## Brazilian Journal of Probability and Statistics

### Weighted Weibull distribution: Bivariate and multivariate cases

#### Abstract

Gupta and Kundu (Statistics 43 (2009) 621–643) introduced a new class of weighted exponential distribution and established its several properties. The probability density function of the proposed weighted exponential distribution is unimodal and it has an increasing hazard function. Following the same line Shahbaz, Shahbaz and Butt (Pak. J. Stat. Oper. Res. VI (2010) 53–59) introduced weighted Weibull distribution, and we derive several new properties of this weighted Weibull distribution. The main aim of this paper is to introduce bivariate and multivariate distributions with weighted Weibull marginals and establish their several properties. It is shown that the hazard function of the weighted Weibull distribution can have increasing, decreasing and inverted bathtub shapes. The proposed multivariate model has been obtained as a hidden truncation model similarly as the univariate weighted Weibull model. It is observed that to compute the maximum likelihood estimators of the unknown parameters for the proposed $p$-variate distribution, one needs to solve $(p+2)$ non-linear equations. We propose to use the EM algorithm to compute the maximum likelihood estimators of the unknown parameters. We obtain the observed Fisher information matrix, which can be used for constructing asymptotic confidence intervals. One data analysis has been performed for illustrative purposes, and it is observed that the proposed EM algorithm is very easy to implement, and the performance is quite satisfactory.

#### Article information

Source
Braz. J. Probab. Stat., Volume 32, Number 1 (2018), 20-43.

Dates
Accepted: July 2016
First available in Project Euclid: 3 March 2018

https://projecteuclid.org/euclid.bjps/1520046133

Digital Object Identifier
doi:10.1214/16-BJPS330

Mathematical Reviews number (MathSciNet)
MR3770862

Zentralblatt MATH identifier
06973947

#### Citation

Al-Mutairi, D. K.; Ghitany, M. E.; Kundu, Debasis. Weighted Weibull distribution: Bivariate and multivariate cases. Braz. J. Probab. Stat. 32 (2018), no. 1, 20--43. doi:10.1214/16-BJPS330. https://projecteuclid.org/euclid.bjps/1520046133

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