  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 the
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.
