• The KIPS Transactions:PartA
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• v.9A no.2
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• pp.267-270
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• 2002

The evaluation of the cumulative distribution function of the noncentral $\chi^2$ distribution is required in approximate determination of the power of the $\chi^2$ test. This article provides an algorithm for evaluating the noncentral $\chi^2$ distribution function in terms of a single "central" $\chi^2$ distribution function and compared various approximations.ximations.

• Journal of the Korean Statistical Society
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• v.24 no.1
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• pp.257-266
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• 1995

Consider a p-variate $(p \geq 4)$ normal distribution with mean $\b{\theta}$ and identity covariance matrix. For estimating $\b{\theta}$ under a quadratic loss we investigate the behavior of risks of Stein-type estimators which shrink the usual estimator toward the mean of observations. By using concavity of the function appearing in the shrinkage factor together with new expectation identities for noncentral chi-squared random variables, a characterization of estimators with nondecreasing risk is obtained.