Statistics and Its Interface

Volume 12 (2019)

Number 4

Normal mean-variance Lindley Birnbaum–Saunders distribution

Pages: 585 – 597



Farzane Hashemi (Department of Statistics, Faculty of Mathematics and Computer, Shahid Bahonar University, Kerman, Iran)

Mehrdad Naderi (Department of Statistics, Faculty of Natural & Agricultural Sciences, University of Pretoria, South Africa)

Ahad Jamalizadeh (Department of Statistics, Faculty of Mathematics and Computer, Shahid Bahonar University, Kerman, Iran)


The generalization of Birnbaum–Saunders (BS) distribution has recently received considerable attention to provide accurate inferential results in dealing with survival data, reliability problems, fatigue life studies and hydrological data. This paper introduces a new extension of the BS distribution based on the normal mean-variance mixture of Lindley distribution. Since the proposed lifetime distribution can take positive and negative skewness and can have decreasing, increasing, upside-down bathtub, increasing-decreasingincreasing and decreasing-increasing-decreasing hazard rate functions, it may provide more flexible model than the existing extensions of BS distribution. Some properties of the new distribution are derived and the computationally analytical EM-type algorithm is developed for computing maximum likelihood estimates. Finally, the performance of the proposed methodology is illustrated through analyzing two real data sets.


Birnbaum–Saunders distribution, ECM-algorithm, Lindley distribution, normal mean-variance mixture distribution

2010 Mathematics Subject Classification

Primary 62E10, 65C60. Secondary 62E15.

The authors deeply thank the associated editor and two anonymous referees for their valuable suggestions, corrections and encouragement, which have improved considerably earlier versions of the manuscript. M. Naderi’s work is based upon research supported by the National Research Foundation, South Africa (Reference: CPRR160403161466 Grant Number: 105840 and STATOMET).

All the graphical and numerical computations presented here have been performed using the R language and the code is available from the first author upon request.

Received 13 September 2018

Published 18 July 2019