Combining independent one-sided noncentral t or normal mean tests (Q1073475)

Let \(T_ 1,T_ 2,...,T_ n\) be independent where the distribution of \(T_ i\) depends on \(\theta_ i,\quad i=1,2,...,n\). The problem is to test the hypothesis \(H_ 0:\theta_ i=0,\quad i=1,...,n\), versus \(H_ A:\theta =(\theta_ 1,...,\theta_ n)\in \Theta_ A=R^ n_+\setminus \{0\}\), all the procedures in question being based on the observed significance levels \(p_ i\) of the individuals \(T_ i,\quad p_ i=P_ 0(T_ i\geq t_ i)\) where \(P_ 0\) represents the null distribution of \(T=(T_ 1,...,T_ n).\) Two cases are investigated. 1) \(T_ i\) has normal distribution; 2) \(T_ i\) has noncentral t-distribution with \(\nu_ i\) degrees of freedom and noncentrality parameter \(\theta_ i,\quad i=1,2,...,n.\) Minimal complete classes of the procedures are found. It is shown particularly that the likelihood ratio tests and Tippett's procedure are admissible in both cases, and Fisher's and the inverse normal procedures are admissible in the normal case but inadmissible in the t-case.