The numerical evaluation of the probability density function of a quadratic form in normal variables

Authors:
Zeng-Hua Lu
Affiliations:
School of Mathematics and Statistics, University of South Australia, City West, GPO Box 2471, SA 5001, Australia
Venue:
Computational Statistics & Data Analysis
Year:
2006

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Abstract

This paper is concerned with numerical techniques for evaluating the probability density function of a quadratic form in normal variables. First, we are interested in the numerical techniques used to evaluate the inversion function of the characteristic function. We derive different truncation bounds for controlling the truncation error. The efficiencies of the bounds are investigated and a practical solution is discussed. We then point out that this technique is not always applicable. Second, we derive an alternative method from the convolution theorem for a special case of the situation where the above technique is not applicable. A number of computational examples are provided, as well as an application to assessing the quality of an approximate distribution by using the Kullback-Leibler information measure.