Modified Sarhan–Balakrishnan singular bivariate distribution

doi:10.1016/j.jspi.2009.07.026

Journal of Statistical Planning and Inference

Volume 140, Issue 2, February 2010, Pages 526-538

https://doi.org/10.1016/j.jspi.2009.07.026 Get rights and content

Abstract

Recently Sarhan and Balakrishnan [2007. A new class of bivariate distribution and its mixture. Journal of Multivariate Analysis 98, 1508–1527] introduced a new bivariate distribution using generalized exponential and exponential distributions. They discussed several interesting properties of this new distribution. Unfortunately, they did not discuss any estimation procedure of the unknown parameters. In this paper using the similar idea as of Sarhan and Balakrishnan [2007. A new class of bivariate distribution and its mixture. Journal of Multivariate Analysis 98, 1508–1527], we have proposed a singular bivariate distribution, which has an extra shape parameter. It is observed that the marginal distributions of the proposed bivariate distribution are more flexible than the corresponding marginal distributions of the Marshall–Olkin bivariate exponential distribution, Sarhan–Balakrishnan's bivariate distribution or the bivariate generalized exponential distribution. Different properties of this new distribution have been discussed. We provide the maximum likelihood estimators of the unknown parameters using EM algorithm. We reported some simulation results and performed two data analysis for illustrative purposes. Finally we propose some generalizations of this bivariate model.

Introduction

Recently Sarhan and Balakrishnan (2007) introduced a new bivariate distribution based on the generalized exponential (GE) and exponential distributions. From now on we call this distribution as the SB distribution. They derived several interesting properties of this new distribution. They obtained the marginal and conditional probability density functions (PDFs) and different moments from the moment generating function (MGF). Unfortunately, they did not discuss any estimation procedure of the unknown parameters of their proposed model. The SB model has four unknown parameters (with the presence of the scale parameter) and it is not immediate how to obtain any estimators of the unknown parameters. Moreover, without a proper estimation procedure it may be difficult to use this distribution in practice.

The main aim of this paper is to introduce a new singular bivariate (SBV) model using the similar idea as of Sarhan and Balakrishnan (2007). This new model has an extra shape parameter than the SB model. It may be mentioned that the four-parameter SB model has two shape and one scale parameters, whereas the proposed SBV model has three shape and one scale parameters. That makes SBV model more flexible than the SB bivariate model. SBV model has an absolute continuous part and a singular part, like the Marshall–Olkin bivariate exponential distribution, SB bivariate model or the bivariate generalized exponential model of Kundu and Gupta (2009). Both the absolute continuous part and the singular part can take various shapes, depending on the parameter values. We discuss various properties of this new distribution and provide different application areas. The SBV model can be observed as a competing risk model or as a shock model also.

The marginal and conditional distributions of the SBV model have been obtained. It is observed that the marginal distributions can be obtained as weighted generalized exponential distributions. Shapes, hazard functions and different moments of the marginal distributions, are discussed. It is observed that the PDFs of the marginals can be decreasing or unimodal and the hazard functions can be increasing, decreasing or bathtub shaped. The shape of the hazard function makes it more flexible than the Marshall–Olkin bivariate exponential distribution, SB bivariate model or the bivariate generalized exponential model. Moment generating functions and different product moments of the SBV model cannot be obtained in closed forms. They can be expressed in terms of infinite series, but for certain restrictions on the shape parameters, the infinite expressions become finite only.

The proposed SBV model has four unknown parameters. To compute the maximum likelihood estimators (MLEs) directly one needs to solve a four dimensional optimization problem. It is not immediate how to solve these four non-linear equations simultaneously. To avoid that we treat this problem as a missing value problem and use the EM algorithm to compute the MLEs of the unknown parameters. For implementing the EM algorithm, at each M-step one needs to solve four one dimensional optimization problems. They are much easier to solve than the direct four dimensional optimization problem. The bootstrap technique can be used very easily to construct the confidence intervals of the unknown parameters. We have performed some simulation studies and for illustrative purposes we have analyzed two data sets using SBV model. It is observed that the proposed model and the EM algorithm work quite well in practice. Finally we discuss some generalizations of the SBV model.

The rest of the paper is organized as follows. In Section 2, we introduce the SBV model and provide two interpretations. Different properties of the SBV model namely, moment generating function, the product moments, marginal distributions, conditional distributions and hazard functions are discussed in Section 3. The implementation of the EM algorithm is provided in Section 4. Simulation results and data analysis have been presented in Section 5. Some generalizations and the conclusions appear in Section 6.

Section snippets

A singular bivariate model

Let $U_{1}, U_{2}, U_{3}$ , be three mutually independent random variables and $U_{i} \sim GE (α_{i}, λ), i = 1, 2, 3$ Here ‘ $\sim$ ’ means follows or has the distribution, and $GE (α, λ)$ denotes the generalized exponential distribution with the shape parameter $α > 0$ and scale parameter $λ > 0$ with the cumulative distribution function (CDF) $F (x; α, λ) = (1 - e^{- λ x})^{α}; x > 0$ and the probability density function (PDF) $f (x; α, λ) = α λ e^{- λ x} (1 - e^{- λ x})^{α - 1}; x > 0$ Define the random variables $X_{1} = \min {U_{1}, U_{3}} and X_{2} = \min {U_{2}, U_{3}}$ Then we say the bivariate vector $(X_{1}, X_{2})$ has the SBV

Different properties

In this section we discuss different properties of the SBV model, namely the moment generating functions, product moments, marginal distributions, conditional distributions and bivariate hazard rate.

Maximum likelihood estimators

In this section we discuss the problem of computing the maximum likelihood estimators (MLEs) of the unknown parameters of the SBV model. It is assumed that we have a sample of size n, of the form ${(x_{11}, x_{12}), \dots, (x_{n 1}, x_{n 2})}$ from SBV( $α_{1}, α_{2}, α_{3}, λ$ ) and our problem is to estimate $α_{1}, α_{2}, α_{3}, λ$ from the given sample. We use the following notations: $I_{1} = {i; x_{i 1} < x_{i 2}}, I_{2} = {i; x_{i 1} > x_{i 2}}, I_{0} = {i; x_{i 1} = x_{i 2} = y_{i}}, I = I_{0} \cup I_{1} \cup I_{2}$ $n_{1} = | I_{0} |, n_{1} = | I_{1} |, n_{2} = | I_{2} |$ Based on the above sample the log-likelihood function becomes $l (α_{1}, α_{2}, α_{3}, λ) = \sum_{i \in I_{1}}$

Simulation and data analysis

In this section we present some simulation results to show how the proposed EM algorithm works for different sample sizes. For illustrative purposes we analyze one simulated data and one real life data.

Conclusions

In this paper we have proposed a new bivariate singular distribution and discussed several properties. This model has been obtained using similar technique as of Sarhan and Balakrishnan (2007) and using generalized exponential models. It is observed that the marginal distributions of the proposed model are more flexible than the corresponding marginal distributions of the Marshall–Olkin bivariate exponential model, Sarhan and Balakrishnan (2007) model or the bivariate generalized exponential

Acknowledgments

The authors would like to thank the associate editor and two referees for their helpful comments which had improved significantly the earlier draft of the paper. Part of the work of the first author has been supported by a grant from the Department of Science and Technology, Government of India. Part of the work of the second author has been supported by a discovery grant from NSERC, Canada.

References (13)

R.C. Gupta et al.
Proportional reversed hazard rate model and its applications
Journal of Statistical Planning and Inference
(2007)
R.D. Gupta et al.
Generalized exponential distributions: existing results and some recent developments
Journal of Statistical Planning and Inference
(2007)
N.L. Johnson et al.
A vector multivariate hazard rate
Journal of Multivariate Analysis
(1975)
D. Kundu et al.
Bivariate generalized exponential distribution
Journal of Multivariate Analysis
(2009)
A. Sarhan et al.
A new class of bivariate distribution and its mixture
Journal of Multivariate Analysis
(2007)
A.P. Basu
Bivariate failure rate
Journal of the American Statistical Association
(1971)

There are more references available in the full text version of this article.

Cited by (33)

Multivariate reliability modelling based on dependent dynamic shock models
2017, Applied Mathematical Modelling
Citation Excerpt :
Note that, in some cases, several components in the system can fail simultaneously due to ‘fatal shocks’. For these cases, multivariate survival distributions with ‘singular parts’ would be appropriate (see, e.g., Marshall and Olkin [32], Kundu and Gupta [25,26]). However, there are clearly situations when such models cannot be applied.
In this paper, we stochastically model positively dependent multivariate reliability distributions based on stochastically dependent dynamic shock models. In the first part, we consider a shock model with delayed failures. This shock model will be used to construct a class of absolutely continuous multivariate reliability distributions. Explicit parametric forms for the multivariate reliability functions are suggested. Multivariate ageing properties and dependence structures of the class are discussed as well. In the second part, we obtain two types of absolutely continuous multivariate exponential distributions based on further generalized shock models.
Mean residual life and inactivity time of a coherent system subjected to Marshall-Olkin type shocks
2016, Journal of Computational and Applied Mathematics
We consider coherent systems subjected to Marshall–Olkin type shocks coming at random times and destroying components of the system. The paper combines two important models, coherent systems and Marshall–Olkin type shocks and studies the mean residual life (MRL) and the mean inactivity time (MIT) functions of coherent systems that is subjected to random shocks. The considered models and theoretical results are supported with examples and graphical representations.
On Marshall-Olkin type distribution with effect of shock magnitude
2014, Journal of Computational and Applied Mathematics
In classical Marshall–Olkin type shock models and their modifications a system of two or more components is subjected to shocks that arrive from different sources at random times and destroy the components of the system. With a distinctive approach to the Marshall–Olkin type shock model, we assume that if the magnitude of the shock exceeds some predefined threshold, then the component, which is subjected to this shock, is destroyed; otherwise it survives. More precisely, we assume that the shock time and the magnitude of the shock are dependent random variables with given bivariate distribution. This approach allows to meet requirements of many real life applications of shock models, where the magnitude of shocks is an important factor that should be taken into account. A new class of bivariate distributions, obtained in this work, involve the joint distributions of shock times and their magnitudes. Dependence properties of new bivariate distributions have been studied. For different examples of underlying bivariate distributions of lifetimes and shock magnitudes, the joint distributions of lifetimes of the components are investigated. The multivariate extension of the proposed model is also discussed.
On bivariate Weibull-Geometric distribution
2014, Journal of Multivariate Analysis
Citation Excerpt :
Moreover, MOBW can be obtained as a special case of the BWG model. Other recent extensions of the MOBE model with a singular component have been discussed in the literature; e.g. see [17,5,11,4]. Due to the presence of five parameters the joint probability density function of BWG distribution is very flexible, and it can take different shapes depending on the shape parameters.
Marshall and Olkin (1997) [14] provided a general method to introduce a parameter into a family of distributions and discussed in details about the exponential and Weibull families. They have also briefly introduced the bivariate extension, although not any properties or inferential issues have been explored, mainly due to analytical intractability of the general model. In this paper we consider the bivariate model with a special emphasis on the Weibull distribution. We call this new distribution as the bivariate Weibull-Geometric distribution. We derive different properties of the proposed distribution. This distribution has five parameters, and the maximum likelihood estimators cannot be obtained in closed form. We propose to use the EM algorithm, and it is observed that the implementation of the EM algorithm is quite straightforward. Two data sets have been analyzed for illustrative purposes, and it is observed that the new model and the proposed EM algorithm work quite well in these cases.
Bivariate marshall-olkin exponential shock model
2021, Probability in the Engineering and Informational Sciences
ON A MULTIVARIATE GENERALIZED POLYA PROCESS without REGULARITY PROPERTY
2020, Probability in the Engineering and Informational Sciences

View all citing articles on Scopus

View full text

Modified Sarhan–Balakrishnan singular bivariate distribution

Abstract

Introduction

Section snippets

A singular bivariate model

Different properties

Maximum likelihood estimators

Simulation and data analysis

Conclusions

Acknowledgments

Journal of Statistical Planning and Inference

Journal of Statistical Planning and Inference

Journal of Multivariate Analysis

Journal of Multivariate Analysis

Journal of Multivariate Analysis

Bivariate failure rate

Journal of the American Statistical Association