A New Asymptotic Expansion for Sums of Random Variables

In this thesis we look to improve upon local Edgeworth expansions for probability distributions of sums of independent identically distributed random variables. Let X be a random variable with finite variance, and X1, X2, …, a sequence of i.i.d. random variables each with the same distribution as X. In addition, suppose that X has probability function p, where p is either a density or a probability mass function. Define the partial sum S1 = X1 + X2 + … + X1, with probability function p^(i). In the standard Edgeworth case, the local expansion for p^(n) is in terms of powers of 1/(n)^(1/2)Your browser may not support display of this image. . This expansion involves the cumulants of X, as well as Hermite polynomials. The expansion defined in this thesis uses a rival sequence of coefficients dependent on the probability function p, which does not require the calculation of cumulants, or the use of Hermite polynomials. The advantages of this expansion are most noticeable when X is a discrete random variable on a small support set. The approach produces better asymptotic results in many cases, especially for symmetric random variables X. The main error term is often simpler (in terms of the value of interest in the support set), and as a result, the asymptotic crossing points of the estimate with the target can be more easily computed (given information on the location of zeros of Hermite polynomials).

Grantor dc:publisher

Wake Forest University

Year dc:date.issued

2010

Author dc:creator

• Chernesky, James Jr.

• Statistics

Language dc:language.iso

en

Handle dc:identifier.uri

http://hdl.handle.net/10339/14870

OAI identifier oai:identifier

oai:wakespace.lib.wfu.edu:10339/14870