Research Article
A Mixture of Inverse Weibull and Inverse Burr Distributions:
Properties, Estimation, and Fitting
A. S. Al-Moisheer,
1
K. S. Sultan,
2
and M. A. Al-Shehri
3
1
Department of Mathematics, College of Science, Aljouf University, P.O. Box 848, Sakaka 42421, Saudi Arabia
2
Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455,
Riyadh 11451, Saudi Arabia
3
Department of Quantitative Analysis, College of Business Administration, King Saud University, P.O. Box 3629,
Riyadh 13214, Saudi Arabia
Correspondence should be addressed to A. S. Al-Moisheer; asalmoisheer@ju.edu.sa
Received 1 June 2017; Revised 15 September 2017; Accepted 19 September 2017; Published 31 October 2017
Academic Editor: Javier Cara
Copyright © 2017 A. S. Al-Moisheer et al. Tis is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
Te new mixture model of the two components of the inverse Weibull and inverse Burr distributions (MIWIBD) is proposed. First,
the properties of the investigated mixture model are introduced and the behaviors of the probability density functions and hazard
rate functions are displayed. Ten, the estimates of the fve-dimensional vector of parameters by using the classical method such as
the maximum likelihood estimation (MLEs) and the approximation method by using Lindley’s approximation are obtained. Finally,
a real data set for the proposed mixture model is applied to illustrate the proposed mixture model.
1. Introduction
Te importance of mixture models comes from the fact that
most available data can be considered as data coming from
a mixture of two or more statistical models; see Sultan et
al. [1]. For books that dealt with the models of the mixture,
see Everitt and Hand [2] and McLachlan and Peel [3].
Because the mixing of statistical distributions gives a new
distribution with the properties of its compounds, we in
this paper propose the two-component mixture models of
inverse Weibull and inverse Burr distributions (MIWIBD).
For the importance of the inverse Weibull distribution (IWD)
as a single component from its uses in physical phenomena,
see Keller et al. [4]. Also, for the importance of the inverse
Burr distribution (IBD) as one component from its uses
in forestry applications, see Lindsay [5]. Tis importance
for each distribution alone has made us merge the two
distributions together to obtain new properties from the
distributive compounds. It should be noted that the mixing
of the IWIBD gives a mixture model with a unimodal and
bimodal peak for the hazard rate functions and these forms
are important in applications which will be displayed in
Section 2. Te probability density function (pdf) from the
MIWIBD is as follows:
(; Θ) =
2
∑
=1
(, Θ
), 0≤
1
≤ 1,
1
+
2
= 1, (1)
where the (pdf) of the frst component (inverse Weibull) is
given by
1
(; Θ
1
)=
1
−
1
1
−(
1
+1)
−(
1
)
−
1
,
≥ 0,
1
,
1
> 0,
(2)
and the (pdf) of the second component (inverse Burr) is given
by
2
(; Θ
2
)=
2
2
−(
2
+1)
(1 +
−
2
)
−(
2
+1)
,
≥ 0,
2
,
2
> 0,
(3)
Hindawi
Mathematical Problems in Engineering
Volume 2017, Article ID 7824323, 11 pages
https://doi.org/10.1155/2017/7824323