Open Journal of Statistics, 2016, 6, 917-930
http://www.scirp.org/journal/ojs
ISSN Online: 2161-7198
ISSN Print: 2161-718X
DOI:
10.4236/ojs.2016.65076
October 27, 2016
On Size-Biased Double Weighted Exponential
Distribution (SDWED)
Zahida Perveen
1*
, Zulfiqar Ahmed
2
, Munir Ahmad
3
1
Lahore Garrison University, Main Campus Sector C, Phase VI, DHA Lahore, Pakistan
2
GIFT University, Gujranwala, Pakistan
3
National College of Business Administration and Economics, Lahore, Pakistan
Abstract
This paper introduces a new distribution based on the exponential distribution,
known as Size-biased Double Weighted Exponential Distribution
(SDWED). Some
characteristics of the new distribution are obtained. Plots for the cumulative distr
i-
bution function, pdf and hazard function, tables with values of skewness and kurtosis
are provided. As a motivation, the statistical application of the results to a problem of
ball bearing data has been provided. It is observed tha
t the new distribution is
skewed to the right and bears most of the properties of skewed distribution. It is
found that our newly proposed distribution fits better than size-
biased Rayleigh and
Maxwell distributions and many other distributions. Since many
researchers have
studied the procedure of the weighted distributions in the estates of forest, biomed
i-
cine and biostatistics etc., we hope in numerous fields of theoretical and applied
sciences, the findings of this paper will be useful for the practitioners.
Keywords
Exponential Distribution, Moments, Moment Ratios, Estimation
1. Introduction
Weighted distributions are suitable in the situation of unequal probability sampling,
such as actuarial sciences, ecology, biomedicine biostatistics and survival data analysis.
These distributions are applicable, when observations are recorded without any expe-
riment, repetition and random process. The notion of weighted distributions has been
used as a device for the collection of suitable model for observed data, during last 25
years. The idea is most applicable when sampling frame is not available and random
sampling is not possible. Firstly the idea of weighted distributions was introduced by
How to cite this paper:
Perveen, Z., Ahmed
,
Z.
and Ahmad, M. (2016) On Size-
Biased
Double Weighted Exponential Distribution
(SDWED)
.
Open Journal of Statistics
,
6
, 917
-
930
.
http://dx.doi.org/10.4236/ojs.2016.65076
Received:
July 25, 2016
Accepted:
October 24, 2016
Published:
October 27, 2016
Copyright © 201
6 by authors and
Scientific
Research Publishing Inc.
This work is licensed under the Creative
Commons Attribution International
License (CC BY
4.0).
http://creativecommons.org/licenses/by/4.0/
Open Access
Z. Perveen et al.
918
Fisher [1]. Cox [2] firstly provided the idea of length-biased sampling and after that Rao
[3] established a unifying method that can be used for several sampling situations and
can be displayed by means of the weighted distributions. Cox [4] estimated the mean of
the original distribution built on length-biased data. Zelen [5] presented the concept of
weighted distribution in studying cell kinetics and early discovery of disease. Warren
[6] applied these distributions in forest product research. Patil and Rao [7] surveyed the
idea and applications of weighted and size-biased sampling distributions. Patil and Rao
[8] also discussed weighted binomial distribution to model the human families and es-
timation of the wildlife family size. Gupta and Keating [9] described the relationship
between reliability measures of original and size-biased distribution. Arnold and Naga-
raja
[10] gave the idea of bivariate weighted distribution whereas Jain and Nanda [11]
extended this idea and discussed multivariate aspect of weighted distribution.
Let
( )
;fx
θ
be the pdf of the random variable
x
and
θ
be the unknown parameter.
The weighted distribution is defined as;
( )
( ) ( )
( )
;
;,,0
wx f x
gx x
Ewx
R
θ
θ θ
∈=
>
(1)
where
w
(
x
) is a weight function. When
( )
m
wx x=
, then these distributions are termed
as size-biased distribution of order
m
. When
1m =
it is called size-biased of order 1 or
say length biased distribution, whereas for
2m =
it is called the area-biased distribu-
tion (Ord and Patil [12], Patil [13] and Mahfound [14]).
In forest product research, equilibrium and length biased distributions have been
used as moment distributions. Kochar and Gupta
[15] discussed the moment distribu-
tional properties in assessment with the actual distributions and derived the bound on
the moments of moment distributions.
Oluyede
[16] described inequalities for the reliability measures of size-biased and the
original distributions. Navarro
et al
. [17] discussed characterization of the original and
the size-biased distribution using reliability measures. Gove
[18] offered the uses of
size-biased distributions in forest science and ecology. Sunoj and Maya
[19] established
relationships among weighted and original distributions in the situation of repairable
system and also characterized the sized-biased and the original distribution. Shen
et al
.
[20] used semi-parametric transformations to model the length biased data. Hussain
and Ahmad
[21] presented misclassification in the size-biased modified power series
distributions and its applications.
Mir and Ahmad [22] derived generalized forms of size-biased discrete distributions
and discussed the practical applications in the field of Medical, Zoology and Accidental
studies. Mir
[23] derived size-biased Geeta distribution and size-biased consul distribu-
tion respectively, different properties are discussed and contrasts with original distribu-
tions are also done. Das and Roy
[24] established size-biased form of generalized Ray-
leigh distribution and apply the consequences to the environmental data. They also ap-
plied the concept of size-biased sampling in the field of environmental studies
Dara
[25] derived reliability measures for size-biased forms of several moment dis-
tributions as the special cases of moment distributions. Iqbal and Ahmad [26] found
Z. Perveen et al.
919
compound scale mixtures of limiting distribution of generalized log Pearson type VII
distribution with different continuous and moment distributions. Hasnain [27] intro-
duced a new family of distributions named as exponentiated moment exponential
(EME) distribution and developed its properties. Iqbal
et al
. [28] found a more general
class for EME distribution and built up different properties including characterization
through conditional moments.
Zahida and Munir
[29] worked on Weighted Weibull Distributions (WWD), Double
Weibull Distributions (DWD), Weighted Double Weibull Distributions (WDWD),
Double Weighted Exponential Distributions (DWED) (both in size-biased and area bi-
ased). Some basic theoretical properties of all these distributions including cumulative
density function, central moments, skewness, kurtosis and moments are studied.
Shannon entropy, Renyi entropy, moment generating function and information gene-
rating function of all these distributions are derived. Reliability measures including sur-
vival function, failure rates, reverse hazard rate function and Mills ratios of these dis-
tributions are also obtained. Parameters are evaluated by using method of maximum
likelihood estimation along with derivation of practical examples.
The exponential distribution has a fundamental role in describing a large class of
phenomena, particularly in the area of reliability theory. This distribution is commonly
used to model waiting times between occurrences of rare events, lifetimes of electrical
or mechanical devices. It is also used to get approximate solutions to difficult distribu-
tion problems.
2. Methodology
2.1. Size-Biased Double Weighted Exponential Distribution (SDWED)
The size-biased double weighted exponential distribution is given by:
( )
( )
( ) ( )
0
0
()
, , 0, 0
d
; ,
f x F cx
gc x c
f xFc
x
xx
λλ
∞
≥= >
∫
(2)
where
f
(
x
) is the first weight and
( )
( ) ( )
( )
0
d
wxgx
fx
xg x x
∞
=
∫
Here
( )
wx x=
and
( )
e
x
gx
λ
λ
−
=
,
0
λ
>
,
0x ≥
is the pdf of exponential distri-
bution.
Thus the pdf of SDWED is
( )
( )
( )
3
2
0
32
1
, e 1 e e , 0, , 0
3
;
x cx cx
c
g c x cx x c
c
x
c
λλ λ
λ
λ λλ
−− −
+
−− ≥ >
+
=
(3)
where
λ
is shape parameter and
c
is scale parameter.
Graphs of Probability Density Function
Figure 1 and Figure 2 show the probability density function of SDWED.
2.2. Distribution Function of SDWED
Distribution function of a density function is defined as:
Z. Perveen et al.
920
Figure 1. The probability density function of SDWED for the indicated values of
c
and
λ.
Figure 2. The probability density function of SDWED for the indicated values of
c
and
λ.
( ) ( )
0
,d;
x
F c htx t
λ
=
∫
( )
( )
( )
2
2
32
0
1
e 1e e d
3
x
t t ct
c
t ct t
cc
Fx
λλ λ
λ
λ
−− −
−=
+
−
+
∫
After some simplification we have:
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( ) ( )
11
3
32 32
2
1 11
22
2
1 1e 1 e
1 1e e
33
1 e 2 1 e 2e 2
3
xc xc
xx
xc xc xc
c xc
cx
Fx
cc cc
cx x c
cc
λλ
λλ
λ λλ
λ
λ
λλ
−+ −+
−−
−+ −+ −+
+− − +
+ −−
= −
++
−+ − + − +
−
+
(4)
Figure 3 shows the commutative distribution function of SDWED.
2.3. Survival Function
The survival function of SDWED is defined as
Z. Perveen et al.
921
( ) ( )
1Sx Fx= −
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( ) ( )
11
3
32 32
2
1 11
22
2
1 1e 1 e
1 1e e
33
1 e 2 1 e 2e 2
3
1
xc xc
xx
xc xc xc
c xc
Sx
cx
cc cc
cx x c
cc
λλ
λλ
λ λλ
λ
λ
λλ
−+ −+
−−
−+ −+ −+
+− − +
+ −−
−
++
−+ − + − +
−
+
= −
(5)
Figure 4 shows the survival function of SDWED.
2.4. Hazard Rate Function of SDWED
The hazard rate function is defined as:
( )
0
()
()
gx
h
S
x
x
=
At
1c =
and
1
λ
=
, the hazard rate function will be:
Figure 3. Cumulative Distribution Function of SDWED for the indicated values of
c
and
λ.
Figure 4. The Survival Function of SDWED for the indicated values of
c
and
λ.