# Accession Number:

## AD0647390

# Title:

## SERIES REPRESENTATIONS OF DISTRIBUTIONS OF QUADRATIC FORMS IN NORMAL VARIABLES, I. CENTRAL CASE,

# Descriptive Note:

# Corporate Author:

## TORONTO UNIV (ONTARIO) DEPT OF INDUSTRIAL ENGINEERING

# Personal Author(s):

# Report Date:

## 1966-11-01

# Pagination or Media Count:

## 37.0

# Abstract:

The probability density function pdf of a positive definite quadratic form in central or non-central normal variables can be represented as a series expansion in a number of different ways. Among these, one of the most important is that of a series of pdfs of non-central chi-squares or central chi-squares with increasing degrees of freedom. These expansions have been discussed by Ruben Ann. Math. Statist. 33 1962 542-570 Ann. Math. Statist. 34 1963 1582-1584 who has given convenient recurrence formulae for determining the coefficients. Expansion in terms of Laguerre series and Maclaurin series powers of the argument have been discussed for central variables by Gurland Ann. Math. Statist. 24 1953 416-427 and Pachares Ann. Math. Statist. 26 1955 128-131 respectively, and in the general non-central case by Shah Ann. Math. Statist. 34 1963 186-190 and Shah and Khatri Ann. Math. Statist. 32 1961 883-887, but the coefficients in their series are not presented in a very convenient form for calculations. It is the purpose of this paper to show how all three kinds of expansion can be derived in a similar way, and incidentally, to obtain convenient recurrence formulae for determining the coefficients in the Laguerre and Maclaurin expansions. In the present paper the central case is discussed. Author

# Descriptors:

# Subject Categories:

- Statistics and Probability