On symmetric semiparametric two‐sample problem

We consider a two‐sample problem where data come from symmetric distributions. Usual two‐sample data with only magnitudes recorded, arising from case‐control studies or logistic discriminant analyses, may constitute a symmetric two‐sample problem. We propose a semiparametric model such that, in addition to symmetry, the log ratio of two unknown density functions is modeled in a known parametric form. The new semiparametric model, tailor‐made for symmetric two‐sample data, can also be viewed as a biased sampling model subject to symmetric constraint. A maximum empirical likelihood estimation approach is adopted to estimate the unknown model parameters, and the corresponding profile empirical likelihood ratio test is utilized to perform hypothesis testing regarding the two population distributions. Symmetry, however, comes with irregularity. It is shown that, under the null hypothesis of equal symmetric distributions, the maximum empirical likelihood estimator has degenerate Fisher information, and the test statistic has a mixture of χ2‐type asymptotic distribution. Extensive simulation studies have been conducted to demonstrate promising statistical powers under correct and misspecified models. We apply the proposed methods to two real examples.