Department of Statistics, George Mason University, Fairfax, Virginia.
National Institute of Allergy and Infectious Diseases, Rockville, Maryland.
Biometrics. 2020 Dec;76(4):1216-1228. doi: 10.1111/biom.13233. Epub 2020 Feb 18.
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 χ -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.
我们考虑一个两样本问题，其中数据来自对称分布。通常情况下，只有幅度记录的两样本数据，如病例对照研究或逻辑判别分析中的数据，可能构成一个对称的两样本问题。我们提出了一个半参数模型，使得除了对称性之外，两个未知密度函数的对数比还可以用已知的参数形式建模。这个新的半参数模型是专门为对称两样本数据设计的，也可以看作是一个受到对称约束的有偏抽样模型。我们采用最大经验似然估计方法来估计未知的模型参数，并利用相应的轮廓经验似然比检验来进行关于两个总体分布的假设检验。然而，对称性带来了不规则性。结果表明，在两个对称分布相等的零假设下，最大经验似然估计量的 Fisher 信息具有退化性，检验统计量具有 χ 型混合渐近分布。我们进行了广泛的模拟研究，以证明在正确和错误模型下都具有有希望的统计功效。我们将所提出的方法应用于两个实际例子。
