Asymptotic Expansions Associated with the $n$th Power of a Density

Abstract

Let $\{\mathbf{X}_n\}^\infty_{n = 1}$ be the sequence of random variables associated with the sequence of densities of the form \begin{equation*}\tag{1.1}C_n(x)f^n(x)\quad n = 1, 2, \cdots, \end{equation*} where $k$ is a positive function and $f$ has a unique mode $m$ at which it is sufficiently smooth. It is known that $n^{\frac{1}{2}} (X_n - m)b$ converges in law to the standard normal distribution when $b$ is a suitably chosen scaling constant (see Laplace (1847), pages 400-403, or von Mises (1964), page 269). In Section 2, it is shown that the cumulative distribution function $F_n$ possesses an asymptotic expansion in powers of $n^{\frac{1}{2}}$ where each coefficient is the product of a polynomial and the standard normal density. The polynomials have coefficients which are expressible in terms of $k$ and $\log f$, together with their derivatives evaluated at $m$. Section 3 shows that the normalizing transformation $\eta_n(\xi) = \Phi^{-1}(F_n(\xi))$, where $\Phi$ is the standard normal cdf, also has an asymptotic expansion in powers of $n^{-\frac{1}{2}}$ and Section 4 makes the same conclusion for the percentiles of $F_n$. The coefficients in each of these expansions are polynomials. Similar theorems are given by Bol'shev (1959), (1963), Dorogovcev (1962), Peiser (1949), and Wasow (1956). Examples of these expansions, namely the $t$-distribution and central order statistics, are given. 1.1. General assumptions. Consideration of the random variables $b(X_n - m)$ where $b^2 = -f''(m)/f(m)$ shows that it is possible, without loss of generality, to specialize to the case where $m = 0, f(0) = 1, f'(0) = 0$, and $f''(0) = -1$. The general assumptions are stated for this case. ASSUMPTION (i). $f(x)$ and $k(x)$, considered as functions of a complex variable, are analytic for $|x| \leqq \delta_1$ where $\delta_1$ is a positive constant. ASSUMPTION (ii). $f(x)$ has an absolute maximum at $x = 0$ and $f(x) \leqq \rho_1 < 1$ for all real $x$ with $|x| \geqq \delta_1$. We further assume that $k(0) \neq 0$ and also that $f(x) \neq 0$ whenever $|x| \leqq \delta_1$. These assumptions are chosen for simplicity and certainly are not the most general under which Laplace's approximation provides an asymptotic expansion. 1.2. Notation. Let $F_n$ denote the cdf of $n^{\frac{1}{2}}X_n$ where $X_n$ has a density satisfying the general assumptions. Denote by $\Phi$ and $\varphi$ the standard normal cdf and density respectively. For any $0 < \alpha < 1$, let $\xi_n$ denote the upper $\alpha$th percentile of $F_n$. It follows from the general assumptions that $\xi_n$ is uniquely determined if $n$ is sufficiently large. When considering percentiles, we shall assume that this is the case. The notation $\thicksim$ is used rather than = for relating a function to its asymptotic series since the series may or may not converge.