Nonparametric goodness-of-fit tests for uniform stochastic ordering (Q682298)

Continuous distribution functions \(F\) and \(G\) satisfy a uniform stochastic order if \(\frac{\bar{F}}{\bar{G}}\) is nondecreasing, where \(\bar{F}\) and \(\bar{G}\) are survivor functions. Let \(X_1,\ldots, X_m\) be i.i.d. from \(F\) and \(Y_1,\ldots, Y_n\) be i.i.d. from \(G,\) and the two samples be independent with unknown both \(F\) and \(G.\) ~\(L^p\) distance-based goodness-of-fit (GOF) tests are proposed for the uniform stochastic ordering of \(F\) and \(G.\) The tests are motivated by the fact that when \(F\) and \(G\) are uniformly stochastic ordered, the ordinal dominance curve \(R=FG^{-1}\) is star-shaped. It is shown that the testing procedure has a unique least favorable configuration of \(F\) and \(G\) for \(p\in [1,~\infty].\) A finite-sample study demonstrates that a one-sample version of the proposed procedure (e.g., with \(G\) known) is more powerful compared with the one-sample GOF test suggested in [\textit{M. A. Arcones} and \textit{F. J. Samaniego}, Ann. Stat. 28, No. 1, 116--150 (2000; Zbl 1106.62332)]. Sample size determination is discussed as well. The methods are illustrated using data from a pharmacology study.