Abstract
The competing risk model based on Lindley distribution is discussed under the progressive type-II censored sample data with binomial removals. The maximum likelihood estimation of the unknown parameters of the distribution is established. Using the Lindley approximation method, we also obtain the Bayesian estimation of the unknown parameters of the distribution under different loss functions. The performance of different estimates is studied in this article. A real practical dataset is analyzed for illustration.
Highlights
Suppose the competing risks failure times follow the Lindley distribution with different parameters θ1 and θ2 under progressive type-II censoring with binomial removals, for θ1 > 0, θ2 > 0 and 0 < p < 1, the MLE of θ j, j = 1, 2 exists and is unique which can be obtained by solving following equations, respectively
Reference [18] introduced how to generate progressive Type-II censored data for continuous distributions. Based on those previous studies and conclusions, we propose the procedures to generate the corresponding random numbers and Algorithm 1 is given by Algorithm 1 Generating the progressively type II censored samples with competing risks
Considering two competing risks, the maximum likelihood estimates of distribution parameters (θ1, θ2 and p) are obtained and the Bayes estimates of these parameters are obtained by using the loss functions of squared error loss function, LINEX loss function and general entropy loss function
Summary
In order to better investigate the problem of product lifetime, Reference [1] proposed a new distribution, known as the Lindley distribution. The Lindley distribution has attracted the attention of many statisticians. Reference [2] explored the mathematical and statistical properties of Lindley distribution. A new two-parameter Lindley distribution was proposed and introduced by Reference [4]. Considering the Lindley distribution, a new life data modeling distribution was developed by Reference [5]. Considering the generalized Lindley distribution, Reference [7] came up with a new bounded domain probability density function, and introduced a distorted premium principle based on the special category of this distribution. Mathematics 2019, 7, 646 θ is the shape parameter of the Lindley distribution and it is a positive real number. The density function of the Lindley distribution has a thin tail, because when x is large, its density function decreases exponentially. In many cases, the Lindley distribution is more flexible than these two distributions
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