The bivariate generalized linear failure rate distribution and its multivariate extension
Introduction
The two-parameter linear failure rate (LFR) distribution, whose hazard function is monotonically increasing in a linear fashion, has been used quite successfully to analyze lifetime data. For some basic properties and for different procedures of estimation of the parameters of the LFR distribution, the readers are referred to Bain (1974), Pandey et al. (1993), Sen and Bhattacharyya (1995), Lin et al., 2003, Lin et al., 2006 and the references cited therein.
Recently, Sarhan and Kundu (2009) introduced a three-parameter generalized linear failure rate (GLFR) distribution by exponentiating the LFR distribution as was done for the exponentiated Weibull distribution by Mudholkar et al. (1995). The exponentiation introduces an extra shape parameter in the model, which may yield more flexibility in the shape of the probability density function (PDF) and hazard function. Several properties of this new distribution are established. It is observed that several known distributions like exponential, Rayleigh and LFR distributions can be obtained as special cases of the GLFR distribution.
The aim of this paper is to introduce a new bivariate generalized linear failure rate (BGLFR) distribution, whose marginals are GLFR distributions. This new five-parameter BGLFR distribution is obtained using a method similar to that used to obtain the Marshall–Olkin bivariate exponential model Marshall and Olkin (1967) and Sarhan and Balakrishnan’s bivariate distribution, Sarhan and Balakrishnan (2007). The proposed BGLFR distribution is constructed from three independent GLFR distributions using a maximization process. Creating a bivariate distribution with given marginals using this technique is nothing new. Alternatively, the same BGLFR distribution can be obtained by coupling the GLFR marginals with the Marshall–Olkin copula (Nelsen, 1999). This new distribution is a singular distribution, and it can be used quite conveniently if there are ties in the data. The joint cumulative distribution function (CDF) can be expressed as a mixture of an absolutely continuous distribution function and a singular distribution function. The joint probability density function (PDF) of the BGLFR distribution can take different shapes and the cumulative distribution function can be expressed in a compact form. The BGLFR distribution can be applied to a maintenance model or a stress model as introduced by Kundu and Gupta (2009).
Several dependency properties of this new distribution are investigated, which will be useful for data analysis purposes. The BGLFR copula has a total positivity of order 2 (TP2) property. Each component is stochastically increasing with respect to the other. This implies that the correlation is always non-negative and the two variables are positively quadrant dependent. Moreover, the correlation between the two variables varies between 0 and 1. Kendall’s tau index can be calculated using the copula property and can be positive. The population version of the medial correlation coefficient as defined by Blomqvist (1950) is always non-negative. The bivariate tail dependence is always positive.
The BGLFR distribution has five parameters, and their estimation is an important problem in practice. The usual maximum likelihood estimators can be obtained by solving five non-linear equations in five unknowns directly, which is not a trivial issue. To avoid difficult computation we treat this problem as a missing value problem and use the EM algorithm, which can be implemented more conveniently than the direct maximization process. Another advantage of the EM algorithm is that it can be used to obtain the observed Fisher information matrix, which is helpful for constructing the asymptotic confidence intervals for the parameters.
Alternatively, it is possible to obtain approximate maximum likelihood estimators by estimating the marginals first and then estimating the dependence parameter through a copula function, as suggested by Joe (1997, Chapter 10), which has the same rate of convergence as the maximum likelihood estimators. This is computationally less involved compared to the MLE calculations. This approach is not pursued here. Analysis of a data set is presented for illustrative purposes. The proposed model provides a better fit than the Marshall–Olkin bivariate exponential model or the recently proposed bivariate generalized exponential model (Kundu and Gupta, 2009).
Although in this paper we mainly discuss the BGLFR, many of our results can be easily extended to the multivariate case. Moreover, the LFR distribution is a proportional reversed hazard model, and our method may be used to introduce other bivariate proportional reversed hazard models.
The rest of the paper is organized as follows. We briefly introduce the GLFR distribution in Section 2. In Section 3 we introduce the BGLFR distribution and study its different properties. The EM algorithm is described in Section 4, and analysis of a data set is presented in Section 5. We discuss the multivariate generalization in Section 6, and finally conclude the paper in Section 7.
Section snippets
Generalized linear failure rate distribution
A random variable has a linear failure rate distribution with parameters and (such that ), if has the following distribution function: for . The exponential distribution with mean () and the Rayleigh distribution with parameter () can be obtained as special cases from the LFR distribution. The PDF of the LFR distribution can be decreasing or unimodal, but the failure rate function is either increasing or constant only (Sen and
Bivariate generalized failure rate distribution
In this section we introduce the BGLFR distribution using a method similar to that which was used by Marshall and Olkin (1967) to define the Marshall–Olkin bivariate exponential (MOBE) distribution.
Suppose and are three independent random variables such that for and 3. Define Then we say that the bivariate vector () has a bivariate GLFR (BGLFR) distribution, with parameters (), and we denote it by BGLFR (
Estimation
In this section we consider the estimation of the unknown parameters of the BGLFR model. It is assumed that we have a sample of size , of the form from , and our problem is to estimate from the given sample. First we obtain the MLEs of the unknown parameters. Since the computation of the MLEs is computationally quite involved, we propose alternative estimators, which can be obtained in a more convenient manner.
For further development we
Data analysis
In this section we present the analysis of a data set mainly to illustrate how the proposed model and the EM algorithm work in practice.
UEFA Champion’s League Data: The data set has been obtained from Meintanis (2007) and is presented in Table 2. It represents soccer data where at least one goal is scored by the home team and at least one goal is scored directly from a penalty kick, foul kick or any other direct kick (all of them will be called kick goals) by any team that has been considered.
Multivariate generalized linear failure rate distribution
In this section we are in a position to define the -variate generalized linear failure rate distribution and provide some of its properties. It may be mentioned that recently Franco and Vivo (2009) provided a multivariate extension of the Sarhan–Balakrishnan bivariate distribution and studied its various properties.
Suppose are independent random variables such that for . Define Then we say that is an -variate GLFR
Conclusions
In this paper we have introduced the bivariate generalized linear failure rate distribution whose marginals are generalized linear failure rate distributions. The proposed bivariate distribution is a singular distribution, and it can be used quite effectively instead of the Marshall–Olkin bivariate exponential model, or the bivariate generalized exponential model when there are ties in the data. Several properties of this new distribution have been established, and also we proposed using the EM
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