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.

我们考虑在数字上评估chi分布随机变量的光滑有界函数的期望值除以自由度数的平方根的问题。在同时推断，总体选择和排序以及对多元t概率进行评估的情况下会出现此问题。它也出现在对某些非标准置信区域的覆盖概率和预期体积属性的评估中。我们使用Mori提出的变换，然后应用梯形规则。该规则具有显着的特性，即对于合适的整数，它是指数收敛的。我们使用它来创建嵌套的正交规则序列，以估计近似误差，从而不会浪费先前对被积数的评估。梯形法则的应用要求用有限和来近似无限和。我们提供了一个新的易于计算的关于该近似误差的上限。我们的总体结论是，该方法非常适合用于计算非标准置信区域的覆盖率和预期体积属性。