On Multi-Sided Permutation Tests

In fields such as clinical trials, the environment, epidemiology, genetics, pharmacology, social experiments, etc., situations in which the effect of a treatment can be positive on some individuals and negative on the rest may sometimes occur. In a two-sample design, this situation can be expressed by a response model where a random effect Δ in the alternative is such that , and . Such a situation is essentially different from the traditional two-sided test, in which the alternative is assumed to be active only on one of two directions. We consider alternatives in which two sub-alternatives (Δ < 0) and (Δ > 0) can be jointly active. In order to deal with such an atypical situation, we suggest to firstly apply two goodness-of-fit tests, one for the positive deviations from the empirical distribution function of control sample and the other for the negative, and then proceed with their nonparametric combination within a permutation framework. A simulation study inspects on properties of such a solution.     

Parameter Estimation for the Tail Distribution of a Random Sequence

The tail Y_t = X_t – u of a random sequence {X_t ∈ , t ∈ } with identically distributed X_t is approximated by the generalized Pareto distribution according to the extreme value theory, wherein Y_t occurs in clusters because of the dependence in the random sequence. Nevertheless, the parameters of the generalized Pareto distribution are estimated by the same methods as in the case of independent and identically distributed Y_t, provided that there is independence between the clusters of Y_t. The estimation variances and confidence intervals can be estimated by the jackknife method. The approaches are theoretically discussed and verified by extensive numerical researches.