Abstract
We introduce a new family of distributions namely inverse truncated discrete Linnik G family of distributions. This family is a generalization of inverse Marshall-Olkin family of distributions, inverse family of distributions generated through truncated negative binomial distribution and inverse family of distributions generated through truncated discrete Mittag-Leffler distribution. A particular member of the family, inverse truncated negative binomial Weibull distribution is studied in detail. The shape properties of the probability density function and hazard rate, model identifiability, moments, median, mean deviation, entropy, distribution of order statistics, stochastic ordering property, mean residual life function and stress-strength properties of the new generalized inverse Weibull distribution are studied. The unknown parameters of the distribution are estimated using maximum likelihood method, product spacing method and least square method. The existence and uniqueness of the maximum likelihood estimates are proved. Simulation is carried out to illustrate the performance of maximum likelihood estimates of model parameters. An AR(1) minification model with this distribution as marginal is developed. The inverse truncated negative binomial Weibull distribution is fitted to a real data set and it is shown that the distribution is more appropriate for modeling in comparison with some other competitive models.
Highlights
In the last two decades researchers have greater intention toward the inversion of univariate probability models and their applicability under inverse transformation
For the situations in which empirical studies indicate that the hazard function might be unimodal, the inverse Weibull (IW) distribution would be an appropriate model
Khan et al (2008) in their theoretical analysis of IW distribution mention that numerous failure characteristics such as wear out periods and infant mortality can be modeled through IW distribution
Summary
In the last two decades researchers have greater intention toward the inversion of univariate probability models and their applicability under inverse transformation. Nadarajah, Jayakumar and Ristić (2013) proposed a new generalization of the Marshal-Olkin family of distributions, by replacing the geometric distribution of N in (2), as truncated negative binomial distribution with pmf given by. Third generalized family of distributions introduced by Jayakuamar and Sankaran (2019) using truncated discrete Linnik family of distributions with parameters β, θ and c have the survival function (1 + c)θ − [1 + cFβ(x)]θ [(1 + c)θ − 1][1 + cFβ(x)]θ. In (5), when θ = 1 and β ≠ 1, we obtain the survival function of the family of distributions generated using truncated discrete Mittag-Leffler distribution. We introduce a new family of distributions which we named as inverse truncated discrete Linnik G family of distributions. X is an inverse truncated discrete Linnik random variable with cumulative distribution function (cdf) G(x) given by.
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