in Figure 3.5. Regardless of the
parameters chosen, we know that an optimal dark sample size of $n_{2}=1$
occurs for at least one $\zeta$-value and so we set the default initial number
of $Y$-images to be one. This, however, cannot be said for the optimal
illuminated sample size $n_{1}$, which always takes on values greater than
one. While the optimal illuminated sample size curve is not always strictly
increasing, we will choose its shot noise-limited value
$2\mathsf{acv}_{0}^{-2}+5$ as a good approximation to the minimum possible
value and make this as our required initial number of initial $X$-images.

Algorithm 3 presents the generic CollectImageData(n1Min, Yrule, Xrule,
HaltRule) procedure used for collecting and processing real-time image data in
the pixel-level conversion gain estimation experiment. The procedure accepts
an initial illuminated sample size, n1Min, which will be
$2\mathsf{acv}_{0}^{-2}+5$ for this experiment, as well as three rules for
determining when to halt $Y$-image collection (Yrule), $X$-image collection
(Xrule), and the entire routine (HaltRule). From Algorithm 3 we observe that
the CollectImageData() procedure begins by initializing the master frames and
updating the master frames until the initial minimum sample sizes are
captured. The procedure then enters a loop whereby the master frames are first
updated if their corresponding rule is false. Once the master frames are
updated, the estimates of $\zeta$, ${\nu^{\smash{\dagger}}}$,
$n_{1}^{\text{opt}}$, and $n_{2}^{\text{opt}}$ are updated for each pixel
group. Lastly, the Yrule, Xrule, and HaltRule are reevaluated to determine if
more images are needed and if the procedure can stop. Upon completion, the
procedure then returns the final sample sizes $n_{1}$ and $n_{2}$, a vector
$V$ containing the final group estimates of ${\nu^{\smash{\dagger}}}$, another
vector $Z$ containing the final group estimates of $\zeta$, and the four
master frames $\bar{Y}$, $\bar{X}$, $\hat{Y}$, and $\hat{X}$.

Algorithm 3 Data collection procedure.

1:procedure CollectImageData(n1Min, Yrule, Xrule, HaltRule)

2: $n_{2}=0$;

3: Capture one $Y$-image;

4: [$n_{2}$, $\bar{Y}$, $\hat{Y}$, $M_{2}^{Y}$]=UpdateStats($Y$, $n_{2}$);

5: $n_{1}=0$;

6: Capture one $X$-image;

7: [$n_{1}$, $\bar{X}$, $\hat{X}$, $M_{2}^{X}$]=UpdateStats($X$, $n_{1}$);

8: while $n_{1}<\text{{n1Min}}$ do

9: Capture one $X$-image;

10: [$n_{1}$, $\bar{X}$, $\hat{X}$, $M_{2}^{X}$]=UpdateStats($X$, $n_{1}$,
$M_{2}^{X}$, $\bar{X}$);

11:

12: Yflag=false;

13: Xflag=false;

14: Stop=false;

15: $N=\text{\\# of groups}$;

16: while Stop=false do

17: if Yflag=false then

18: Capture one $Y$-image;

19: [$n_{2}$, $\bar{Y}$, $\hat{Y}$, $M_{2}^{Y}$]=UpdateStats($Y$, $n_{2}$,
$M_{2}^{Y}$, $\bar{Y}$);

20: if Xflag=false then

21: Capture one $X$-image;

22: [$n_{1}$, $\bar{X}$, $\hat{X}$, $M_{2}^{X}$]=UpdateStats($X$, $n_{1}$,
$M_{2}^{X}$, $\bar{X}$);

23: for $i=1:N$ do

24: $m_{i}=\text{\\# of pixels in group }i$;

25: $Z_{i}=\frac{m_{i}n_{1}-3}{m_{i}n_{1}-1}(\sum_{\text{group
}i}\hat{Y})/(\sum_{\text{group }i}\hat{X})$;

26: $V_{i}={\nu^{\smash{\dagger}}}(Z_{i})$;

27:
$(\hat{n}_{1}^{\text{opt}},\hat{n}_{2}^{\text{opt}})_{i}=(n_{1}^{\text{opt}},n_{2}^{\text{opt}})(Z_{i})$;

28: Yflag=Yrule;

29: Xflag=Xrule;

30: Stop=HaltRule;

31: return $n_{2}$, $n_{1}$, $V$, $\bar{Y}$, $\bar{X}$, $\hat{Y}$, $\hat{X}$

For the KAI-0407M pixel-level conversion gain estimation experiment the
following three simple rules were chosen:

$\text{Yrule}=\begin{cases}\text{true},&\text{if}\
\hat{n}_{2}^{\text{opt}}\geq n_{2}\ \text{for all groups}\\\
\text{false},&\text{otherwise},\end{cases}$
$\text{Xrule}=\begin{cases}\text{true},&\text{if}\
\hat{n}_{1}^{\text{opt}}\geq n_{1}\ \text{for all groups}\\\
\text{false},&\text{otherwise},\end{cases}$
$\text{HaltRule}=\begin{cases}\text{true},&\text{if}\
(\hat{n}_{1}^{\text{opt}}\geq n_{1}\land\hat{n}_{2}^{\text{opt}}\geq n_{2})\
\text{for}\ \geq 95\%\ \text{of all groups}\\\
\text{false},&\text{otherwise}.\end{cases}$

We see that Yrule and Xrule direct the algorithm to keep collecting images if
the estimated optimal sample sizes are less than the current sample sizes for
any of the column groups. Furthermore, HaltRule halted the procedure when the
current sample sizes exceeded the estimates for at least $95\%$ of the column
groups.

####  3.6.5 Summary of results

The data collection algorithm was executed on the KAI-0407M ccd sensor
resulting in dark and illuminated sample sizes of $n_{2}=862$ and $n_{1}=2469$
images, respectively. Figure 3.8 plots the group estimates of $\zeta$ and
${\nu^{\smash{\dagger}}}$ versus iteration number of the algorithm for fifteen
randomly selected columns. We can see that these estimates converge to
different values, which confirms our choice of pixel grouping is appropriate.
Furthermore, we observe that the variance of these estimates in the final
iterations of the algorithm is practically zero showing that the size (i.e.
number of pixels) of the chosen pixel groups was also sufficiently large.

Figure 3.8: Estimates of $\zeta$ (top) and ${\nu^{\smash{\dagger}}}$ (bottom)
versus iteration number for fifteen randomly selected columns.

To compute the pixel-level conversion gain array, a.k.a. the _$g$ -map_, we
first created a pixel-level ${\nu^{\smash{\dagger}}}$ array via

$V^{\dagger}=\mathbf{1}_{512\times 512}\times\operatorname{diag}(V),$

where $\mathbf{1}_{512\times 512}$ is a $512\times 512$ array of ones, $V$ is
the $512\times 1$ vector of group estimates for ${\nu^{\smash{\dagger}}}$, and
$\operatorname{diag}(V)$ is a $512\times 512$ diagonal matrix with diagonal
elements equal to elements of $V$. The master frames $\bar{Y}$, $\bar{X}$,
$\hat{Y}$, $\hat{X}$, along with $V^{\dagger}$, and the final sample size
values $n_{1}$ and $n_{2}$ were then imported into a Mathematica. For each set
of pixel coordinates $(i,j)$, $1\leq i,j\leq 512$, the conversion gain was
estimated with

$(\mathscr{G}_{{\nu^{\smash{\dagger}}},K})_{ij}=(\bar{X}_{ij}-\bar{Y}_{ij})\times\mathscr{T}_{V_{ij}^{\dagger},K}(\hat{X}_{ij},\hat{Y}_{ij},\alpha_{1},\alpha_{2}),$

where $\mathscr{T}_{{\nu^{\smash{\dagger}}},K}$ is the $K$th order asymptotic
approximation of $\mathscr{T}_{\nu^{\smash{\dagger}}}$ given in (2.6) and
$\alpha_{i}=(n_{i}-1)/2$, $i=1,2$. Comparing the histograms and sample
statistics of $\mathscr{G}_{{\nu^{\smash{\dagger}}},1}$ and
$\mathscr{G}_{{\nu^{\smash{\dagger}}},2}$ showed negligible difference
indicating that a $K=2$ order approximation was sufficient to accurately
compute $\mathscr{G}_{\nu^{\smash{\dagger}}}$ for each pixel.

Figure 3.9 displays the final $g$-map for the KAI-0407M ccd along with its
histogram. First comparing the histogram to the provided normal reference we
see it exhibits a positive skewness, which was estimated to be

$\widehat{\mathsf{Skew}}\,\mathscr{G}_{\nu^{\smash{\dagger}}}=0.2426.$

As for the $g$-map itself, we observe what appears to be purely random noise
with no noteworthy features or patterns. This behavior is expected since we
deliberately inhibited gain nonuniformity by transferring image data from the
sensor pixels through a single readout register. Since the fluctuations in the
$g$-map for this particular case should be almost entirely due to statistical
noise, we can see how effective the data collection algorithm was by comparing
the sample absolute coefficient of variation to the target value of
$\mathsf{acv}_{0}=0.05$. Computing the sample mean
$\hat{\mathsf{E}}\mathscr{G}_{\nu^{\smash{\dagger}}}$ and sample variance
$\widehat{\mathsf{Var}}\mathscr{G}_{\nu^{\smash{\dagger}}}$ of the $g$-map
data we found for the sample absolute coefficient of variation

$\widehat{\mathsf{ACV}}\mathscr{G}_{\nu^{\smash{\dagger}}}=\frac{\sqrt{\widehat{\mathsf{Var}}\mathscr{G}_{\nu^{\smash{\dagger}}}}}{\hat{\mathsf{E}}\mathscr{G}_{\nu^{\smash{\dagger}}}}=0.0497,$

which differs from the target value by only $0.7\%$. This small discrepancy
indicates the data collection algorithm was able to adequately control the
experiment and halt data capture at the appropriate time.

Figure 3.9: Histogram of $g$-map data with best normal approximation (top) and
image plot of the final $g$-map (bottom).

To see how $\mathscr{G}_{\nu^{\smash{\dagger}}}$ compares to the traditional
conversion gain estimator we also computed the traditional $g$-map with
elements

$G_{ij}=(\bar{X}_{ij}-\bar{Y}_{ij})(\hat{X}_{ij}-\hat{Y}_{ij})^{-1}.$

Table 3.5 presents tabulated values for select sample statistics of both maps.
While none of these sample statistics are able to compare the bias of each
estimator we know that $\mathsf{ARB}\mathscr{G}_{\nu^{\smash{\dagger}}}\approx
0.01$. Furthermore, a quick comparison of the histogram for $\hat{X}-\hat{Y}$
against its normal fit shows excellent agreement so that $\mathsf{ARB}G$ can
be approximated by [12, c.f. Corollary 3.2]

$\mathsf{ARB}G=\left\lvert\frac{2(1-\zeta)}{\sqrt{2(\frac{1}{\alpha_{1}}+\frac{1}{\alpha_{2}}\zeta^{2})}}\mathcal{D}\left(\frac{1-\zeta}{\sqrt{2(\frac{1}{\alpha_{1}}+\frac{1}{\alpha_{2}}\zeta^{2})}}\right)-1\right\rvert.$
(3.21)

Substituting the global estimate

$Z=\left(\frac{512^{2}n_{1}-3}{512^{2}n_{1}-1}\right)\frac{\sum_{i=1}^{512}\sum_{y=1}^{512}\hat{Y}_{ij}}{\sum_{i=1}^{512}\sum_{y=1}^{512}\hat{X}_{ij}}$

for $\zeta$ in (3.21) gives $\mathsf{ARB}G\approx 0.0027$ and so we conclude
$G$ incurs less relative bias than
$\mathscr{G}_{\nu^{\smash{\dagger}}}$777Technically speaking $\mathsf{ARB}G$
is undefined; however, the expression in (3.21) does give a useful measure of
relative bias for $G$ whenever $\mathsf{P}(\hat{X}-\hat{Y}>0)\approx 1$.
Inspection of the $\hat{X}$ and $\hat{Y}$ data for the KAI-0407M ccd reveals
all $512^{2}$ values of $\hat{X}-\hat{Y}$ are positive; thus, the comparison
of $\mathsf{ARB}\mathscr{G}_{\nu^{\smash{\dagger}}}$ and $\mathsf{ARB}G$ is
informative.. However, note that this small increase in relative bias affords
the estimator $\mathscr{G}_{\nu^{\smash{\dagger}}}$ a decrease in sample
variance of about $11.4\%$ when compared to that of $G$.

| $\mathscr{G}_{\nu^{\smash{\dagger}}}$ | $G$ | Unit
---|---|---|---
$\hat{\mathsf{E}}(\cdot)$ | $2.1697$ | $2.1975$ | $e\mathchar 45\relax/\mathrm{DN}$
$\widehat{\mathsf{Var}}(\cdot)$ | $0.0116$ | $0.0131$ | $(e\mathchar 45\relax/\mathrm{DN})^{2}$
$\widehat{\mathsf{ACV}}(\cdot)$ | $0.0497$ | $0.0521$ | $-$
Table 3.5: Comparison of sample statistics for the
$\mathscr{G}_{\nu^{\smash{\dagger}}}$\- and $G$-maps.

##  Chapter 4 Conclusions

In this work we covered a lot of ground in understanding the estimation of the
reciprocal difference of normal variances,
$\tau=(\sigma_{1}^{2}-\sigma_{2}^{2})^{-1}$, and how this is applicable to the
photon transfer method of image sensor characterization.

We began in Theorem 1 by showing that no unbiased, finite-variance estimator
of $\tau$ existed under the normal model. Appealing to the principle of bias-
variance tradeoff a biased yet finite-variance estimator, $\mathscr{T}_{n}$,
was produced, which estimated the first $n$ terms of the Taylor series for
$\tau$. Working with the methods of _Summability Calculus_ the domain of
$\mathscr{T}_{n}$ was then extended to include complex-valued $n$ resulting in
the generalized estimator $\mathscr{T}_{\nu}$. Many properties of this
generalized estimator were discovered including a reflection formula as well
as its first two moments. The absolute relative bias
$\mathsf{ARB}\mathscr{T}_{\nu}$ and absolute coefficient of variation
$\mathsf{ACV}\mathscr{T}_{\nu}$ were also derived along with their exact
confidence intervals. An asymptotic expansion of $\mathscr{T}_{\nu}$ for large
sample sizes was then given, which played a critical role in applications
presented in latter sections.

Equipped with a substantial theoretical foundation, Chapter 3 tackled the
problem of applying the results of the preceding analysis to construct a novel
estimator, $\mathscr{G}_{\nu}$, of the photon transfer conversion gain
measurement. As a corollary to Theorem 1 it was proven that no unbiased,
finite-variance estimator of the conversion gain, $g$, existed under the
normal model of pixel noise. Mirroring the analysis of $\mathscr{T}_{\nu}$,
the first two moments of $\mathscr{G}_{\nu}$ were derived and used to
construct expressions for $\mathsf{ARB}\mathscr{G}_{\nu}$ and
$\mathsf{ACV}\mathscr{G}_{\nu}$. Then, using long standing observations from
the literature as a clue, the function $\mathscr{E}$ was introduced as a sort
of normalized metric for determining when the dispersion of
$\mathscr{G}_{\nu}$ was dominated by the dispersion of $\mathscr{T}_{\nu}$. It
was shown that $\mathscr{E}$ approaches unity in the shot noise limit
$\zeta\searrow 1$, which supported the observations in the literature and
showed that $\mathsf{ACV}\mathscr{G}_{\nu}$ could be very well approximated by
$\mathsf{ACV}\mathscr{T}_{\nu}$ given the level of illumination was
sufficiently large. These observations were subsequently utilized in a Monte
Carlo simulation to demonstrate the process of conversion gain estimation with
$\mathscr{G}_{\nu}$ as well as in the computation confidence intervals for
$\mathsf{ARB}\mathscr{G}_{\nu}$ and $\mathsf{ACV}\mathscr{G}_{\nu}$. A short
discussion followed the Monte Carlo experiment, which introduced the notion of
manipulating the parameter $\nu$ and sample sizes as to achieved desired
values for absolute relative bias and absolute coefficient of variation.

The questions following the Monte Carlo experiment in Section 3.4 served as a
springboard into the study of bias control and optimal sample sizes. To
control bias, two new estimators $\mathscr{T}_{\nu^{\smash{\dagger}}}$ and
$\mathscr{G}_{\nu^{\smash{\dagger}}}$ were defined, which varied
${\nu^{\smash{\dagger}}}$ in such a manner as to force estimator bias to
follow a prescribed profile. A definition of optimal sample sizes for was then
introduced followed by a study of the optimal sample sizes for
$\mathscr{T}_{\nu^{\smash{\dagger}}}$ and
$\mathscr{G}_{\nu^{\smash{\dagger}}}$ in the low illumination limit
$\zeta\nearrow 1$. It was demonstrated that as the illumination level
decreased, the optimal sample sizes,
$(n_{1}^{\text{opt}},n_{2}^{\text{opt}})$, for both estimators were
asymptotically equal and proportional to $(1-\zeta)^{-2}$. Perhaps the most
significant finding of this investigation at low illumination came in
Corollary 15, which showed that substituting the optimal sample sizes for
$\mathscr{T}_{\nu^{\smash{\dagger}}}$ into the metric $\mathscr{E}$ produced
nonzero values in the limit $\zeta\nearrow 1$. Furthermore, for sufficiently
large magnitudes of dark noise, $\sigma_{\mathrm{d}}g$, this limiting value of
$\mathscr{E}$ could be near unity meaning that the optimal sample sizes of
$\mathscr{G}_{\nu^{\smash{\dagger}}}$ can be very closely approximated to
those of $\mathscr{T}_{\nu^{\smash{\dagger}}}$ even if the illumination level
was near zero.

The fact that the optimal sample sizes of
$\mathscr{T}_{\nu^{\smash{\dagger}}}$ served as such good approximations to
those of $\mathscr{G}_{\nu^{\smash{\dagger}}}$ provided the justification
needed to perform a detailed investigation of their properties. The
investigation began by showing exact solutions for the optimal sample sizes of
$\mathscr{T}_{\nu^{\smash{\dagger}}}$ could be derived at at two special point
of interest in the $\zeta$-domain. For all remaining values of $\zeta$
explicit approximations for the optimal sample sizes were found and these
approximations were shown to perform very well for a wide range of parameters.
To compute the optimal sample sizes, a numerical routine was implemented,
which using the explicit approximations as a starting point. A brief analysis
of this numerical routine was conducted as to highlight potential weaknesses
that could be improved in future work.

With a means for computing optimal sample sizes, Section 3.6 concluded this
work with an introduction to _pixel-level_ photon transfer conversion gain
estimation. The concept of sensor nonuniformity was introduced as a motivation
for a pixel-level approach to photon transfer characterization and the
challenges of such an approach were discussed. Using a real image sensor a
first attempt at pixel-level conversion gain estimation was presented. In
particular, a focus was given on how to design a pixel-level conversion gain
estimation experiment using the optimal sample size curves and the quantity
$\mathscr{E}$ as tools for selecting an appropriate illumination level for
collecting data. An algorithm for collecting and halting data capture was
discussed and then executed on the chosen image sensor. A summary of the
experimental result then ensued, which presented the pixel-level $g$-map as
the primary data product of the experiment. Details of how the $g$-map was
computed as well as analysis of how the data collection algorithm performed
were conducted; revealing the algorithms were successful in controlling the
experiment and halting data capture at the appropriate time.

The successfulness of the theoretical results presented herein and their
application to pixel-level conversion gain estimation subsequently open the
door to developing a much needed comprehensive approach to pixel-level photon
transfer characterization.

##  Chapter 5 Appendices

###  A Definitions and relations

######  Definition 17 (Sign Function).

$\operatorname{sign}(x)\coloneqq\begin{cases}x/|x|,&x\neq 0\\\
0,&x=0.\end{cases}$

######  Definition 18 (Indicator Function).

$\mathds{1}_{A}\coloneqq\begin{cases}1,&A\text{ is true}\\\ 0,&A\text{ is
false}.\end{cases}$

######  Definition 19 (Gamma Function).

$\Gamma(z)\coloneqq\int_{0}^{\infty}t^{z-1}e^{-t}\,\mathrm{d}t,\quad\Re z>0$

######  Definition 20 (Beta Function).

$\operatorname{B}(s,z)\coloneqq\frac{\Gamma(s)\Gamma(z)}{\Gamma(s+z)}$

######  Definition 21 (Pochhammer Symbol (rising factorial)).

$(s)_{z}\coloneqq\frac{\Gamma(s+z)}{\Gamma(s)}$

######  Definition 22 (Factorial Power (falling factorial)).

$(s)^{(z)}\coloneqq\frac{\Gamma(s+1)}{\Gamma(s-z+1)}$

######  Relation 3 (Rising and falling factorical connection formula).

$(s)^{(n)}=(-1)^{n}(-s)_{n},\quad n\in\mathbb{Z}$

######  Relation 4 (Falling factorical product representation).

$(s)_{n}=\begin{cases}1,&n=0\\\
\prod_{k=0}^{n-1}(s+k)&n\in\mathbb{N}\end{cases}$

######  Relation 5 (Falling factorical product representation).

$(s)^{(n)}=\begin{cases}1,&n=0\\\
\prod_{k=0}^{n-1}(s-k)&n\in\mathbb{N}\end{cases}$

######  Definition 23 (Generating function of Stirling numbers of the $1$st-
kind $\mathcal{S}_{n}^{(k)}$).

$(s)_{n}\coloneqq\sum_{k=0}^{n}(-1)^{n-k}\mathcal{S}_{n}^{(k)}s^{k}$

######  Definition 24 (Generating function of generalized Nørlund polynomial
$B_{k}^{(\ell)}(z)$).

$\left(\frac{t}{e^{t}-1}\right)^{\ell}e^{zt}\coloneqq\sum_{k=0}^{\infty}B_{k}^{(\ell)}(z)\frac{t^{k}}{k!}$

######  Definition 25 (Nørlund polynomial).

$B_{k}^{(\ell)}\coloneqq B_{k}^{(\ell)}(0)$

######  Definition 26 (Bernoulli polynomial).

$B_{k}(z)\coloneqq B_{k}^{(1)}(z)$

######  Definition 27 (Lerch’s Transcendent).

$\Phi(z,s,\omega)\coloneqq\sum_{k=0}^{\infty}\frac{z^{k}}{(k+\omega)^{s}}$

######  Definition 28 (Generalized Hypergeometric Series).

${{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop
b_{1},\dots,b_{q}};z\right)\coloneqq\sum_{k=0}^{\infty}\frac{(a_{1})_{k}\cdots(a_{p})_{k}}{(b_{1})_{k}\cdots(b_{q})_{k}}\frac{z^{k}}{k!}$

######  Definition 29 (Regularized Generalized Hypergeometric Function).

${{}_{p}\mathbf{F}_{q}}\left({a_{1},\dots,a_{p}\atop
b_{1},\dots,b_{q}};z\right)\coloneqq\frac{1}{\prod_{k=1}^{q}\Gamma(b_{k})}{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop
b_{1},\dots,b_{q}};z\right)$

######  Definition 30 (Appell $F_{1}$ Hypergeometric Series).

$F_{1}(a;b,b^{\prime};c;s,z)\coloneqq\sum_{k,\ell=0}^{\infty}\frac{(a)_{k+\ell}(b)_{k}(b^{\prime})_{\ell}}{(c)_{k+\ell}\,k!\,\ell!}s^{k}z^{\ell},\quad\max\\{|s|,|z|\\}<1$

######  Definition 31 (Appell $F_{2}$ Hypergeometric Series).

$F_{2}(a;b,b^{\prime};c,c^{\prime};s,z)\coloneqq\sum_{k,\ell=0}^{\infty}\frac{(a)_{k+\ell}(b)_{k}(b^{\prime})_{\ell}}{(c)_{k}(c^{\prime})_{\ell}\,k!\,\ell!}s^{k}z^{\ell},\quad|s|+|z|<1$

######  Definition 32 (Incomplete Beta Function).

$\operatorname{B}_{z}(\alpha,\beta)\coloneqq\Gamma(\alpha)z^{\alpha}{\mathbf{F}}\left({\alpha,1-\beta\atop\alpha+1};z\right),\quad-\alpha\notin\mathbb{N}_{0}$

######  Definition 33 (Regularized Incomplete Beta Function).

$\operatorname{I}_{z}(\alpha,\beta)\coloneqq\frac{\operatorname{B}_{z}(\alpha,\beta)}{\operatorname{B}(\alpha,\beta)}$

######  Relation 6.

[27, Eq. $07.23.03.0122.01$].

$F(1,\beta;\gamma;s)=(\gamma-1)s^{1-\gamma}(1-s)^{-(\beta-\gamma+1)}\operatorname{B}_{s}(\gamma-1,\beta-\gamma+1).$

######  Relation 7 (Gamma reflection formula).

$\Gamma(z)\Gamma(1-z)=\pi\csc\pi z,\quad z\notin\mathbb{Z}$

######  Relation 8.

$(1-z-n)_{n}=(-1)^{n}(z)_{n},\quad n\in\mathbb{Z}$

###### Proof.

Using the gamma reflection formula in Relation 7 one writes

$(1-z-n)_{n}=\frac{\Gamma(1-z)}{\Gamma(1-z-n)}=\frac{\csc\pi
z}{\csc\pi(z+n)}\frac{\Gamma(z+n)}{\Gamma(z)}=(-1)^{n}(z)_{n}$

∎

######  Relation 9.

$\frac{1}{(1-z)_{-n}}=(-1)^{n}(z)_{n},\quad n\in\mathbb{Z}$

###### Proof.

Using Relation 8 one writes

$\frac{1}{(1-z)_{-n}}=\frac{\Gamma(1-z)}{\Gamma(1-z-n)}=\frac{\Gamma(1-z-n+n)}{\Gamma(1-z-n)}=(1-z-n)_{n}=(-1)^{n}(z)_{n}$

∎

######  Relation 10.

$(z)_{n}=(z)_{m}(z+m)_{n-m}$

###### Proof.

$(z)_{n}=\frac{\Gamma(z+n)}{\Gamma(z)}=\frac{\Gamma(z+m)}{\Gamma(z+m)}\frac{\Gamma(z+m+n-m)}{\Gamma(z)}=(z)_{m}(z+m)_{n-m}$

∎

######  Relation 11.

$\binom{n}{k}=\frac{(-1)^{k}(-n)_{k}}{k!}$

######  Relation 12.

[27, Eq. $07.23.03.0002.01$].

${F}\left({a,b\atop
c};1\right)=\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)},\quad\Re\\{c-a-b\\}>0$

######  Relation 13.

For $2k=0,2,4,\dots$

$(1-a-2k)_{2k}=2^{2k}\left(\tfrac{a}{2}\right)_{k}\left(\tfrac{a+1}{2}\right)_{k}.$

###### Proof.

Using the gamma reflection formula in Relation 7 one writes

$(1-a-2k)_{2k}=\frac{\Gamma(1-a)}{\Gamma(1-a-2k)}=\frac{\csc\pi
a}{\csc\pi(a+2k)}\frac{\Gamma(a+2k)}{\Gamma(a)}=\frac{\Gamma(a+2k)}{\Gamma(a)}.$

Then applying the gamma duplication formula in Relation
LABEL:rel:gamma_duplication yields

$\frac{\Gamma(a+2k)}{\Gamma(a)}=\frac{\Gamma(\frac{a}{2}+k)\Gamma(\frac{a}{2}+k+\frac{1}{2})2^{2k}}{\Gamma(\frac{a}{2})\Gamma(\frac{a}{2}+\frac{1}{2})}=2^{2k}\left(\tfrac{a}{2}\right)_{k}\left(\tfrac{a+1}{2}\right)_{k}.$

∎

######  Relation 14.

$(a)_{n+k}=(a)_{n}(a+n)_{k}.$

###### Proof.

$(a)_{n+k}=\frac{\Gamma(a+n+k)}{\Gamma(a)}=\frac{\Gamma(a+n)}{\Gamma(a)}\frac{\Gamma(a+n+k)}{\Gamma(a+n)}=(a)_{n}(a+n)_{k}.$

∎

###  B Limiting properties of $\mathscr{T}_{\nu}$ as $|\nu|\to\infty$

One curiosity that remains is what happens to $\mathscr{T}_{\nu}$ and its
moments as $|\nu|\to\infty$. The following theorem presents these results.

######  Theorem 19.

Let $Y_{1}\sim\mathcal{G}(\alpha_{1},\beta_{1})$ and
$Y_{2}\sim\mathcal{G}(\alpha_{2},\beta_{2})$ be gamma random variables
parameterized in terms of a known shape $\alpha_{i}$ and unknown rate of the
form $\beta_{i}=\alpha_{i}/\kappa_{i}$. Then,

$\displaystyle\mathscr{U}$
$\displaystyle=\hphantom{-}\frac{1}{\kappa_{1}}{{}_{1}F_{1}}\left({1\atop\alpha_{2}};\frac{\alpha_{2}Y_{2}}{\kappa_{1}}\right),$
$\displaystyle\text{with}\ \kappa_{1}>\kappa_{2}\ \text{and}\ \kappa_{1}\
\text{known}$ $\displaystyle\mathscr{V}$
$\displaystyle=-\frac{1}{\kappa_{2}}{{}_{1}F_{1}}\left({1\atop\alpha_{1}};\frac{\alpha_{1}Y_{1}}{\kappa_{2}}\right),$
$\displaystyle\text{with}\ \kappa_{1}<\kappa_{2}\ \text{and}\ \kappa_{2}\
\text{known}$

are unbiased estimators of $\tau=(\kappa_{1}-\kappa_{2})^{-1}$.

###### Proof.

Without loss of generality we consider the discrete estimator of Lemma 3
expressed by

$\mathscr{T}_{n}=\frac{1}{Y_{1}}\sum_{k=0}^{n-1}\frac{\alpha_{1}^{-k-1}}{(\alpha_{1})_{-k-1}(\alpha_{2})_{k}}\left(\frac{\alpha_{2}Y_{2}}{Y_{1}}\right)^{k},$

where we recall that $Y_{1}\sim\mathcal{G}(\alpha_{1},\alpha_{1}/\kappa_{1})$
and $Y_{2}\sim\mathcal{G}(\alpha_{2},\alpha_{2}/\kappa_{2})$ are independent.
From Theorem 4 we know that $\mathsf{E}\mathscr{T}_{n}<\infty$ if and only if
$n<\alpha_{1}$; thus; in order for
$\mathsf{E}(\lim_{n\to\infty}\mathscr{T}_{n})<\infty$ we require
$\alpha_{1}\to\infty$. To evalute the limit in $\alpha_{1}$ note that if
$\alpha_{1}\in\mathbb{N}$ then in terms of distribution

$Y_{1}=\frac{1}{\alpha_{1}}\sum_{k=1}^{\alpha_{1}}Y_{k}^{\prime},\quad
Y_{k}^{\prime}\sim\mathcal{G}(1,1/\kappa_{1})$

such that by the strong law of large numbers
$Y_{1}\overset{\mathrm{a.s.}}{\to}\kappa_{1}$ as $\alpha_{1}\to\infty$.
Furthermore, by [7, Eq. $5.11.13$] as $\alpha_{1}\to\infty$ we have
$\alpha_{1}^{-k-1}/(\alpha_{1})_{-k-1}\sim 1+\mathcal{O}\\{1/\alpha_{1}\\}$;
thus, passing to the limit $\alpha_{1}\to\infty$:

$\mathscr{T}_{n}\overset{\mathrm{a.s.}}{\to}\mathscr{U}_{n}\ \text{where}\
\mathscr{U}_{n}=\frac{1}{\kappa_{1}}\sum_{k=0}^{n-1}\frac{1}{(\alpha_{2})_{k}}\left(\frac{\alpha_{2}Y_{2}}{\kappa_{1}}\right)^{k}.$

As a consequence of taking $\alpha_{1}\to\infty$ there is no longer any
restriction on how large $n$ can be. Taking the limit $n\to\infty$ in the
previous result subsequently yields

$\lim_{n\to\infty}\mathscr{U}_{n}=\frac{1}{\kappa_{1}}\sum_{k=0}^{\infty}\frac{(1)_{k}}{(\alpha_{2})_{k}\,k!}\left(\frac{\alpha_{2}Y_{2}}{\kappa_{1}}\right)^{k},$

which is the estimator $\mathscr{U}$. In addition to $\kappa_{1}$ being known
if we assume $\kappa_{1}>\kappa_{2}$ then one can easily confirm

$\mathsf{E}\mathscr{U}=\lim_{n\to\infty}\mathsf{E}\mathscr{T}_{n}=\frac{1}{\kappa_{1}-\kappa_{2}},$

which completes the proof for the estimator $\mathscr{U}$. To obtain the
corresponding proof for the estimator $\mathscr{V}$ one can use the reflection
formula in Theorem 5 to write

$\lim_{n\to-\infty}\mathscr{T}_{n}(Y_{1},Y_{2},\alpha_{1},\alpha_{2})=-\lim_{n\to\infty}\mathscr{T}_{n}(Y_{2},Y_{1},\alpha_{2},\alpha_{1}),$

where we require $\alpha_{2}\to\infty$, i.e. $\kappa_{2}$ known, and
$\kappa_{1}<\kappa_{2}$ in order for the limiting expected value to converge
to the desired quantity. ∎

######  Remark 4.

The estimator $\mathscr{U}$ is the solution to the integral equation
$(\ref{eq:double_int_eq})$ when $\kappa_{1}$ is known, that is, it satisfies

$\mathcal{L}\\{y_{2}^{\alpha_{2}-1}\mathscr{U}(y_{2})\\}(\beta_{2})=\frac{\Gamma(\alpha_{2})\beta_{2}^{-\alpha_{2}}}{\kappa_{1}-\alpha_{2}/\beta_{2}}.$

Likewise,

$\mathcal{L}\\{y_{1}^{\alpha_{1}-1}\mathscr{V}(y_{1})\\}(\beta_{1})=\frac{\Gamma(\alpha_{1})\beta_{1}^{-\alpha_{1}}}{\alpha_{1}/\beta_{1}-\kappa_{2}}.$

These results agree with [19, Eq. $5.4.9$].

######  Lemma 39.

$\mathsf{Var}\mathscr{U}=\frac{1}{(\kappa_{1}-\kappa_{2})^{2}}\left({F}\left({1,1\atop\alpha_{2}};\frac{\kappa_{2}^{2}}{(\kappa_{1}-\kappa_{2})^{2}}\right)-1\right)$

if $\kappa_{1}>2\kappa_{2}$ and infinite otherwise. Likewise,

$\mathsf{Var}\mathscr{V}=\frac{1}{(\kappa_{1}-\kappa_{2})^{2}}\left({F}\left({1,1\atop\alpha_{1}};\frac{\kappa_{1}^{2}}{(\kappa_{1}-\kappa_{2})^{2}}\right)-1\right)$

if $\kappa_{2}>2\kappa_{1}$ and infinite otherwise.

###### Proof.

We will present the proof for $\mathsf{Var}\mathscr{U}$ with the proof for
$\mathsf{Var}\mathscr{V}$ being essentially the same. We begin by evaluating
$\lim_{n,\alpha_{1}\to\infty}\mathsf{E}\mathscr{T}_{n}^{2}$ from the
expression in Lemma 20. According to [7, Eq. $15.12.2$] we have the asymptotic
relation

${F}\left({k+1,\ell+1\atop\alpha_{1}};1\right)\sim
1+\mathcal{O}\\{1/\alpha_{1}\\},\quad\alpha_{1}\to\infty.$

Thus, upon taking the appropriate limits and writing
${{}_{2}}F_{1}(-k,-\ell;\alpha_{1};1)$ in terms of Pochhammer symbols one has

$\mathsf{E}\mathscr{U}^{2}=\frac{1}{\kappa_{1}^{2}}\sum_{k,\ell=0}^{\infty}\frac{(\alpha_{2})_{k+\ell}(1)_{k}(1)_{\ell}}{(\alpha_{2})_{k}(\alpha_{2})_{\ell}\,k!\,\ell!}\,\zeta^{k}\,\zeta^{\ell}.$
(5.1)

where we again use the shorthand $\zeta=\kappa_{2}/\kappa_{1}$. According to
Definition 31, this double series can be expressed in terms of Appell’s second
hypergeometric function and is absolutely convergent if
$\zeta<1/2\implies\kappa_{1}>2\kappa_{2}$. To simplify this result we apply
the reduction formula in [7, Eq. $16.16.3$] to find

$\displaystyle\mathsf{E}\mathscr{U}^{2}$
$\displaystyle=\frac{1}{\kappa_{1}^{2}}F_{2}(\alpha_{2};1,1;\alpha_{2},\alpha_{2};\zeta,\zeta)$
$\displaystyle=\frac{1}{\kappa_{1}^{2}}(1-\zeta)^{-1}F_{1}(1;\alpha_{2}-1,1;\alpha_{2};\zeta,\zeta(1-\zeta)^{-1}),$

where $F_{1}(\cdot)$ is Appell’s first hypergeometric function as defined in
Definition 30. Taking advantage of the symmetry
$F_{1}(a;b,b^{\prime};c;s,z)=F_{1}(a;b^{\prime},b;c;z,s)$ and applying the
reduction formula [7, Eq. $15.12.2$] then yields

$\mathsf{E}\mathscr{U}^{2}=\frac{1}{\kappa_{1}^{2}}(1-\zeta)^{-2}{F}\left({1,1\atop\alpha_{2}};\frac{\zeta^{2}}{(1-\zeta)^{2}}\right).$

Subtracting $(\mathsf{E}\mathscr{U})^{2}$ and simplifying yields the desired
result. ∎

######  Remark 5.

The divergence of $\mathsf{Var}\mathscr{U}$ for $\kappa_{1}<2\kappa_{2}$ is
not just a consequence of the double series representation
$(\ref{eq:EUsqrd_double_series})$. Indeed, using the integral definition of
$\mathsf{E}\mathscr{U}^{2}$ and $\zeta=\kappa_{2}/\kappa_{1}$ one may deduce
with a simple substitution

$\mathsf{E}\mathscr{U}^{2}\propto\int_{0}^{\infty}[{{}_{1}F_{1}}(1;\alpha_{2};t)]^{2}t^{\alpha_{2}-1}e^{-t/\zeta}\,\mathrm{d}t.$

But according to [7, Eq. $13.7.1$], as $t\to\infty$

$[{{}_{1}F_{1}}(1;\alpha_{2};t)]^{2}t^{\alpha_{2}-1}e^{-t/\zeta}\sim\Gamma^{2}(\alpha_{2})t^{1-\alpha_{2}}e^{(2-1/\zeta)t},$

which diverges for $\zeta^{-1}<2$, i.e. $\kappa_{1}<2\kappa_{2}$. At the
boundary $\kappa_{1}=2\kappa_{2}$ we use Relation 12 to further note that

$\mathsf{E}\mathscr{U}^{2}=\begin{cases}4\kappa_{1}^{-2}(\alpha_{2}-1)/(\alpha_{2}-2),&\alpha_{2}>2\\\
\infty,&\alpha_{2}\leq 2.\end{cases}$

###### Proof of Theorem 1.

Assume $\mathscr{T}$ exists and let $\zeta=\kappa_{2}/\kappa_{1}$ with
$\zeta<1$. Then by the law of total variance we have

$\mathsf{Var}\mathscr{T}(Y_{1},Y_{2})=\mathsf{E}(\mathsf{Var}(\mathscr{T}(Y_{1},Y_{2})|Y_{2}))+\mathsf{Var}g(\kappa_{1},Y_{2}),$

where $g(\kappa_{1},Y_{2})=\mathsf{E}(\mathscr{T}(Y_{1},Y_{2})|Y_{2})$ and by
assumption $\mathsf{E}g(\kappa_{1},Y_{2})=(\kappa_{1}-\kappa_{2})^{-1}$. Now,
from Lemma 1 we know $Y_{2}$ is a complete-sufficient statistic for
$\kappa_{2}$ such the Lehmann-Scheffé theorem asserts $g(\kappa_{1},Y_{2})$ is
the unique umvue of its expected value when $\kappa_{1}$ is known. But if
$g(\kappa_{1},Y_{2})$ is unique and $\zeta<1$ then Theorem 19 proves
$g(\kappa_{1},Y_{2})=\mathscr{U}$. Given
$\mathsf{E}(\mathsf{Var}(\mathscr{T}(Y_{1},Y_{2})|Y_{2}))\geq 0$ it follows
that $\mathsf{Var}\mathscr{U}\leq\mathsf{Var}\mathscr{T}$. Furthermore, Lemma
39 tells us that $\mathsf{Var}\mathscr{U}=\infty$ if $\zeta>1/2$; thus, it
must be that $\mathsf{Var}\mathscr{T}=\infty$ on $1/2<\zeta<1$. By a similar
argument if $\zeta>1$ then $\mathsf{Var}\mathscr{V}\leq\mathscr{T}$ with
$\mathscr{V}$ also given in Theorem 19. Since $\mathsf{Var}\mathscr{V}=\infty$
for $\zeta<2$ we can combine the previous result to conclude
$\mathsf{Var}\mathscr{T}=\infty$ on $1/2<\zeta<2$ which completes the proof. ∎

###  C Proofs of differential operator identities

Here present the proofs associated with the differential operator identities
in Lemma 7.

###### Proof of Lemma 7 $(i)$.

Begin by using the Definition 6 to write

$\Lambda_{\omega}z^{s}=\omega z^{s}+z\partial_{z}z^{s}.$

Then by the chain rule we find

$\Lambda_{\omega}z^{s}=\omega
z^{s}+z(sz^{s-1}+z^{s}\partial_{z})=z^{s}(\omega+s+z\partial_{z})=z^{s}\Lambda_{\omega+s},$

which completes the proof. ∎

###### Proof of Lemma 7 $(ii)$.

See [10]. ∎

###### Proof of Lemma 7 $(iii)$.

Let $P(n):(z\vartheta)^{n}=z(\vartheta z)^{n}z^{-1}$. It is trivial to show
that $P(0)$ holds; thus, assuming $P(n)$ we have for $P(n+1)$

$(z\vartheta)^{n+1}=(z\vartheta)z(\vartheta z)^{n}z^{-1}=z(\vartheta
z)(\vartheta z)^{n}z^{-1}=z(\vartheta z)^{n+1}z^{-1},$

thus, $P(n)\implies P(n+1)$. Substituting the result in $(ii)$ for $(\vartheta
z)^{n}$ subsequently produces the desired result. ∎

###### Proof of Lemma 7 $(iv)$.

Let $P(n):\Lambda_{\omega}^{n}=z^{-\omega}\vartheta^{n}z^{\omega}$ and note
that $P(0)$ trivially holds. Assuming $P(n)$ we have for $P(n+1)$

$\Lambda_{\omega}^{n+1}=(\omega+z\partial_{z})\Lambda_{\omega}^{n}=(\omega+z\partial_{z})z^{-\omega}\vartheta^{n}z^{\omega}=z^{-\omega}(z\partial_{z})\vartheta^{n}z^{\omega}=z^{-\omega}\vartheta^{n+1}z^{\omega},$

where the second to last equality is due to result $(i)$. Thus, $P(n)\implies
P(n+1)$ which completes the proof. ∎

###### Proof of Lemma 7 $(v)$.

Let $P(n):(\vartheta)^{(n)}=\frac{1}{z}(z\vartheta)^{n}z^{1-n}$. By Definition
7, it immediately follows that $P(0)$ holds. Assuming $P(n)$ we have for
$P(n+1)$

$(\vartheta)^{(n+1)}=(\vartheta)^{(n)}(\vartheta-n)=\frac{1}{z}(z\vartheta)^{n}z^{1-n}\Lambda_{-n}.$

Then, making use of $(iv)$ we find

$(\vartheta)^{(n+1)}=\frac{1}{z}(z\vartheta)^{n}z^{1-n}z^{n}\vartheta
z^{-n}=\frac{1}{z}(z\vartheta)^{n+1}z^{1-(n+1)}.$

Therefore, $P(n)\implies P(n+1)$. Substituting the result of $(iii)$ for
$(z\vartheta)^{n}$ then completes the proof. ∎

###### Proof of Lemma 7 $(vi)$.

Let $P(n):(\Lambda_{\omega})^{(n)}=z^{-\omega}(\vartheta)^{(n)}z^{\omega}$. By
Definition 7, it immediately follows that $P(0)$ holds. Now use Definition 7
to write

$(\Lambda_{\omega})^{(n+1)}=(\omega-n+\vartheta)(\Lambda_{\omega})^{(n)}.$

Assuming $P(n)$ we have for $P(n+1)$

$(\Lambda_{\omega})^{(n+1)}=(\omega-n+\vartheta)z^{-\omega}(\vartheta)^{(n)}z^{\omega}=z^{-\omega}(-n+\vartheta)(\vartheta)^{(n)}z^{\omega}=z^{-\omega}(\vartheta)^{(n+1)}z^{\omega},$

where the second to last equality is due to result $(i)$. Thus, $P(n)\implies
P(n+1)$ and upon substituting the result of $(v)$ for $(\vartheta)^{(n)}$ the
desired result is obtained. ∎

###### Proof of Lemma 7 $(vii)$.

From [7, Eq. $26.8.10$] we have

$\vartheta^{n}=\sum_{k=0}^{n}{{}_{2}\mathcal{S}}_{n}^{(k)}(\vartheta)^{(k)}.$

Substituting the result of $(v)$ for $(\vartheta)^{(k)}$ produces the desired
result. ∎

###  D Estimate of truncation error for the $\mathsf{ACV}\mathscr{T}_{\nu}$
series expansion

An important ingredient in the computation of confidence intervals in Section
3.4 and optimal sample sizes in Section 3.5.2 is the ability to evaluate
$\mathsf{ACV}^{2}\mathscr{T}_{\nu}$ via its series expansion. To evaluate the
series expansion we write

$\mathsf{ACV}^{2}\mathscr{T}_{\nu}=\mathsf{ACV}_{n}^{2}\mathscr{T}_{\nu}+E_{n},$

where

$\mathsf{ACV}_{n}^{2}\mathscr{T}_{\nu}=\sum_{k=1}^{n}\sum_{\ell=0}^{k}\frac{\tilde{g}_{k,\ell}^{2}(z,\nu)}{(\alpha_{1})_{\ell}(\alpha_{2})_{k-\ell}\,\ell!\,(k-\ell)!}$

and

$E_{n}=\sum_{k=n+1}^{\infty}\sum_{\ell=0}^{k}\frac{\tilde{g}_{k,\ell}^{2}(z,\nu)}{(\alpha_{1})_{\ell}(\alpha_{2})_{k-\ell}\,\ell!\,(k-\ell)!}$

and then determine the number $n$ such that the error incurred in the
approximation
$\mathsf{ACV}^{2}\mathscr{T}_{\nu}\approx\mathsf{ACV}_{n}^{2}\mathscr{T}_{\nu}$
falls within some specified tolerance. Of course, we do not know a closed form
for $E_{n}$ and so whatever procedure used to find $n$ will ultimately require
finding a useful upper bound.

######  Definition 34 (Complementary incomplete Hypergeometric function).

For $\nu\in\mathbb{C}$

$F(\alpha,\beta;\gamma;z)_{\nu}^{c}\coloneqq
F(\alpha,\beta;\gamma;z)-F(\alpha,\beta;\gamma;z)_{\nu},$

where $F(\alpha,\beta;\gamma;z)_{\nu}$ is the incomplete hypergeometric
function.

######  Proposition 9.

For $\nu\in\mathbb{C}$

$F(\alpha,\beta;\gamma;z)_{\nu}^{c}=\frac{(\alpha)_{\nu}(\beta)_{\nu}}{(\gamma)_{\nu}}\frac{z^{\nu}}{\Gamma(1+\nu)}{{}_{3}F_{2}}\left({1,\alpha+\nu,\beta+\nu\atop
1+\nu,\gamma+\nu};z\right).$

Likewise, if $\nu=n\in\mathbb{N}_{0}$

$F(\alpha,\beta;\gamma;z)_{n}^{c}=F(\alpha,\beta;\gamma;z)-\frac{(\alpha)_{n-1}(\beta)_{n-1}}{(\gamma)_{n-1}}\frac{z^{n-1}}{\Gamma(n)}{{}_{3}F_{2}}\left({1,1-n,2-n-\gamma\atop
2-n-\alpha,2-n-\beta};\frac{1}{z}\right).$

######  Theorem 20.

Let $E_{n}$ denote the truncation error incurred in approximating
$\mathsf{ACV}^{2}\mathscr{T}_{\nu}$ with
$\mathsf{ACV}_{n}^{2}\mathscr{T}_{\nu}$. Then, $E_{n}\leq E_{n,m}^{\ast}$,
where

$E_{n,m}^{\ast}=\sum_{j=0}^{m-1}(\mathsf{E}h_{k^{\prime}+j,k^{\prime}})^{2}b_{j}{F}\left({1+\nu,1+\nu\atop\alpha_{1}};1\right)_{k^{\prime}}^{c}\\\
+(\mathsf{E}h_{k^{\prime\prime}+m,k^{\prime\prime}\mathds{1}_{z\in[0,1]}})^{2}\Biggl{(}{F}\left({1+\nu,1+\nu\atop\alpha_{1}};1\right){F}\left({1-\nu,1-\nu\atop\alpha_{2}};1\right)_{m}^{c}\\\
-b_{m}\sum_{k=0}^{k^{\prime\prime}-1}a_{k}{{}_{5}F_{4}}\left({-k,1-k-\alpha_{1},1,1-\nu+m,1-\nu+m\atop
1+m,m+\alpha_{2},-k-\nu,-k-\nu};1\right)\Biggr{)},$

$k^{\prime}=\max(0,n+1-j)$, $k^{\prime\prime}=\max(0,n+1-m)$,
$a_{j}=(1+\nu)_{j}^{2}/((\alpha_{1})_{j}j!)$,
$b_{j}=(1-\nu)_{j}^{2}/((\alpha_{2})_{j}j!)$,
$F(\alpha,\beta;\gamma;z)_{\nu}^{c}$ is the complementary incomplete
hypergeometric function of Definition 34, and $\mathsf{E}h_{n,\omega}$ is
given in (2.10). Furthermore, $E_{n,m}^{\ast}<\infty$ if $\alpha_{1}>2(1+\nu)$
and $\alpha_{2}>2(1-\nu)$ and infinite otherwise.

###### Proof.

We begin by writing

$E_{n}=\sum_{k=n+1}^{\infty}\sum_{\ell=0}^{k}a_{\ell}b_{k-\ell}(\mathsf{E}h_{k,\ell})^{2},$

where $a_{j}=(1+\nu)_{j}^{2}/((\alpha_{1})_{j}j!)$ and
$b_{j}=(1-\nu)_{j}^{2}/((\alpha_{2})_{j}j!)$. The series $E_{n}$ is absolutely
convergent and so we may rearrange its terms as

$E_{n}=\sum_{j=0}^{m-1}b_{j}\sum_{k=k^{\prime}}^{\infty}a_{k}(\mathsf{E}h_{k+j,k})^{2}+\sum_{k=k^{\prime\prime}}^{\infty}\sum_{\ell=0}^{k}a_{\ell}b_{k-\ell+m}(\mathsf{E}h_{k+m,\ell})^{2},$

where $k^{\prime}=\max(0,n+1-j)$ and $k^{\prime\prime}=\max(0,n+1-m)$. Since
all terms are nonnegative we may call on Lemma 18 to obtain the upper bound
$E_{n}\leq E_{n,m}^{\ast}$, where

$E_{n,m}^{\ast}=\sum_{j=0}^{m-1}b_{j}(\mathsf{E}h_{k^{\prime}+j,k^{\prime}})^{2}\sum_{k=k^{\prime}}^{\infty}a_{k}+(\mathsf{E}h_{k^{\prime\prime}+m,k^{\prime\prime}\mathds{1}_{z\in[0,1]}})^{2}\sum_{k=k^{\prime\prime}}^{\infty}\sum_{\ell=0}^{k}a_{\ell}b_{k-\ell+m}.$

Upon inspection,

$\sum_{k=k^{\prime}}^{\infty}a_{k}={F}\left({1+\nu,1+\nu\atop\alpha_{1}};1\right)_{k^{\prime}}^{c},$

which converges when $\alpha_{1}>2(1+\nu)$. Furthermore,

$\sum_{k=k^{\prime\prime}}^{\infty}\sum_{\ell=0}^{k}a_{\ell}b_{k-\ell+m}=\sum_{k=0}^{\infty}a_{k}\sum_{k=m}^{\infty}b_{k}-\sum_{k=0}^{k^{\prime\prime}-1}\sum_{\ell=0}^{k}a_{\ell}b_{k-\ell+m},$

where

$\sum_{k=0}^{\infty}a_{k}={F}\left({1+\nu,1+\nu\atop\alpha_{1}};1\right),$

which converges for $\alpha_{1}>2(1+\nu)$ and

$\sum_{k=m}^{\infty}b_{k}={F}\left({1-\nu,1-\nu\atop\alpha_{2}};1\right)_{m}^{c},$

which converges for $\alpha_{2}>2(1-\nu)$. Lastly, with a bit of algebra and
working with the properties of the Pochhammer symbol we may use [7, Eq.
$16.2.4$] to write

$\sum_{\ell=0}^{k}a_{\ell}b_{k-\ell+m}=a_{k}b_{m}{{}_{5}F_{4}}\left({-k,1-k-\alpha_{1},1,1-\nu+m,1-\nu+m\atop
1+m,m+\alpha_{2},-k-\nu,-k-\nu};1\right).$

Bringing all results together gives the desired form for $E_{n,m}^{\ast}$. The
proof is now complete. ∎

######  Corollary 17.

Let, $p>0$ and

$R_{n,p}=\left\lvert\frac{\mathsf{ACV}_{n}^{p}\mathscr{T}_{\nu}}{\mathsf{ACV}^{p}\mathscr{T}_{\nu}}-1\right\rvert$

denote the absolute relative error incurred when approximating
$\mathsf{ACV}^{p}\mathscr{T}_{\nu}$ with
$\mathsf{ACV}_{n}^{p}\mathscr{T}_{\nu}$. Then, $R_{n,p}\leq R_{n,m,p}^{\ast}$
where

$R_{n,m,p}^{\ast}=\left\lvert\left(1+\frac{E_{n,m}^{\ast}}{\mathsf{ACV}_{n}^{2}\mathscr{T}_{\nu}}\right)^{-p/2}-1\right\rvert$

and $E_{n,m}^{\ast}$ is given in Theorem 20. Furthermore, if $\nu>1$ is
constant then for all $z\in Z\subset\mathbb{R}^{+}$, $R_{n,p}^{\ast}\leq
R_{n,m,p}^{\star}$ where

$R_{n,m,p}^{\star}=\left\lvert\left(1+\frac{E_{n,m}^{\ast}|_{z=\sup
Z}}{\mathsf{ACV}_{n}^{2}\mathscr{T}_{\nu}|_{z=\inf
Z}}\right)^{-p/2}-1\right\rvert\times 100\%.$

If instead $\nu<-1$ interchange $\inf Z$ and $\sup Z$.

###### Proof.

The first result follows from writing
$\mathsf{ACV}^{p}\mathscr{T}_{\nu}=\left(\mathsf{ACV}_{n}^{2}\mathscr{T}_{\nu}+E_{n}\right)^{p/2}$
and then substituting $E_{n}\mapsto E_{n,m}^{\ast}$ to obtain an upper bound
on $R_{n,p}$. For the second result we may use Lemma 19 to claim that
$\mathsf{E}h_{k^{\prime}+j,k^{\prime}}$,
$\mathsf{E}h_{k^{\prime\prime}+m,k^{\prime\prime}}$, and
$\mathsf{E}h_{k^{\prime\prime}+m,0}$ are all increasing functions of $z$ for
some constant $\nu>1$ on $z\in\mathbb{R}^{+}$. Furthermore, since
$\mathsf{E}h_{k^{\prime\prime}+m,k^{\prime\prime}}=\mathsf{E}h_{k^{\prime\prime}+m,0}=(k^{\prime\prime}+m+1)^{-1}$
at $z=1$ it follows that
$\mathsf{E}h_{k^{\prime\prime}+m,k^{\prime\prime}\mathds{1}_{z\in[0,1]}}$ must
also be increasing in $z$ on $z\in\mathbb{R}^{+}$. Combinbing these
observations with the fact that $E_{n,m}^{\ast}$ and
$\mathsf{ACV}_{n}^{2}\mathscr{T}_{\nu}$ are sums of nonnegative terms implies
that both of these functions must also be increasing in $z$ on
$z\in\mathbb{R}^{+}$; hence, for all $z\in Z$: $E_{n}^{\ast}\leq
E_{n}^{\ast}|_{z=\sup Z}$ and
$\mathsf{ACV}_{n}^{-2}\mathscr{T}_{\nu}\leq\mathsf{ACV}_{n}^{-2}\mathscr{T}_{\nu}|_{z=\inf
Z}$, which gives the upper bound on $R_{n,m,p}^{\ast}$. Noting the
monotonicity of these functions is reversed if $\nu<-1$ gives the
complementary bound. ∎

###  E Bounding functions for $|\partial_{\beta}^{n}F(\alpha,\alpha;\beta;x)|$

In this section we derive several results that will be used to determine
bounding functions for $|\partial_{\beta}^{n}F(\alpha,\alpha;\beta;x)|$ on
$x\in[0,1]$ when $\alpha\in\mathbb{R}$ and $\beta\in\mathbb{R}^{+}$. The main
results are found in Theorems 21 and 22.

Before we start deriving the bounding functions we must find a suitable
expression for higher order derivatives of the hypergeometric function w.r.t.
its bottom parameter. The following Lemma turns out to be key for doing just
that.

######  Lemma 40.

For $a,b>0$, $n\in\mathbb{N}_{0}$, and $X\sim\operatorname{Beta}(a,b)$

$(a)_{b}\partial_{a}^{n}(a)_{b}^{-1}=\mathsf{E}\log^{n}X.$

###### Proof.

Denoting $P(n):(a)_{b}\partial_{a}^{n}(a)_{b}^{-1}=\mathsf{E}\log^{n}X$ we
immediately conclude that $P(0)$ holds. Assuming $P(n)$ we find

$\displaystyle P(n)\implies\partial_{a}^{n+1}(a)_{b}^{-1}$
$\displaystyle=\partial_{a}(a)_{b}^{-1}\mathsf{E}\log^{n}X$
$\displaystyle=\partial_{a}\int_{0}^{1}(\log^{n}t)\,\frac{t^{a-1}(1-t)^{b-1}}{\Gamma(b)}\,\mathrm{d}t$
$\displaystyle=\int_{0}^{1}(\log^{n+1}t)\,\frac{t^{a-1}(1-t)^{b-1}}{\Gamma(b)}\,\mathrm{d}t$
$\displaystyle=(a)_{b}^{-1}\mathsf{E}\log^{n+1}X.$

Thus, $P(n)\implies P(n+1)$ and the proof is complete. ∎

Using this probabilistic interpretation for derivatives of the Pochhammer
symbol now facilitates a simple integral representation for
$\partial_{\gamma}^{n}F(\alpha,\beta;\gamma;x)$ as seen in the following
result.

######  Lemma 41.

For $\gamma>0$, $|x|<1$, and $n\in\mathbb{N}_{0}$

$\partial_{\gamma}^{n}F(\alpha,\beta;\gamma;x)=\int_{0}^{1}\log^{n}(1-t)\,(1-t)^{\gamma-1}\partial_{t}F(\alpha,\beta;1;xt)\,\mathrm{d}t+\mathds{1}_{n=0}$

###### Proof.

From Lemma 40 we write

$\partial_{\gamma}^{n}F(\alpha,\beta;\gamma;x)=\sum_{k=1}^{\infty}\frac{(\alpha)_{k}(\beta)_{k}}{(\gamma)_{k}}\frac{x^{k}}{k!}\mathsf{E}\log^{n}X+\mathds{1}_{n=0},\quad
X\sim\operatorname{Beta}(\gamma,k).$

Now writing the expected value in integral form subsequently gives

$\partial_{\gamma}^{n}F(\alpha,\beta;\gamma;x)=\lim_{m\to\infty}\int_{0}^{1}(\log^{n}t)t^{\gamma-1}\sum_{k=1}^{m}\frac{(\alpha)_{k}(\beta)_{k}}{\Gamma(k)}\frac{x^{k}}{k!}(1-t)^{k-1}\,\mathrm{d}t+\mathds{1}_{n=0}.$

Since $|x|<1$ the magnitude of the integrand is bounded above by $C(-\log
t)^{n}t^{\gamma-1}$ for some $C>0$, which is integrable for all
$n\in\mathbb{N}_{0}$ and $\gamma>0$. Consequently, we have by argument of
dominated convergence

$\partial_{\gamma}^{n}F(\alpha,\beta;\gamma;x)=-\int_{0}^{1}(\log^{n}t)t^{\gamma-1}\partial_{t}F(\alpha,\beta;1;x(1-t))\,\mathrm{d}t+\mathds{1}_{n=0},$

which upon substituting $t\mapsto 1-t$ gives the desired form. The proof is
now complete. ∎

Upon inspection of the integral representation in Lemma 40, one bounding
function immediately stands out.

######  Theorem 21.

Let $\alpha\in\mathbb{R}$, $\beta>0$, $x\in[0,1]$, and $n\in\mathbb{N}_{0}$.
Then for any $d>1$

$|\partial_{\beta}^{n}F(\alpha,\alpha;\beta;x)|<e^{-n}\left(\frac{dn}{\beta}\right)^{n}(F(\alpha,\alpha;(1-d^{-1})\beta;x)-1)+\mathds{1}_{n=0},$

where for the case $n=0$ we define $0^{0}\coloneqq 1$.

###### Proof.

For brevity denote $\beta^{\prime}=(1-d^{-1})\beta$. Then by Lemma 41 we have

$|\partial_{\beta}^{n}F(\alpha,\alpha;\beta;x)|=\int_{0}^{1}(1-t)^{\beta/d}(-\log(1-t))^{n}\,(1-t)^{\beta^{\prime}-1}\partial_{t}F(\alpha,\alpha;1;xt)\,\mathrm{d}t+\mathds{1}_{n=0}.$

Now, $0\leq(1-t)^{\beta/d}(-\log(1-t))^{n}\leq e^{-n}(dn/\beta)^{n}$; thus,
with the help of [11, Eq. $7.512.12$] if $d>0$ then $\beta^{\prime}>0$ and we
have

$|\partial_{\beta}^{n}F(\alpha,\alpha;\beta;x)|<e^{-n}\left(\frac{dn}{\beta}\right)^{n}\frac{\alpha^{2}x}{\beta^{\prime}}{{}_{3}F_{2}}\left({1,\alpha+1,\alpha+1\atop
2,\beta^{\prime}+1};x\right)+\mathds{1}_{n=0}.$

The contigious relation in [27, Eq. $07.27.17.0012.01$] then reduces this
result to its final form. The proof is now complete. ∎

Our ultimate goal in this effort is to find bounding function for the purposes
of studying the convergence of higher order derivatives of
$\mathsf{ACV}^{2}\mathscr{T}_{\nu}$ w.r.t. $\alpha_{1}$ and $\alpha_{2}$.
However, observe that a key feature of the bounding function given in Theorem
21 is a reduction of the bottom parameter by a multiplicative factor of
$(1-d^{-1})$, $d>1$. Using this bounding function for studying higher order
derivatives of $\mathsf{ACV}^{2}\mathscr{T}_{\nu}$ would effectively reduce
$\alpha_{1}$ and $\alpha_{2}$ by this same factor and thus would alter the
convergence properties of the integral representing the higher order
derivatives of $\mathsf{ACV}^{2}\mathscr{T}_{\nu}$. As a consequence, we would
not determine the full range of possible values for $\alpha_{1}$ and
$\alpha_{2}$ and must turn to finding a more suitable bounding function. We
proceed with two more preliminary results and then onto Theorem 22 which gives
us the result were looking for.

######  Lemma 42.

$\int(1-t)^{s-1}(1-xt)^{-s}\,\mathrm{d}t=-\frac{1}{s}(1-t)^{s}(1-xt)^{-s}{F}\left({1,s\atop
1+s};x\frac{1-t}{1-xt}\right)+C.$

###### Proof.

Let $I$ denote the integral in question. Substituting $s=(1-t)/(1-xt)$ yields

$I=-\int(1-xs)^{-1}s^{c-1}\,\mathrm{d}s=-\int_{0}^{s}(1-xz)^{-1}z^{c-1}\,\mathrm{d}z.$

Another substitution of $z=sw$ subsequently allows us to write the resulting
integral in the form of the integral representation for the hypergeometric
function in Relation 1. The proof is now complete. ∎

######  Lemma 43 ([2, Lem. $2.3$]).

For $\alpha,\beta\in\mathbb{R}^{+}$ and $x\in[0,1]$

$F(\alpha,\beta;\alpha+\beta;x)\leq
1-\frac{1}{\operatorname{B}(\alpha,\beta)}\log(1-x).$

We now present the

######  Theorem 22.

For $\alpha\in\mathbb{R}$, $\beta\in\mathbb{R}^{+}$, $x\in[0,1]$, and
$n\in\mathbb{N}_{0}$:

$|\partial_{\beta}^{n}F(\alpha,\alpha;\beta;x)|\leq\upsilon_{n}(\alpha,\beta,x),$

where

$\upsilon_{n}(\alpha,\beta,x)=\begin{cases}\frac{n!}{\operatorname{B}(|\alpha|,|\alpha|)}\frac{1}{(\beta-2\alpha\mathds{1}_{\alpha>0})^{n+1}}+\mathds{1}_{n=0},&\beta>2\alpha\\\\[4.30554pt]
\frac{n!}{\operatorname{B}(\alpha,\alpha)}(\frac{1}{2\alpha}-\log(1-x))^{n+1}+\mathds{1}_{n=0},&\beta=2\alpha\\\\[4.30554pt]
\frac{n!}{\operatorname{B}(\alpha,\alpha)}(1-x)^{\beta-2\alpha}(\frac{1}{\beta}-\log(1-x))^{n+1}+\mathds{1}_{n=0},&\beta<2\alpha.\end{cases}$

###### Proof.

We begin with the integral representation of Lemma 41 and write

$|\partial_{\beta}^{n}F(\alpha,\alpha;\beta;x)|\leq\alpha^{2}(-1)^{n}\int_{0}^{1}\log^{n}(1-t)\,(1-t)^{\beta-1}F(\alpha+1,\alpha+1;2;xt)\,\mathrm{d}t+\mathds{1}_{n=0}.$

Each case will now be proven from this integral representation.

  1. $(1)$

$\beta>2\alpha$.

###### Proof.

The case $\beta>2\alpha$ must itself be broken down into three subcases:
$\alpha<0$, $\alpha=0$, and $\alpha>0$. If $\alpha<0$ then we have from Lemma
23 that

$F(\alpha+1,\alpha+1;2;xt)\leq\frac{1}{\alpha^{2}\operatorname{B}(-\alpha,-\alpha)};$

hence,

$\displaystyle|\partial_{\beta}^{n}F(\alpha,\alpha;\beta;x)|$
$\displaystyle\leq\frac{(-1)^{n}}{\operatorname{B}(-\alpha,-\alpha)}\int_{0}^{1}\log^{n}(1-t)\,(1-t)^{\beta-1}\,\mathrm{d}t+\mathds{1}_{n=0}$
$\displaystyle=\frac{n!}{\operatorname{B}(-\alpha,-\alpha)}\frac{1}{\beta^{n+1}}+\mathds{1}_{n=0}.$

For $\alpha=0$ we have the trivial calculation

$|\partial_{\beta}^{n}F(0,0;\beta;x)|=\mathds{1}_{n=0}.$

Now if $\alpha>0$ we again call on Lemma 23 to deduce

$F(\alpha+1,\alpha+1;2;xt)=(1-xt)^{-2\alpha}F(1-\alpha,1-\alpha;2;xt)\leq\frac{(1-t)^{-2\alpha}}{\alpha^{2}\operatorname{B}(\alpha,\alpha)}.$

It follows that

$\displaystyle|\partial_{\beta}^{n}F(\alpha,\alpha;\beta;x)|$
$\displaystyle\leq\frac{(-1)^{n}}{\operatorname{B}(\alpha,\alpha)}\int_{0}^{1}\log^{n}(1-t)\,(1-t)^{\beta-2\alpha-1}\,\mathrm{d}t+\mathds{1}_{n=0}$
$\displaystyle=\frac{n!}{\operatorname{B}(\alpha,\alpha)}\frac{1}{(\beta-2\alpha)^{n+1}}+\mathds{1}_{n=0}.$

Combining these three results yields the desired form for the case
$\beta>2\alpha$. ∎

  2. $(2)$

$\beta\leq 2\alpha$.

###### Proof.

First observe that $\beta>0\land\beta\leq 2\alpha\implies\alpha>0$ and so we
have from Lemma 23

$F(\alpha+1,\alpha+1;2;xt)\leq\frac{(1-xt)^{-2\alpha}}{\alpha^{2}\operatorname{B}(\alpha,\alpha)}\leq\frac{(1-x)^{\beta-2\alpha}(1-xt)^{-\beta}}{\alpha^{2}\operatorname{B}(\alpha,\alpha)}$

and

$|\partial_{\beta}^{n}F(\alpha,\alpha;\beta;x)|\leq\frac{(1-x)^{\beta-2\alpha}}{\operatorname{B}(\alpha,\alpha)}I_{n}+\mathds{1}_{n=0},$

where

$I_{n}=\int_{0}^{1}(-1)^{n}\log^{n}(1-t)\,(1-t)^{\beta-1}(1-xt)^{-\beta}\,\mathrm{d}t.$

Performing integration by parts with $u=(-1)^{n}\log^{n}(1-t)$ and
$\mathrm{d}v=(1-t)^{\beta-1}(1-xt)^{-\beta}\,\mathrm{d}t$ we use Lemma 42 to
write

$I_{n}=-\frac{(-1)^{n}}{\beta}\log^{n}(1-t)(1-t)^{\beta}(1-xt)^{-\beta}{F}\left({1,\beta\atop
1+\beta};x\frac{1-t}{1-xt}\right)\bigg{|}_{t=0}^{1}\\\
+\frac{n}{\beta}\int_{0}^{1}(-1)^{n-1}\log^{n-1}(1-t)(1-t)^{\beta-1}(1-xt)^{-\beta}{F}\left({1,\beta\atop
1+\beta};x\frac{1-t}{1-xt}\right)\,\mathrm{d}t.$

Evaluating the limit term gives

$uv|_{t=0}^{1}=\begin{cases}\frac{1}{\beta}F(1,\beta;1+\beta;x),&n=0\\\
0,&n\in\mathbb{N}\end{cases}$ (5.2)

with the latter case being due to $\log^{n}(1-t)(1-t)^{\beta}\to 0$ in both
limits. Furthermore, calling in Lemma 43 we find

${F}\left({1,\beta\atop 1+\beta};x\frac{1-t}{1-xt}\right)\leq
1-\beta\log\left(1-\frac{x(1-t)}{1-xt}\right)\leq 1-\beta\log(1-x)$

Hence, $I_{n}\leq\frac{n}{\beta}(1-\beta\log(1-x))I_{n-1}$ and

$I_{n}\leq\frac{n!}{\beta^{n}}(1-\beta\log(1-x))^{n}I_{0}.$

Note that $I_{0}$ is given by the $n=0$ case of $(\ref{eq:uv_term_cases})$ and
$I_{0}\leq\frac{1}{\beta}(1-\beta\log(1-x))$. These observations lead us to
conclude

$|\partial_{\beta}^{n}F(\alpha,\alpha;\beta;x)|\leq\frac{(1-x)^{\beta-2\alpha}}{\operatorname{B}(\alpha,\alpha)}\frac{n!}{\beta^{n+1}}(1-\beta\log(1-x))^{n+1}+\mathds{1}_{n=0},$

which is the final result for the $\beta<2\alpha$ case. Substituting
$\beta\mapsto 2\alpha$ into the r.h.s. of the inequality then gives the
desired result for the remaining case of $\beta=2\alpha$. The proof is now
complete. ∎

∎

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