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Computation of the expected value of a function of a chi-distributed random variable

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  • Paul Kabaila

    (La Trobe University)

  • Nishika Ranathunga

    (La Trobe University)

Abstract

We consider the problem of numerically evaluating the expected value of a smooth bounded function of a chi-distributed random variable, divided by the square root of the number of degrees of freedom. This problem arises in the contexts of simultaneous inference, the selection and ranking of populations and in the evaluation of multivariate t probabilities. It also arises in the assessment of the coverage probability and expected volume properties of some non-standard confidence regions. We use a transformation put forward by Mori, followed by the application of the trapezoidal rule. This rule has the remarkable property that, for suitable integrands, it is exponentially convergent. We use it to create a nested sequence of quadrature rules, for the estimation of the approximation error, so that previous evaluations of the integrand are not wasted. The application of the trapezoidal rule requires the approximation of an infinite sum by a finite sum. We provide a new easily computed upper bound on the error of this approximation. Our overall conclusion is that this method is a very suitable candidate for the computation of the coverage and expected volume properties of non-standard confidence regions.

Suggested Citation

  • Paul Kabaila & Nishika Ranathunga, 2021. "Computation of the expected value of a function of a chi-distributed random variable," Computational Statistics, Springer, vol. 36(1), pages 313-332, March.
    • Handle: RePEc:spr:compst:v:36:y:2021:i:1:d:10.1007_s00180-020-01005-y
    DOI: 10.1007/s00180-020-01005-y
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    References listed on IDEAS

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    1. Farchione, David & Kabaila, Paul, 2008. "Confidence intervals for the normal mean utilizing prior information," Statistics & Probability Letters, Elsevier, vol. 78(9), pages 1094-1100, July.
    2. Paul Kabaila & A. H. Welsh & Rheanna Mainzer, 2017. "The performance of model averaged tail area confidence intervals," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 46(21), pages 10718-10732, November.
    3. Charles W. Dunnett, 1989. "Multivariate Normal Probability Integrals with Product Correlation Structure," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 38(3), pages 564-579, November.
    4. Paul Kabaila & A. H. Welsh & Waruni Abeysekera, 2016. "Model-Averaged Confidence Intervals," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 43(1), pages 35-48, March.
    5. Tetsuhisa Miwa & A. J. Hayter & Satoshi Kuriki, 2003. "The evaluation of general non‐centred orthant probabilities," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(1), pages 223-234, February.
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