Abstract
A new scale-invariant extension of the Lindley distribution and its power generalization has been introduced. The moments and the moment-generating functions of the proposed models have closed forms. The failure rate, the mean residual life, and the α -quantile residual life functions have been explored. The failure rate function of these models accommodates increasing, bathtub-shaped, and increasing then bathtub-shaped forms. The parameters of the models have been estimated by the maximum likelihood method for the complete and right-censored data. In a simulation study, the efficiency and consistency of the maximum likelihood estimator have been investigated. Then, the proposed models were fitted to four data sets to show their flexibility and applicability.
Highlights
Introduction e Lindley distribution introduced byLindley [1] has been attracted much interest in recent years. e probability density function (PDF) and the cumulative distribution function (CDF) of the Lindley are, respectively, as follows: f(x) θ2 (1 + x)e− θx, θ > 0, x ≥ 0, 1+θ (1)F(x) 1 − 1 + θ + θxe− θx, θ > 0, x ≥ 0. 1+θClearly the Lindley distribution is a mixture of two gamma distributions G(1, θ), gamma with shape parameter 1 and scale θ and G(2, θ) with weights θ/1 + θ and 1/1 + θ, respectively.Ghitany et al [2] explored some properties of the Lindley model
We introduce one extended Lindley (EL) distribution with parameters k > 0, m > 0, k ≠ m, δ > 0, and θ > 0, namely, EL(k, m; δ, θ), with the PDF as follows: f(x)
We introduce the power Lindley extended (PEL) distribution with parameters k > 0, m > 0, k ≠ m, δ > 0, θ > 0, and c > 0, namely, PEL(k, m; δ, θ, c), with the PDF as follows: f(x)
Summary
We introduce one extended Lindley (EL) distribution with parameters k > 0, m > 0, k ≠ m, δ > 0, and θ > 0, namely, EL(k, m; δ, θ), with the PDF as follows: f(x) θ. We can consider a suitable sequence of pairs (k, m) and search for a good model by estimating (δ, θ) In this way, we avoid headaches of optimizing four dimensional functions. Since the failure rate function may be increasing, bathtub-shaped, or increasing, bathtub-shaped, the MRL function, and the α-QRL function may be decreasing, upside down bathtub-shaped, or decreasing upside down bathtub-shaped, respectively (refer to Lai and Xie [25]). E point which maximizes the MRL/median residual life function is known as the burn-in time and has been attracted interest of many authors (see Mi [26]).
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