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arXiv · 1411.5305

Accurate distribution of X^{T}X with singular, idempotent variance-covariance matrix

Hao Yuan Zhang·Jan Vrbik
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Abstract

Assume that X is a set of sample statistics which follow a special case Central Limit Theorem, namely: as the sample size n increases the corresponding distribution becomes multivariate Normal with the mean (of each X) equal to zero and with an idempotent variance-covariance matrix V. It is well known that X^{T}X has (in the same limit), a chi-squared distribution with degrees of freedom equal to the trace of V. In this article we extend the above result to include the corresponding (1/n)-proportional corrections, making the new approximation substantially more accurate and extending its range of applicability to small-size samples.

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Hao Yuan Zhang, Jan Vrbik. 2014-11-20. Accurate distribution of X^{T}X with singular, idempotent variance-covariance matrix. https://arxiv.org/abs/1411.5305

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