A New Weibull–Pareto Distribution: Properties and Applications

Many distributions have been used as lifetime models. In this article, we propose a new three-parameter Weibull–Pareto distribution, which can produce the most important hazard rate shapes, namely, constant, increasing, decreasing, bathtub, and upsidedown bathtub. Various structural properties of the new distribution are derived including explicit expressions for the moments and incomplete moments, Bonferroni and Lorenz curves, mean deviations, mean residual life, mean waiting time, and generating and quantile functions. The Rényi and q entropies are also derived. We obtain the density function of the order statistics and their moments. The model parameters are estimated by maximum likelihood and the observed information matrix is determined. The usefulness of the new model is illustrated by means of two real datasets on Wheaton river flood and bladder cancer. In the two applications, the new model provides better fits than the Kumaraswamy–Pareto, beta-exponentiated Pareto, beta-Pareto, exponentiated Pareto, and Pareto models.     

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.     

Proportional hazards models based on biased samples and estimated selection probabilities

In non‐randomized biomedical studies using the proportional hazards model, the data often constitute an unrepresentative sample of the underlying target population, which results in biased regression coefficients. The bias can be avoided by weighting included subjects by the inverse of their respective selection probabilities, as proposed by Horvitz & Thompson (1952) and extended to the proportional hazards setting for use in surveys by Binder (1992) and Lin (2000). In practice, the weights are often estimated and must be treated as such in order for the resulting inference to be accurate. The authors propose a two‐stage weighted proportional hazards model in which, at the first stage, weights are estimated through a logistic regression model fitted to a representative sample from the target population. At the second stage, a weighted Cox model is fitted to the biased sample. The authors propose estimators for the regression parameter and cumulative baseline hazard. They derive the asymptotic properties of the parameter estimators, accounting for the difference in the variance introduced by the randomness of the weights. They evaluate the accuracy of the asymptotic approximations in finite samples through simulation. They illustrate their approach in an analysis of renal transplant patients using data obtained from the Scientific Registry of Transplant Recipients     

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.     

Likelihood ratio ordering of order statistics

This paper provides an improvement on the work of Bapat and Kochar (1994, Linear Algebra Appl., 199, 281–291) and strengthens the literature on the likelihood ratio ordering of order statistics. For independent (but possibly nonidentically distributed) absolutely continuous random variables X₁,…,X_n, it is shown under some weak conditions that

X_1:n^lrlrX_n:n,

where ^lr stands for the likelihood ratio ordering and X_k:n represents the kth-order statistic.     

Rényi entropy properties of records

We provide bounds for Rényi entropy of records. We also show that the Rényi entropy ordering of random variables determines the Rényi entropy ordering of their respective records. We characterize exponential distribution by maximization of Rényi entropy under some conditions. We show that Rényi distance between distribution of records and parent distribution is distribution free.     

Properties of the elasticity of a continuous random variable. A special look at its behavior and speed of change

Belzunce et al. (1995 Belzunce, F., Candel, J., Ruiz, J.M. (1995). Ordering of truncated distributions through concentration curves. Sankhya 57:375–383. [Google Scholar]) define the elasticity for non negative random variables as the reversed proportional failure rate (RPFR). Veres-Ferrer and Pavía (2012 Veres-Ferrer, E.J., Pavía, J.M. (2012). La elasticidad: una nueva herramienta para caracterizar distribuciones de probabilidad. Rect@ 13:145–158. [Google Scholar], 2014b Veres-Ferrer, E.J., Pavía, J.M. (2014b). On the relationship between the reversed hazard rate and elasticity. Stat. Pap. 55:275–284.[Crossref], [Web of Science ®] , [Google Scholar]) interpret it in economic terms, extending its definition to variables that can also take negative values, and briefly present the role of elasticity in characterizing probability distributions. This paper highlights a set of properties demonstrated by elasticity, which shows many similar properties to the reverse hazard function. This paper pays particular attention to studying the increase/decrease and the speed of change of the elasticity function. These are important properties because of the characterizing role of elasticity, which makes it possible to introduce our hypotheses and knowledge about the random process in a more meaningful and intuitive way. As a general rule, it is observed the need for distinguishing between positive and negative areas of the support.