Poisson Generated Family of Distributions: A Review
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Poisson Generated Family of Distributions: A Review Sandeep Kumar Maurya Central University of South Bihar, Gaya, India
Saralees Nadarajah University of Manchester, Manchester, UK Abstract The present article represents a survey on Poisson generated family of distributions. Based on this family of distribution, several transformations and distributions have been proposed. Out of which, some of them are proposed by referencing it, and some are independent. The family can be proposed by using the compounding concept of zero truncated Poisson distribution with any other model or family of distributions. Here, we provide a complete survey on this family of distributions and list the contributory related research works. We also address 12 power series distributions, 77 distributions based on the Poisson family of distribution, and 23 distributions, based on different ten transformation methods based on this family of distribution. These numbers show the importance of the Poisson family of distribution. AMS (2000) subject classification. Primary: 60-02, 60E05; Secondary: 62F10. Keywords and phrases. Lifetime models, Poisson distribution, Power series distribution, Compounding of distribution function, Transformation method
1 Introduction In statistical literature, we suppose that every real phenomenon is governed by some lifetime model. If we know the model, we can completely specify our problem or phenomenon as various lifetime models have been developed for this purpose. Poisson distribution is one of the famous models that also provide a family of distribution. By using that family, a number of lifetime models have been proposed and studied their properties by several authors. The Poisson family of distribution (PFD) can be constructed by using the concept of compounding. In the compounding method, there are two different ways available; one is by using zero truncated power series distribution and others by using zero truncated Poisson distribution directly with
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S. K. Maurya and S. Nadarajah
other continuous lifetime models. There are two basic reasons behind using zero truncated Poisson distribution or power series distribution. The first one is that in a real dataset, there are negligible instances of the value zero, which compels one to consider the value zero to be excluded, and the second one is that the compounding process is based on the number of complementary risk for failures of components. Hence, it is more appealing that this number must be greater than or equal to one. In this paper, we provide a detailed study of both approaches for constructing PFD. The main idea of compounding is that the lifetime of a system with N (a discrete random variable) components and the failure time of ith component Ti (say), independent of N , follow some continuous lifetime distribution. Then the maximum or minimum time of failures of components of the system depending on the condition whether they are parallel or in series, respectively. One of the base paper that in-light in the field of compounding was proposed by Adam
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