Characterisation of marginal distribution and density functions is of interest where data on a pair of random variables (X, Y) are observed. Stochastic orderings between (X, Y) have been studied in statistics and economics. Likelihood-ratio ordering is useful in understanding the behaviour of the random variables. In this article, tests based on U-statistics are proposed to test for equality of marginal density functions against the alternative of likelihood-ratio ordered when (X, Y) are dependent. The tests can be used when the data are either completely observed or subjected to independent univariate right censoring. The asymptotic variances of these tests are complicated and hence, are estimated using jackknife variance estimators. Validity of the jackknife variance estimators in statistical inference based on the proposed tests is demonstrated using simulation studies. The test for uncensored setting has desired size and good power for small sample. The performance of the tests for censored case depends on the sample size, proportion of censoring and the measure of dependence between X and Y. The tests are illustrated on three real data sets chosen in order to bring out various aspects of the tests.

在观察到一对随机变量 ( X ,  Y ) 的数据时，对边际分布和密度函数的表征很感兴趣。( X ,  Y )之间的随机排序已在统计学和经济学中进行了研究。似然比排序有助于理解随机变量的行为。在本文中，提出了基于 U 统计量的检验来检验边际密度函数与在 ( X ,  Y) 是依赖的。当数据被完全观察或经过独立的单变量右删失时，可以使用测试。这些测试的渐近方差很复杂，因此使用折刀方差估计器进行估计。使用模拟研究证明了基于所提出的测试的统计推断中折刀方差估计量的有效性。未经审查设置的测试对于小样本具有所需的大小和良好的功效。删失案例的测试性能取决于样本大小、删失比例以及X和Y之间的相关性度量。为了展示测试的各个方面，选择了三个真实数据集来说明测试。