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Register a new userHolonomic gradient method for distribution function of a weighted sum of noncentral chi-square random variables
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Added by
Akimichi Takemura
Publication date
2015
fields
Mathematical Statistics
and research's language is
English
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We apply the holonomic gradient method to compute the distribution function of a weighted sum of independent noncentral chi-square random variables. It is the distribution function of the squared length of a multivariate normal random vector. We treat this distribution as an integral of the normalizing constant of the Fisher-Bingham distribution on the unit sphere and make use of the partial differential equations for the Fisher-Bingham distribution.
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We show that the distribution of the scalar Schur complement in a noncentral Wishart matrix is a mixture of central chi-square distributions with different degrees of freedom. For the case of a rank-1 noncentrality matrix, the weights of the mixture representation arise from a noncentral beta mixture of Poisson distributions.
The exact distribution of the square sum of Dirichlet random variables is given by two different univariate integral representations. Alternatively, three representations by orthogonal series with Jacobi or Legendre polynomials are derived. As a special case the distribution of the square sum of spacings - also called Greenwoods statistic - is obtained. Nine quantiles of this statistic are tabulated with eight digits where the number of squares ranges from 10 to 100.
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