Edgeworth Expansions When the Parameter Dimension Increases with Sample Size

Suppose that we have a statistical model with $q=q_n$ unknown parameters, $w=(1_1,\dots,w_q)'$, estimated by $\hat{w}$, based on a sample of size $n$. Suppose also, that we have Edgeworth expansions for the density and distribution of $X_n=n^{1/2} (\hat{w}-w)$. %We ask the question: How fast can $q=q_n$ increase with $n$ for the three main Edgeworth expansions to remain valid? We show that it is sufficient that $q_n=o(n^{1/6})$, if the estimate $\hat{w}$ is a standard estimate. That is, $E\ \hat{w}\rightarrow w$ as $n\rightarrow w$, and for $r\geq 1$, its $r$th order cumulants have magnitude $n^{1-r}$ and can be expanded in powers of $n^{-1}$. This very large class of estimates has a huge range of potential applications. When $\hat{w}=t(\bar{X})$ for $t:R^q\rightarrow R^p$ a smooth function of a sample mean $\bar{X}$ from a distribution on $R^q$, and $p_nq_n=pq\rightarrow\infty$ as $n\rightarrow\infty$, I show that the Edgeworth expansions for $\hat{w}$ remain valid if $q_n^8 p_n^6=o(n)$. For example, this holds for fixed $p=p_n$ if $q_n=o(n^{1/8})$. I also give a method that greatly reduces the number of terms needed for the 2nd and 3rd order terms in the Edgeworth expansions, that is, for the 1st and 2nd order corrections to the Central Limit Theorems (CLTs).