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# Accession Number:

## AD0672346

# Title:

## FOURIER SERIES AND CHEBYSHEV POLYNOMIALS IN STATISTICAL DISTRIBUTION THEORY.

# Descriptive Note:

## Research rept.,

# Corporate Author:

## CONNECTICUT UNIV STORRS DEPT OF STATISTICS

# Report Date:

## 1968-07-01

# Pagination or Media Count:

##
144.0

# Abstract:

## After the elementary functions, the Fourier series are the most important functions in applied mathematics. Nevertheless, they have been somewhat neglected in statistical distribution theory. In this paper, the reasons for this omission are investigated and certain modifications of the Fourier series proposed. These results are presented in the form of representation theorems. In addition to the basic theorems, computational algorithms and procedures are developed. As an illustration, a useful representation of the incomplete beta function ratio is developed. Although the representation theorems have been developed for those random variables whose range is contained in the intervals 0.1 and -1,1, methods of using the theorems for other intervals are discussed. In addition, multivariate analogues of the theorems are presented. The usefulness of the representation theorems extends beyond the evaluation of a distribution function. In particular, they are useful in investigating the accuracy of an approximation to a distribution function and can be used to improve the accuracy of such an approximation. As a final application of the procedures, three important distribution problems are discussed. These are the likelihood ratio tests, products of independent beta variables, and quadratic forms. Author

# Distribution Statement:

## APPROVED FOR PUBLIC RELEASE

#