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.
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Acknowledgements
This work was supported by an Australian Government Research Training Program Scholarship. The authors thank the reviewers for their comments and suggestions as these led to an improved paper.
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Kabaila, P., Ranathunga, N. Computation of the expected value of a function of a chi-distributed random variable. Comput Stat 36, 313–332 (2021). https://doi.org/10.1007/s00180-020-01005-y
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DOI: https://doi.org/10.1007/s00180-020-01005-y