Rearranging Edgeworth-Cornish-Fisher Expansions

Approximations to the distribution of sample statistics of higher order than the order n -1/2 provided by the central limit theorem are of central interest in the theory of asymptotic statistics. See, e.g., Bhattacharya and Ranga Rao (1976), Rothenberg (1984), Hall (1992), Blinnikov and Moessner (1998), van der Vaart (1998), andCramér (1999). An important tool for performing these refinements is provided by the Edgeworth expansion (Edgeworth (1905), Edgeworth (1907)), which approximates the distribution of the statistics of interest around the limit distribution (often the normal distribution) using a combination of Hermite polynomials with coefficients defined in terms of population moments. Inverting the expansion yields a related higher order approximation, the Cornish-Fisher expansion (Cornish and Fisher (1938), Fisher and Cornish (1960)), to the quantiles of the statistic around the quantiles of the limiting distribution.

One important shortcoming of either the Edgeworth or Cornish-Fisher expansions is that the resulting approximations to the distribution and quantile functions are not necessarily increasing, which violates an obvious monotonicity requirement. This comes from the fact that the polynomials involved in the expansion are not monotone. Here we propose to use a procedure, called the rearrangement, to restore the monotonicity of the approximations and, perhaps more importantly, to improve the estimation properties of these approximations. The resulting improvement is due to the fact that the rearrangement necessarily brings the non-monotone approximations closer to the true monotone target function.

The main findings of the paper can be illustrated through a single picture given in Figure 1, where we plot the true distribution function of a standardized sample mean X based on a small sample, a third order Edgeworth approximation to that distribution, and the rearrangement of the third order approximation. We see that the Edgeworth approximation is sharply non-monotone and provides a rather poor approximation to the distribution function. The rearrangement merely sorts the value of the approximate distribution function in increasing order. One can see that the rearranged approximation, in addition to being monotonic, is a much better approximation to the true function than the original approximation.

We organize the rest of the paper as follows. In Section 2, we describe the rearrangement and qualify the approximation property it provides for monotonic functions. In Section 3, we introduce the rearranged Edgeworth-Cornish-Fisher expansions and explain how they produce better approximations to distributions and quantiles of sample statistics. In Section 4, we illustrate the procedure with several additional examples.

In what follows, let X be a compact interval. We first consider an interval of the form

be the quantile function of F f (y). Thus,

This function f * is called the increasing rearrangement of the function f . The rearrangement is a tool extensively used in functional analysis and optimal transportation (see, e.g., Hardy, Littlewood, and Pólya (1952) and Villani (2003).) It originates in the work of Chebyshev, who used it to prove a set of inequalities (Bronshtein, Semendyayev, Musiol, Muehlig, and Mühlig (2004), p. 31). Here, we employ this tool to improve approximations of monotone functions, such as the Edgeworth-Cornish-Fisher approximations to the distribution and quantile functions of sample statistics.

The rearrangement operator simply transforms a function f to its quantile function f * . That is, x → f * (x) is the quantile function of the random variable f (X) when X ∼ U (0, 1). Another convenient way to think of the rearrangement is as a sorting operation: Given values of the function f (x) evaluated at x in a fine enough mesh of equidistant points, we simply sort the values in increasing order. The function created in this way is the rearrangement of

The following result establishes that the rearrangement always improves the quality of the approximation to a monotone target function.

Proposition 1 (Improving Approximation of Monotone Functions) Let f 0 : X = [a, b] → K be a weakly increasing measurable function in x, where K is a bounded subset of R. This is the target function that we want to approximate. Let f : X → K be another measurable function, an initial approximation to the target function f 0 . 1. For any p ∈ [1, ∞], the rearrangement of f , denoted by f * , weakly reduces the estimation error:

(1)

2. Suppose that there exist regions X 0 and X ′ 0 , each of measure greater than δ > 0, such that for all x ∈ X 0 and x ′ ∈ X ′ 0 we have that (i) x ′ > x, (ii) f (x) > f (x ′ ) + ǫ, and (iii) f 0 (x ′ ) > f 0 (x) + ǫ, for some ǫ > 0. Then the gain in the quality of approximation is strict for p ∈ (1, ∞). Namely, for any

where

Corollary 1 (Strict Improvement) If the target function f 0 is increasing over X and f is decreasing over a subset of X that has

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