Bahadur’s Stochastic Comparison of Combining infinitely Independent Tests in Case of Extreme Value Distribution

Abstract

For simple null hypothesis, given any non-parametric combination method which has a monotone increasing acceptance region, there exists a problem for which this method is most powerful against some alternative. Starting from this perspective and recasting each method of combining pvalues as a likelihood ratio test, we present theoretical results for some of the standard combiners which provide guidance about how a powerful combiner might be chosen in practice. In this paper we consider the problem of combining n independent tests as n → ∞ for testing a simple hypothesis in case of extreme value distribution (EV(θ,1)). We study the six free-distribution combination test producers namely; Fisher, logistic, sum of p-values, inverse normal, Tippett’s method and maximum of p-values. Moreover, we studying the behavior of these tests via the exact Bahadur slope. The limits of the ratios of every pair of these slopes are discussed as the parameter θ → 0 and θ → ∞. As θ → 0, the logistic procedure is better than all other methods, followed in decreasing order by the inverse normal, the sum of p-values, Fisher, maximum of p-values and Tippett’s procedure. Whereas, θ → ∞ the logistic and the sum of p-values procedures are equivalent and better than all other methods, followed in decreasing order by Fisher, the inverse normal, maximum of p-values and Tippett’s procedure.