Computational Statistics & Data Analysis 51 (2007) 5377 – 5387
www.elsevier.com/locate/csda
Mixture of two inverse Weibull distributions: Properties and
estimation
K.S. Sultan
∗
, M.A. Ismail, A.S. Al-Moisheer
Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, SaudiArabia
Available online 9 October 2006
Abstract
The mixture model of two Inverse Weibull distributions (MTIWD) is investigated. First, some properties of the model with some
graphs of the density and hazard function are discussed. Next, the identifiability property of the MTIWD is proved. In addition, the
estimates of the unknown parameters via the EM Algorithm are obtained. The performance of the findings in the paper is showed
by demonstrating some numerical illustrations through Monte Carlo simulations.
© 2006 Elsevier B.V.All rights reserved.
Keywords: Finite mixtures; Mixing proportion; Mode; Median; Reliability and failure rate functions; Identifiability; EM Algorithm; Mean square
error; Bias; Monte Carlo method
1. Introduction
Mixture models play a vital role in many practical applications. For example, direct applications of finite mixture
models are in fisheries research, economics, medicine, psychology, palaeoanthropology, botany, agriculture, zoology,
life testing and reliability, among others. Indirect applications include outliers, Gaussian sums, cluster analysis, latent
structure models, modeling prior densities, empirical Bayes method and nonparametric (kernel) density estimation.
In many applications, the available data can be considered as data coming from a mixture population of two or more
distributions. This idea enables us to mix statistical distributions to get a new distribution carrying the properties of its
components.
The mixture of two Inverse Weibull distribution (MTIWD) has its pdf as
f (t ; ) = p
1
f
1
(t ;
1
) + p
2
f
2
(t ;
2
), p
1
+ p
2
= 1, (1.1)
where = (p
1
,
1
,
2
,
1
,
2
),
i
= (
i
,
i
), i = 1, 2, and f
i
(t ;
i
), the density function of the i th component, is
given by
f
i
(t ;
i
) =
i
-
i
i
t
-(
i
+1)
e
-(
i
t)
-
i
, t 0,
i
,
i
> 0, i = 1, 2. (1.2)
The cdf of the MTIWD is given by
F (t ; ) = p
1
F
1
(t ;
1
) + p
2
F
2
(t ;
2
), (1.3)
∗
Corresponding author. Tel.: +966 01 4676263; fax: +966 01 4676274.
E-mail address: ksultan@ksu.edu.sa (K.S. Sultan).
0167-9473/$ - see front matter © 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.csda.2006.09.016