Abstract
The analysis and modeling of zero truncated count data is of primary interest in many elds such as engineering, public health, sociology, psychology, epidemiology. Therefore, in this article we have proposed a new and simple structure model, named a zero truncated discrete Lindley distribution. Thedistribution contains some submodels and represents a two-component mixture of a zero truncated geometric distribution and a zero truncated negative binomial distribution with certain parameters. Several properties of the distribution are obtained such as mean residual life function, probability generating function, factorial moments, negative moments, moments of residual life function, Bonferroni and Lorenz curves, estimation of parameters, Shannon and Renyi entropies, order statistics with the asymptotic distribution of their extremes and range, a characterization, stochastic ordering and stress-strength parameter. Moreover, the collective risk model is discussed by considering theproposed distribution as primary distribution and exponential and Erlang distributions as secondary ones. Test and evaluation statistics as well as three real data applications are considered to assess the peformance of the distribution among the most frequently zero truncated discrete probability models.
Highlights
An important statistical property which is used in many fields, such as reliability theory, queueing systems, medicine, industry and many others, is the truncation of probability distributions
For example: length of hospital stay is recorded as a minimum of at least one day, number of times a voter has voted during the general election, number of journal articles published in different disciplines, number of tickets received by teenagers as predicted by school performance, number of children ever born to sample of mothers over 40 years of age, number of eggs and gall-cells were counted in flowers heads in two years, number of occupants in passenger cars, number of stressful events reported by patients, number ofhouses replying to a postal survey etc
The flexibility of the model is investigated by its consonance with some real data sets as revealed through techniques such as empirical distribution function plots, empirical hazard function plots, Anderson-Darling, KolmogrovSimnorov (KS), Akaike information criterion (AIC), Bayesian information criterion (BIC) and Akaike information criterion corrected (AICc) statistics
Summary
An important statistical property which is used in many fields, such as reliability theory, queueing systems, medicine, industry and many others, is the truncation of probability distributions. Proposition 2.1: If Y~ZTDL(p, β) the probability mass function (pmf), of the random variable Y is log-concave for all choices of β and independent of p. Corollary 2.1: The following results are the direct consequence of log-concavity (see Chakraborty and Ong, 2016): i) ZTDL has an increasing failure rate function. Proposition 2.3: If Y~ZTDL(p, β) the probability generating function (pmf), of the random variable Y is expressed as t(1 − p)2(1 − pt + β) GY(t) = (1 − pt)2(1 − p + β) , where β ≥ 0, 0 < p < 1 for 0 < pt < 1. Residual life random variable is used much in risk analysis and so we investigate some of its properties, like survival function, mean, variance and rth moment for the ZTDL distribution.
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