Necessary and sufficient conditions for stochastic orders between (n − r + 1)-out-of-n systems in proportional hazard (reversed hazard) rates model

Consider two (n ? r + 1)-out-of-n systems, one with independent and non-identically distributed components and another with independent and identically distributed components. When the lifetimes of components follow the proportional hazard rates model, we establish a necessary and sufficient condition for the usual stochastic order to hold between the lifetimes of these two systems. For the special case of r = 2, some generalized forms of this result to the hazard rate, dispersive and likelihood ratio orders are also obtained. Moreover, for the case when the lifetimes of components follow the proportional reversed hazard rates model, we derive some similar results for comparing the lifetimes of two systems . Applications of the established results to different situations are finally illustrated.     

A Mixture Model of Proportional Reversed Hazard Rate

This article investigates properties of mixture model of proportional reversed hazard rate. Firstly, the mixing random variable and the overall population variable are proved to be positively likelihood dependent. Secondly, lower bounds for the distribution function as well as the conditional distribution are established in the case that the mixing variable belongs to certain nonparametric classes. Finally, some stochastic orders on the mixing (baseline) variables are proved to be translated to the corresponding overall population variables.     

On the Mixture Proportional Mean Residual Life Model

In this article, we propose a new mixture model induced by the model of proportional mean residual life. Under some appropriate assumptions, it is shown that the mixing and overall variables in the model admit the positive likelihood ratio dependence structure. To see how the overall variable is affected by the stochastic variation of the mixing variable, we study some stochastic comparisons using these variables. Finally, some useful bounds for tail probability of the overall variable for large values of the mixing variable are derived.     

Scalable statistical inference for averaged implicit stochastic gradient descent

Stochastic gradient descent (SGD) provides a scalable way to compute parameter estimates in applications involving large‐scale data or streaming data. As an alternative version, averaged implicit SGD (AI‐SGD) has been shown to be more stable and more efficient. Although the asymptotic properties of AI‐SGD have been well established, statistical inferences based on it such as interval estimation remain unexplored. The bootstrap method is not computationally feasible because it requires to repeatedly resample from the entire data set. In addition, the plug‐in method is not applicable when there is no explicit covariance matrix formula. In this paper, we propose a scalable statistical inference procedure, which can be used for conducting inferences based on the AI‐SGD estimator. The proposed procedure updates the AI‐SGD estimate as well as many randomly perturbed AI‐SGD estimates, upon the arrival of each observation. We derive some large‐sample theoretical properties of the proposed procedure and examine its performance via simulation studies.     

Testing the rate ratio under inverse sampling based on gradient statistic

Inverse sampling is widely applied in studies with dichotomous outcomes, especially when the subjects arrive sequentially or the response of interest is difficult to obtain. In this paper, we investigate the rate ratio test problem under inverse sampling based on gradient statistic with the asymptotic method and parametric bootstrap technique. The gradient statistic has many advantages, for example, it is simple to calculate and competitive with Wald-type, score and likelihood ratio tests in terms of local power. Numerical studies are carried out to evaluate the performance of our gradient test and the existing tests, namely Wald-type, score and likelihood ratio tests. The simulation results suggest that the gradient test based on the parametric bootstrap method has excellent type I error control and large powers even in small sample design. Two real examples, from a heart disease study and a drug comparison study, are applied to illustrate our methods.     

Properties of systems with two exchangeable Pareto components

The life lengths of the units in a system can be modelled by a bivariate distribution. In this paper, we suppose that the joint distribution of the units is a symmetric bivariate Pareto (Lomax) distribution. For this model, we obtain basic reliability properties for series and parallel systems. J. M. Ruiz Partially Supported by Ministerio de Ciencia y Tecnologia under grant BFM2003-02947 and Fundacion Seneca under grant 00698/PI/04.     

On a generalization to Marshall–Olkin scheme and its application to Burr type XII distribution

Marshall–Olkin semi-Burr and Marshall–Olkin Burr distributions are introduced and studied. Their various characteristics in reliability analysis are derived. Applications in time series analysis are discussed.     

On proportional odds models

Recently, Marshall and Olkin (Biometrika 84(3):641–652 1997) introduced a family of distributions by adding a new parameter to a survival function. In this paper, we give physical interpretation of the family using odds function. It is shown that the family of distributions satisfies the property of proportional odds function. We, then, develop a generalized family and study its properties. Further, we give various definitions of proportional odds model in the bivariate set up. Based on these, we introduce new families of bivariate distributions and study their properties.     

The percentile residual life up to time t0: Ordering and aging properties

Motivated by practical issues, a new stochastic order for random variables is introduced by comparing all their percentile residual life functions until a certain instant. Some interpretations of these stochastic orders are given, and various properties of them are derived. The relationships to other stochastic orders are studied and also an application in reliability theory is described. Finally, we present some characterization results of the decreasing percentile residual life up to time t₀ aging notion.     

Partial Ordering and Aging Properties of Order Statistics When the Sample Size is Random: A Brief Review

In many practical situations, order statistics arise naturally with random sample size. In this article, we review results on partial orderings and aging properties of such order statistics. The comparison of order statistics for different sample sizes is also discussed here.