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On characterizing mixtures of some life distributions

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Abstract

There are many practical problems in which finite mixture of probability distributions arise as models of life length. Although, the role of failure rate and mean residual life in modelling life lengths are well established, much work has not been done to characterize mixture distributions in terms of these concepts. In the present paper we establish an identity connecting the failure rate and mean residual life that characterizes two component mixtures of exponential, Lomax and beta densities. We also prove a similar result connecting the failure rate and the second moment of residual life.
... The relationship between the failure rate (reversed failure rate) and the mean residual life (reversed mean residual life) or left (right) truncated expectations of functions of X were found to be quite useful in studying the comparative behaviour of these functions and in characterizing the probability distributions, especially when any of these functions does not have a simple closed form for analytic treatment. Important contributions to characterizations of this kind include those relating to specific distributions [1][2][3]15,21,26,29], families of distributions [5,20,31] and their generalizations to wider class of distributions [8,10,22,28,30]. Other than the mean, a second characteristic of the residual life that plays a similar role in identifying life distributions and distinguishing them is the variance residual life (VRL), defined as ...
... The rest of the paper is organized into three sections. In Section 2 we derive an expression for the covariance between an absolutely continuous function and h(X ) on the condition that X > x, and deduce a characterization for models represented by (1) ...
... The Lomax and beta mixtures of [1] and the gamma mixture of [2] also admit similar identities by adopting this procedure. ...
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In the present paper we study the properties of the left and right truncated variance of a function of a non-negative random variable, that characterize a class of continuous distributions. These properties include characterizations by the relationships the conditional variance has with the truncated expectations and/or the failure rate as well as the lower bound to the conditional variance. It is shown that the characteristic properties are linked to those based on the relationship between the conditional means and the failure rates, discussed in the literature. The lower bound developed here compares favourably with that given by the Cramer–Rao inequality.
... The price paid for simplicity in such cases is the limitation to the range of applicability. Starting with individual distributions like gamma and negative binomial (Osaki and Li [475]), the work in this direction progressed to characterization of Pearson family (Nair and Sankaran [442]), the exponential family (Consul [155]), mixtures of distributions (Abraham and Nair [12]), generalized Pearson system (Sankaran et al. [516]) and other generalizations (Gupta and Bradley [238]). The general relationship ...
... Since there are members of the family with support on the positive real line, the model will be useful for describing lifetime data. In this context, the hazard quantile function is 12 10 8 ...
Book
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Quantile-Based Reliability Analysis presents a novel approach to reliability theory using quantile functions in contrast to the traditional approach based on distribution functions. Quantile functions and distribution functions are mathematically equivalent ways to define a probability distribution. However, quantile functions have several advantages over distribution functions. First, many data sets with non-elementary distribution functions can be modeled by quantile functions with simple forms. Second, most quantile functions approximate many of the standard models in reliability analysis quite well. Consequently, if physical conditions do not suggest a plausible model, an arbitrary quantile function will be a good first approximation. Finally, the inference procedures for quantile models need less information and are more robust to outliers. Quantile-Based Reliability Analysis's innovative methodology is laid out in a well-organized sequence of topics, including: Definitions and properties of reliability concepts in terms of quantile functions; Ageing concepts and their interrelationships; Total time on test transforms; L-moments of residual life; Score and tail exponent functions and relevant applications; Modeling problems and stochastic orders connecting quantile-based reliability functions. An ideal text for advanced undergraduate and graduate courses in reliability and statistics, Quantile-Based Reliability Analysis also contains many unique topics for study and research in survival analysis, engineering, economics, and the medical sciences. In addition, its illuminating discussion of the general theory of quantile functions is germane to many contexts involving statistical analysis. © Springer Science+Business Media New York 2013. All rights reserved.
... , see for example Fagiuoli and Pellerey (1994), Hesselager et al. (1997), Gupta (2007), and Nair and Preeth (2008). Recently, Feizjavadian and Hashemi (2015) studied the weighted function w(x) = m X (x), where m X (x) is the mean residual lifetime denoted in (1). In particular, they introduced and studied the mean residual weighted (MRW) density ...
... There are many practical problems in which finite mixtures of probability distributions arise as models of life length; see Abraham and Nair (2001) and references therein. In this section, we constract some finite mixture representations among the GCREW n distributions and length biased of the parent distribution. ...
Article
Recently, Feizjavadian and Hashemi (2015 Feizjavadian, S.H. and Hashemi, R. (2015). Mean residual weighted versus the length-biased Rayleigh distribution. Journal of Statistical Computation and Simulation 85, 2823–2838.[Taylor & Francis Online], [Web of Science ®]) introduced and studied the mean residual weighted (MRW) distribution as an alternative to the length biased distribution, by using the concepts of the mean residual lifetime and the cumulative residual entropy. In this paper, a new sequence of weighted distributions is introduced based on the generalized cumulative residual entropy. This sequence includes the MRW distribution. Properties of this sequence are obtained generalizing and extending previous results on the MRW distribution. Moreover, expressions for some known distributions are given, and finite mixtures among the new sequence of weighted distributions and the length biased distribution are studied. Numerical examples are given to illustrate the new results.
... The price paid for simplicity in such cases is the limitation to the range of applicability. Starting with individual distributions like gamma and negative binomial (Osaki and Li [475]), the work in this direction progressed to characterization of Pearson family (Nair and Sankaran [442]), the exponential family (Consul [155]), mixtures of distributions (Abraham and Nair [12]), generalized Pearson system (Sankaran et al. [516]) and other generalizations (Gupta and Bradley [238]). The general relationship ...
... Since there are members of the family with support on the positive real line, the model will be useful for describing lifetime data. In this context, the hazard quantile function is 12 10 8 ...
Chapter
A probability distribution can be specified either in terms of the distribution function or by the quantile function. This chapter addresses the problem of describing the various characteristics of a distribution through its quantile function. We give a brief summary of the important milestones in the development of this area of research. The definition and properties of the quantile function with examples are presented. In Table 1.1, quantile functions of various life distributions, representing different data situations, are included. Descriptive measures of the distributions such as location, dispersion and skewness are traditionally expressed in terms of the moments. The limitations of such measures are pointed out and some alternative quantile-based measures are discussed. Order statistics play an important role in statistical analysis. Distributions of order statistics in quantile forms, their properties and role in reliability analysis form the next topic in the chapter. There are many problems associated with the use of conventional moments in modelling and analysis. Exploring these, and as an alternative, the definition, properties and application of L-moments in describing a distribution are presented. Finally, the role of certain graphical representations like the Q-Q plot, box-plot and leaf-plot are shown to be useful tools for a preliminary analysis of the data.
... For an analogous discussion based on residual life, one may refer to Nagaraja (1975), Dallas (1979), Gupta and Gupta (1983), Galambos and Hagwood (1992), Navarro et al. (1998), Lin (2003, Sunoj (2004) and Huang and Su (2012), among others. Further related results for left truncated random variables can be found in Ruiz and Navarro (1995), Su and Huang (2000), Sankaran and Nair (2000), Abraham and Nair (2001) and El-Arishy (2005). Some reliability aspects of partial moments using quantile functions have also been studied in residual life by Nair et al. (2013). ...
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Due to the important role in theory of reliability, forensic science, study of risk analysis, actuarial science, survival analysis, and many other areas of applied probability, the study of inactivity time and its higher order and partial moments has received considerable attention from many authors. In the present paper we give some further insight on the subject. We provide characterization of some continuous distributions based on higher order moments of inactivity time. It is shown that the distribution function can be obtained uniquely through the higher order moments of inactivity time. Furthermore, we study some characterizations based on partial moments of the inactivity time.
... For this reason, mixture distributions have been investigated for use in modeling complex failure distributions. Jiang and Murthy [16] and Abraham and Nair [1] derived some reliability properties of certain finite mixture distributions. However, the analytic difficulties encountered when inferring mixture model parameters from data prevent their widespread use in reliability assessment. ...
Article
ACKNOWLEDGEMENTS This research could not have been completed without the help of several people whom I wish to thank. I thank Dr. Andrew Makeev for his guidance and ideas during the course of this research. Through our conversations we came upon the idea that became the topic of this study. Furthermore, his advice was very helpful in overcoming the setbacks I faced while conducting this research. I thank Dr. Erian Armanios for his support as my adviser and friend. He has greatly eased my transition to life and work here at Georgia Tech. My thanks go to Dr. George Kardomateas as well for his help as a member of my thesis committee. I also thank my lab mates Samer, Serkan, Yihong, and Yuan for welcoming me into the research group. Finally, I thank my fiancée Christiane for her love and patience. She gives me the happiness that allows me to confidently face each day’s challenges.
Chapter
There are several functions in reliability theory used to describe the patterns of failure in different mechanisms or systems as a function of age. The functional forms of many of these concepts characterize the life distribution and therefore enable the identification of the appropriate model. In this chapter, we discuss these basic concepts, first using the distribution function approach and then introduce their analogues in terms of quantile functions. Various important concepts introduced here include the hazard rate, mean residual life, variance residual life, percentile residual life, coefficient of variation of residual life, and their counterparts in reversed time. The expressions for all these functions for standard life distributions are given in the form of tables to facilitate easy reference. Formulas for the determination of the distribution from these functions, their characteristic properties and characterization theorems for different life distributions by relationships between various functions are reviewed. Many of the quantile functions in the literature do not have closed-form expressions for their distributions, and they have to be evaluated numerically. This renders analytic manipulation of these reliability functions based on the distribution function rather difficult. Accordingly, we introduce equivalent definitions and properties of the traditional concepts in terms of quantile functions. This leads to hazard quantile function, mean residual quantile function and so on. The interrelationships between these functions are presented along with characterizations. Various examples given in the sequel illustrate how the quantile based reliability functions can be found directly from the quantile functions of life distributions. Expressions of such functions for standard life distributions can also be read from the tables provided in each case.
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