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Due to the importance of generalized order statistics (GOS) in many branches of Statistics, a wide interest has been shown in investigating stochastic comparisons of GOS. In this article, we study the likelihood ratio ordering of $p$ -spacings of GOS, establishing some flexible and applicable results. We also settle certain unresolved related problems by providing some useful lemmas. Since we do not impose restrictions on the model parameters (as previous studies did), our findings yield new results for comparison of various useful models of ordered random variables including order statistics, sequential order statistics, $k$ -record values, Pfeifer's record values, and progressive Type-II censored order statistics with arbitrary censoring plans. Some results on preservation of logconvexity properties among spacings are provided as well.

New bivariate models are obtained with conditional distributions (in two different senses) satisfying the proportional generalized odds rate (PGOR) model. The PGOR semi-parametric model includes as particular cases the Cox proportional hazard rate (PHR) model and the proportional odds rate (POR) model. Thus the new bivariate models are very flexible and include, as particular cases, the bivariate extensions of PHR and POR models. Moreover, some well known parametric bivariate models are also included in these general models. The basic theoretical properties of the new models are obtained. An application to fit a real data set is also provided.