Inference for the means of a gamma distribution with unknown shape parameter
研究了形状参数未知时伽马分布均值的假设检验问题，通过鞍点技术近似条件密度和分布函数，避免了数值积分或模拟，适用于单样本和多样本情形。
The construction of a similar test region for the hypothesis that the mean of a gamma distribution has a specified value requires numerical integration of a conditional density. This conditional density is not explicitly known and we give here a simple and accurate approximation to the density, using saddlepoint techniques for approximating an inversion integral. To avoid the numerical integration of the approximative density we also derive a simple expansion of the distribution function, which reduces to aF-distribution in certain limits. For the case of several samples with the same shape parameter, we consider the hypothesis that the means are equal. The problems that arise in the construction of a similar test region are the same as for the one sample case, and the solutions given are basically the same too. In this paper we study inference for the mean of a gamma distribution with unknown shape parameter. Both the one sample case and the multisample case will be examined. In the one sample case we consider the hypothesis that the mean has a specified value and in the multisample case we assume that the shape parameter is the same for all samples and consider the hypothesis that all the means are equal. In both cases an exact (or similar) test can be obtained by conditioning on the minimal sufficient statistic under the null hypothesis. However, the conditional distributions needed in the construction of the tests involve func- tions that are not known in any explicit form. It is the purpose of this paper to obtain simple and accurate approximations to these functions, so that an approximation to the exact tests can be calculated without the use of simulation. If we lety denote the mean of the gamma distribution and let w be the shape parameter, the gamma distribution has density