An Efficiency Robust Nonparametric Test for Scale Change for Data From a Gamma Distribution

When testing whether two samples came from a common distribution against the alternative that they differ in scale, the asymptotically most powerful rank test depends on the density function underlying the data. In some applications it is only known that the data are from a gamma distribution with increasing hazard rate (i.e., the value of the shape parameter, k, is known to be ≥ 1). If the optimum test (Savage 1956) for data from an exponential law (k = 1) is used when the true value of k is much larger, a loss of power or asymptotic relative efliciency results. Indeed, the ARE of the exponential scores test relative to the optimum rank test for large k(∞) is about 81.4%. This article presents the maximin efficiency robust test for the gamma (k ≥ 1) family that has an asymptotic efficiency of at least 95% relative to the optimum test for each member of the family. The test can be calculated using standard statistical packages and is illustrated on real data.