Abstract
An important property that any lifetime model should satisfy is scale invariance. In this paper, a new scale-invariant quasi-inverse Lindley (QIL) model is presented and studied. Its basic properties, including moments, quantiles, skewness, kurtosis, and Lorenz curve, have been investigated. In addition, the well-known dynamic reliability measures, such as failure rate (FR), reversed failure rate (RFR), mean residual life (MRL), mean inactivity time (MIT), quantile residual life (QRL), and quantile inactivity time (QIT) are discussed. The FR function considers the decreasing or upside-down bathtub-shaped, and the MRL and median residual lifetime may have a bathtub-shaped form. The parameters of the model are estimated by applying the maximum likelihood method and the expectation-maximization (EM) algorithm. The EM algorithm is an iterative method suitable for models with a latent variable, for example, when we have mixture or competing risk models. A simulation study is then conducted to examine the consistency and efficiency of the estimators and compare them. The simulation study shows that the EM approach provides a better estimation of the parameters. Finally, the proposed model is fitted to a reliability engineering data set along with some alternatives. The Akaike information criterion (AIC), Kolmogorov-Smirnov (K-S), Cramer-von Mises (CVM), and Anderson Darling (AD) statistics are used to compare the considered models.
Highlights
Lindley [1] and inverse Lindley models have attracted much attention in the last decade
By differentiating from λ(x) with respect to x, we find that the failure rate (FR) function shows a unimodal form
The quasi-inverse Lindley (QIL)(α, θ) is a mixture of two inverse gamma distributions IG(1, θ ) and IG(2, θ) and α determines the weights of the mixture
Summary
Lindley [1] and inverse Lindley models have attracted much attention in the last decade. Ghitany et al [4] considered the distribution introduced by Sankaran [3] to study a compound Poisson-Lindley model truncated to zero. Al-babtain et al [8] generalized the Lindley model to a distribution with five parameters. Shanker et al [12] introduced and studied a new quasi-Lindley model. Broderick et al [17] proposed a generalization of the Lindley model with four parameters. Sharma et al [18] introduced the inverse Lindley distribution and considered it as a stress-strength model. We present and study a scale-invariant extension of the inverse Lindley distribution.
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