Generalized Asymptotic Expansions of Cornish-Fisher Type

Abstract

Let $\{F_n(x)\}$ be a sequence of distribution functions depending on a parameter $n$, and converging to a limiting distribution $\Phi(x)$ as $n$ increases. Then a generalized expansion of Cornish-Fisher type is an asymptotic relation between the quantiles of $F_n$ and $\Phi$. The original Cornish-Fisher formulae [3], [5] provided leading terms of these expansions in the case of normal $\Phi$, expressing a normal deviate in terms of the corresponding quantile of $F_n$ and its cumulants (the "normalizing" expansion) and, conversely, the quantiles of $F_n$ in terms of its cumulants and the corresponding quantiles of $\Phi$ (the "inverse" expansion). The value of both these asymptotic formulae has been well illustrated by their use in approximating the quantiles of complicated distributions (Johnson and Welch [9], Fisher [4], Goldberg and Levine [6]), and for obtaining random quantiles for distribution sampling applications (Teichroew [13], Bol'shev [2]). For a survey of the literature on Cornish-Fisher expansions, and some discussion of their validity, see Wallace ([14], Section 4). In Sections 2, 3 of the present paper, formal expansions are obtained which generalize the Cornish-Fisher relations to arbitrary analytic $\Phi$. Essentially, these expansions provide algorithms for transforming an asymptotic expansion of $F_n$ in terms of the "standard" distribution $\Phi$ into asymptotic relations between the quantiles of these distributions. The "standardizing" expansion of the quantile $u$ of $\Phi$ in terms of the corresponding quantile $x$ of $F_n$ is expressed (Section 2) in terms of a sequence of functions defined by a differential recurrence operator. A similar differential operator appears in the generalized "inverse" expansion for $x$ in terms of $u$ (Section 3), which arises from the application of Lagrange's inversion formula to the equation of quantiles. An asymptotic expansion for quantiles of the Wilks likelihood ratio criterion is given as an example. Formal expansions in terms of the cumulants of $F_n$ and $\Phi$ are obtained in Section 4 by developing $F_n$ about $\Phi$ as a Charlier differential series and collecting terms of like degree in the resulting exponential series. For known cumulants and for normal $\Phi$ these formal expressions reduce, as shown in Section 5, to a general form of the Cornish-Fisher expansions, in which the polynomial terms are represented as sums of products of Hermite polynomials. This representation is shown in Section 6 to account for some properties of the Cornish-Fisher polynomials.