Volume 9, Issue 4
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Moment-Based Density Approximants
This article is concerned with the problem of approximating a density function from the moments (or cumulants) of a given distribution. Approximants of this type can be obtained, for example, by making use of Pearson or Johnson curves, see [1, 2, 3], or saddlepoint approximations as discussed in . These methodologies can provide adequate approximations in a variety of applications involving unimodal distributions. However, they may prove difficult to implement and their applicability can be subject to restrictive conditions. The approximants proposed in this article are expressed in terms of relatively simple formulae and apply to a very wide array of distributions. Moreover, their accuracy can be improved by use of additional moments. Interestingly, another technique called the inverse Mellin transform, which is based on the complex moments of certain distributions, provides representations of their exact density functions in terms of generalized hypergeometric functions; for theoretical considerations as well as various applications, see [5, 6].
First, it should be noted that the th moment of a statistic, , whose exact density is unknown, can be determined exactly or numerically by integrating the product over the range of integration of the 's, where denotes the joint density of the 's, ; for instance, this approach is used in Example 1. Alternatively, the moments of a random variable can be obtained from the derivatives of its moment-generating function as is done in Example 4 or by making use of a relationship between the moments and the cumulants when the latter are known . Moments can also be derived recursively as is the case, for instance, in connection with certain queueing models. When the moments of a statistic uniquely determine its distribution and a sufficient number of moments are known, we can often approximate its density function in terms of sums involving orthogonal polynomials of a certain type. Conveniently, such polynomials are available as built-in Mathematica functions.
Density approximants based on Legendre and Laguerre polynomials are discussed in Sections 2 and 3, respectively, for random variables having finite and semi-infinite supports. The main formulae which allows for the direct evaluation of the density approximants are equations (15), derived in Section 2, and (29), obtained in Section 3. The approximant that is expressed in terms of Laguerre polynomials applies to a wide class of statistics which includes those whose asymptotic distribution is chi-square, such as , where denotes a likelihood ratio statistic, as well as those that are distributed as quadratic forms in normal variables, such as the sample serial covariance. It should be noted that an indefinite quadratic form can be expressed as the difference of two independent nonnegative definite quadratic forms whose cumulants are well known. As for distributions having compact supports, there are, for example, the Durbin-Watson statistic, Wilks' likelihood ratio criterion, the sample correlation coefficient, as well as many other useful statistics that can be expressed as the ratio of two quadratic forms, as discussed in .
In Section 4, we propose a unified density estimation methodology which only requires the moments of the distribution to be approximated and those of a suitable 'base density function'. As it turns out, this approach yields density approximants that are identical to those obtained from certain orthogonal polynomials--namely, the Legendre, Laguerre, Jacobi, and Hermite polynomials--whose associated weight function is proportional to the corresponding base density function. Several examples illustrate the various results. The Mathematica code utilized for implementing the main formulae and plotting the graphs is supplied in the Appendix.
For results in connection with the convergence of approximating sums that are expressed in terms of orthogonal polynomials, see [9, 10, 11, 12]. Since the proposed methodology allows for the use of a large number of theoretical moments and the functions being approximated are nonnegative, the approximants can be regarded as nearly exact bona fide density functions, and quantiles can thereupon easily be estimated with great accuracy. As well, the representations of the approximants make them easy to report and amenable to complex calculations.
Until now, orthogonal polynomials have been scarcely discussed in the statistical literature in connection with the approximation of distributions. This might have been due to difficulties encountered in deriving moments of high orders or in obtaining accurate results from high-degree polynomials. In any case, given the powerful computational resources that are widely available these days, such complications can hardly be viewed as impediments any longer. It should be pointed out that the simple semiparametric technique proposed in Section 4 eliminates some of the complications associated with the use of orthogonal polynomials while yielding identical density approximants. This article is self-contained, and the results presented herein potentially have a host of applications. Being that the subject matter of this article is density approximation as opposed to density estimation, it ought to be emphasized that the techniques presented herein are meant to be used in conjunction with exact moments rather than sample moments.
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