T_{a_b}} = \left\{
          \begin{array}{ll} O(n^{(b - 1)/2})\,, & \text{if $b$ is odd} \\
    O(n^{b/2})\,, & \text{if $b$ is even}
          \end{array} \right. \, ,$$ where $\nu_{R_{a_1}, S_{a_2}, \ldots, T_{a_b}} = \expect_G
    (H_{R_{a_1}} H_{S_{a_2}} \cdots H_{T_{a_b}})$ are joint central moments of estimating functions and their derivatives, with $R_{a_1}, S_{a_2}, \ldots, T_{a_b}$ being subsets of $a_1, a_2,
    \ldots, a_b > 0$ integers, respectively.

5.  \[invertibility\] The matrix with elements $\mu_{rs}$ $(r,s = 1,\ldots,p)$ is invertible.

Assumption \[consistency\] is a working assumption that we make about the unbiasedness of the estimating functions and the consistency of the $M$-estimators. Consistency can sometimes be shown to hold under weak assumptions about $G$ and the asymptotic unbiasedness of the estimating functions; see, for example, @vaart:1998 [Section 5.2] and @huber+ronchetti:2009 [Section 6.2], for theorems on the consistency of $M$-estimators. We assume that there is an index $n$, which is typically, but not necessarily, the number of observations, that measures the rate the information about the parameter ${{}\theta}$ accumulates, and that the difference $\hat{{}\theta} - \bar{{}\theta}$ is of order $n^{-1/2}$ in probability.

Assumption \[smoothness\] allows taking derivatives of the estimating functions across the parameter space when constructing the stochastic Taylor expansions required for the derivation of the empirical bias-reducing 