Saddlepoint Expansions for Directional Test Probabilities
推导了多元随机变量给定方向条件下的条件尾概率展开式，用于简单假设检验，其近似误差在独立重复次数平均下为O(n^{-1})，低维时小样本效果良好。
SUMMARY An expansion is derived for the conditional tail probability of a multivariate random variable given its direction from the expected value. In particular, if the score function of a statistical model is chosen as this variable, such a conditional probability gives a test for a simple hypothesis, which is asymptotically equivalent to the likelihood ratio test. The expansion is of the large deviation type and is derived through the use of saddlepoint methods. Thus the relative error of the test probability is O(n –1) uniformly in a bounded set for an average of n independent replications. The approximation is based on the cumulant transform for the random variable, but is otherwise a simple expression in terms of chi-squared distributions. In the one-dimensional case it reduces to the expansion obtained by Lugannani and Rice. Numerical examples show an excellent fit, comparable with other saddlepoint expansions, when the dimension is low, even for very small sample sizes, but for higher dimensions more replications are required to give a similar approximation.