Abstract
In this paper, we propose a new lifetime distribution by compounding the gamma and Lindley distributions. Construction of it can be interpreted in the viewpoint of the reliability analysis and Bayesian inference. Moreover, the distribution has decreasing and unimodal hazard rate shapes. Several properties of the distribution are obtained, involving characteristics of the (reverse) hazard rate function, quantiles, moments, extreme order statistics and some stochastic order relations. Estimating the distribution parameters is discussed by some estimation methods and their performance is evaluated by a simulation study. Also, estimation of the stress-strength parameter is investigated. Usefulness of the distribution among other models is illustrated by fitting two failure data sets and using some goodness-of-fit measures.
Highlights
Different nature of the lifetime data requires introducing new distributions with various failure rate shapes to capture the analysis of such data
The maximum likelihood estimators (MLEs) are asymptotically unbiased and consistent. ii) Table 1 shows that for most of cases considered, the biases and Mean square error (MSE) of the least squares estimators (LSEs) and weighted least squares estimators (WLSEs) decrease as n increases. iii) With respect to the bias, the LSEs work better than the MLEs and WLSEs. iv) With respect to the MSEs, the MLEs work better than the LSEs and WLSEs for α < 1 but for α ≥ 1, the LSEs and WLSEs work better than the MLEs
We study some of its mathematical properties such as characteristics of the pdf, hrf, rhrf, quantiles, moments, extreme order statistics and their limiting distributions, and stochastic ordering
Summary
Different nature of the lifetime data requires introducing new distributions with various failure rate shapes to capture the analysis of such data. Among those shapes, we mention: the decreasing and unimodal (upside-down bathtub) hazard (failure) rates. We mention: the decreasing and unimodal (upside-down bathtub) hazard (failure) rates The former rates have many applications in reliability and survival analysis. The gamma Lindley distribution was proposed by (Zeghdoudi and Nedjar 2015) as a mixture of gamma(2, θ) and one-parameter Lindley distribution The corresponding reliability (survival) function is given by (β + x)α+1(1 + β) − xα(1 + β)(β + x) − βαxα
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