International Journal of Industrial Engineering & Production Research (2019) 30: 133-147
DOI: 10.22068/ijiepr.30.2.133
International Journal of Industrial Engineering & Production Research, June 2019, Vol. 30, No. 2
The New Distribution of Exponential Singh-Maddala, Some
Properties, and Its Application in Reliability
Zahra Karimi Ezmareh
1
& Gholamhossein Yari
2*
Received 20 October 2018; Revised 06 March 2019; Accepted 28 April 2019; Published online 20 June 2019
© Iran University of Science and Technology 2019
ABSTRACT
In this paper, a new distribution that is highly applicable in the fields of reliability and economics is
introduced. The parameters of this distribution are estimated by using two methods of Maximum
Likelihood and Bayes with two prior distributions, Weibull and Uniform. These two methods are
compared using Monte-Carlo simulation. Finally, this new model is fitted on the real data (with the
failure time of 84 aircraft), and some of comparative criteria are calculated to confirm the superiority
of the proposed model to others.
KEYWORDS Failure time, Exponential distribution, Singh-Maddala distribution, Estimation
parameters, Monte-Carlo simulation, Fitting the model.
1. Introduction1
Reliability is used in various fields of science,
insurance, economics, medicine, engineering, etc.
In the past, reliability was a concern for sensitive
and complex industries such as military, nuclear,
and aerospace industries, whose lack of reliability
could exert irreparably damage; however, now,
this has become a general concern.
The history of reliability growth can be
referenced to the period before the 1930s. During
that period, due to concerns about the proper
functioning of the products, studies were
conducted on the designing of systems with
parallel components. The exact history of
reliability was stated by Knight (1991) [10]
and Andrew [2].
Now, two important numerical quantities are
defined here to measure non-repairable
reliability.
Definition 1-1. The mean residual life ( ࡹࡾࡸ )
function of a lifetime random variable ࢄ is given
by
ߤ(ݔ)=
ଵ
ி
ത
(௫)
∫ ݐ(ݐ) ݐ
ஶ
௫
− ݔ,ݔ> 0. (1)
Corresponding author: Gholamhossein Yari
*
yari@iust.ac.ir
1. School of Mathematics, Iran University of Science and
Technology, Tehran, Iran.
2. School of Mathematics, Iran University of Science and
Technology, Tehran, Iran.
Definition 1-2. The mean time to failure (ࡹࢀࢀࡲ)
of a lifetime random variable is defined as:
ܯܧ=ܨ() = ∫ ݐ(ݐ) ݐ
ஶ
. (2)
A commonly used distribution in the analysis of
lifetime data is the Generalized Beta distribution
of second kind (II) (GB(II)). In addition, by
increasing the skewness in the income data, in
order to achieve more flexible distribution in fit,
four-parameter distributions with more shape
parameters are introduced in economic modeling
by increasing the skewness in the income data.
One of these distributions is the GB(II)
distribution, which was first proposed by
McDonld (1984) [14]. The probability density
function (pdf) of this distribution is as follows:
(ݔ)=
ఈ௫
ഃషభ
ఉ
ഃ
(ఋ,ఒ)[ଵା(
)
]
ഊశഃ
ߜ,ߣ,ߚ,ߙ,ݔ,> 0. (3)
This distribution involves many statistical
distributions as special or limited. One of the
most important distributions, which is very useful
in the fields of reliability, economics, and
finance, is the distribution of the three parameters
of Singh Maddala (SM), obtained by placing
ࢾ=. McDonld (1984)[14] showed that the SM
distribution provided better fits than gamma and
lognormal. Shahzad and Asghar (2013) [16] used
the L-moments and TL-moments methods to
derive the point estimators of the parameters for
SM distribution. The family of distributions
RESEARCH PAPER
[ DOI: 10.22068/ijiepr.30.2.133 ] [ Downloaded from ijiepr.iust.ac.ir on 2021-12-11 ]
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