Download Volume 7, Issue 2, May-August-2016, pp. 01–10, Article ID: IJITMIS_07_02_001

International Journal of Information Technology & Management Information System
(IJITMIS)
Volume 7, Issue 2, May-August-2016, pp. 01–10, Article ID: IJITMIS_07_02_001
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ISSN Print: 0976 – 6405 and ISSN Online: 0976 – 6413
© IAEME Publication
___________________________________________________________________________
CONSTRUCTING A NEW WEIGHTED
PROBABILITY DISTRIBUTION WITH
APPLICATION
Dhwyia Salman Hassan
Department of Business Information Technology,
University of Information Technology and Communications, Iraq
ABSTRACT:
The weighted probability distributions are used when an investigator
records an observation by nature according to a certain stochastic model. the
recorded observation will not have the original distribution unless every
observation is given an equal chance of begin recorded .A number of research
using the concepts of weighted and size – biased sampling distribution due to
importance of this kind of distribution and its application in many fields such
as such as medicine, ecology, reliability and human populations. So we
introduce this paper to explain one of double weighted distribution derived
from inverse Weibull distribution using two types of weight functions. We work
first on deriving the probability distribution and then to derive its cumulative
distribution function and required moments of new weighted probability
distribution. We provide a data analysis to see how the new model works in
practice, the parameters and reliability function estimated using maximum
likelihood method.
Key words: Inverse Weibull distribution, weighted distribution, doubles weighted
distribution, reliability (DWIWD), mode, MLE, MOM.
Cite this Article Dhwyia Salman Hassan, Constructing a New Weighted
Probability Distribution with Application.International Journal of Information
Technology & Management Information System, 7(2), 2016, pp. 01–10.
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1. INTRODUCTION
When an investigator records an observation of a sample according to a certain
stochastic model,this recorded observation will not have original distribution unless
every observation is given an equal chance of being recorded, i.e it may be recorded
with some weight function as indicated by Patil and Rao (1978) also Cook and Martin
(1974) use weighted distribution for estimating wildlife population density. Cook and
Martin introduce various applications on the animals in a group has a specified
distribution. Also Gupta and Keating (1985) apply the weighted distribution for
reliability measures under length Biased sampling. Many research about the weighted
distribution are introduced by Gupta, R.C and Kirmani (1990), Patil and Ord
(1997),Khan and Pasha(2008) and Jose (2009) Use length–Biased distribution with
application about water quality. Vikas and Taneja (2012) introduce a measure on
length –Biased distribution using dynamic measure of past in accuracy data. We
introduce here the derivation of double weighted distribution which is the inverse
Weibull distribution, which is used to model a variety of failure characteristics such
as infant mortality, cost of effectiveness and maintenance periods of reliability
activities the inverse weighted Weibull distribution have many applications in
Encyclopedia of Bio-statistics.
2. MATERIAL AND METHODS
The inverse Weibull (IW) cumulative distribution function (C.D.F) is defined by:
(1)
The probability density function (
) of (IW) is defined as:
(2)
When (
) equation (2) refer to a new distribution called inverse Exponential
(
). And when
equation (2) refers to a new distribution called inverse
Rayleigh distribution. Also, we can define the mean, variance and coefficient of
variation of (IW) as:
(3)
(4)
Coefficient Variation (C.V) is;
(5)
A mathematical definition of the weighted distribution is defined as;
Let (
) be a probability space,
be a random variable ( ) when
[
] be an interval on real line with (
) can be finite or infinite.
When the distribution function
of
is absolutely continuous with a
(
),
,
be anon – negative weighted function satisfying;
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Constructing a New Weighted Probability Distribution with Application
Then the
having the
;
(6)
Equation (6) define the weighted distribution, depending upon the choice of the
weighted function
, we have different weighted models, which can be used as a
tool for the selection of suitable models for observed data.
The paper considered the double weighted inverse Weibull (DWIW), using
different weighted functions and explaining how to derive its
,
, reliability
hazard, and how to estimate the parameters.
3. DEFINITION OF THE DOUBLE WEIGHTED DISTRIBUTION
The double weighted distribution (DWD) is derived from weighted distribution (WD),
we consider the double weighted inverse Weibull (DWIW), using different weight
functions.
Definition1:
The double weighted distribution is given as;
(7)
(8)
Where
is first weight function
is the second weight function depend
and
.
Let first weighted function
and since the
of inverse Weibull
has been shown in (2) .And define the distribution function;
on
(9)
Then;
(10)
Let;
Therefore;
(11)
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Then the probability density function of the double weighted inverse Weibull
distribution (DWIWD) can be found from;
(12)
(13)
Let;
The
of DWIWD become;
(14)
When
,
The cumulative distribution function of DWIWD is giving by;
After some transformation, we get;
..
(15)
Where;
is incomplete gamma function.
And;
(16)
Also we can prove that;
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Constructing a New Weighted Probability Distribution with Application
This indicates that:

The
of DWIWD when

The
of DWIWD when
has one mode say
has one mode say
To find the mode of
, proceed as follows;
(17)
Differentiating equation (17) with respect to
we get;
The mode of
..
Also for
(18)
, the mode of this function is obtained from;
...
The reliability function DWIWD with weighted function
(19)
is;
Which is equal to?
(20)
And;
(21)
4. MOMENTS OF DWIWD
For
the
moment of DWIWD is;
(22)
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The mean is;
(23)
Let
And the variance is;
(24)
Since coefficient of variation;
Depend on parameter
so, it can be used to estimate , were;
The second case where;
(25)
The mean and the variance of
are;
(26)
(27)
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Constructing a New Weighted Probability Distribution with Application
5. ESTIMATION THE PARAMETERS OF
In this section we introduce maximum likelihood method methods to estimate
parameters of
, then we apply it on real data set.
5.1 Maximum Likelihood Method
The estimator of parameters by this method work on maximizing the logarithm of
likelihood function;
Which is obtained from equation (13).
The partial derivatives;
(29)
Solving equations,
, this yields the maximum likelihood (ML)
estimators of (
By assuming
are known;
..
And by assuming
(31)
are known;
…
While the MLE for
).
(32)
can be obtained by solving equation (29) numerically.
5.2 Moment Estimation Method
The moment estimator of parameter obtained from equating sample moments
with population moment
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defined in equation (22), and since
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Dhwyia Salman Hassan
coefficient of variation is independent of (
used to estimate the parameter ( ) were;
), it is only a function of ( ), it can be
(33)
Here the parameter
can be estimated directly by moments and from equation
(22), we use the equations (33) & (34) as follows;
Since
(34)
(34)
Depending on (
) from equation (32) we can find [
(33 & 34) simultaneously.
] by solving
6. APPLICATION
In this section we provide a data analysis to see how the new model works in practice.
The data have been obtained from Electrical Manufacturing Company in Baghdad and
it is provided below. It represents the lifetimes of 34 Electrical Bulb.
0.3633 1.0977 0.9397 2.1993 1.2381 2.1536 1.0970
0.3205 0.3048 0.3354 0.9316 1.6939 0.3236 0.7210
1.4760 0.7527 0.8443 0.3166 0.9706 2.8862 2.7788
0.9811 1.5591 0.4398 4.1935 1.4255 0.1387 0.2379
1.2947 0.7003 0.3575 0.3219 0.3823 1.5740.
The MLE(s) of the unknown parameter(s) for different values of c parameter and
the corresponding estimated reliability plot are given below:
c
1
4.5239
1.9916
2
3.5751
1.9916
3
3.3693
1.9916
4
3.3752
1.9915
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Constructing a New Weighted Probability Distribution with Application
7. CONCLUSION
The double weighted distribution from inverse Weibull distribution has been
constructed. At first, the pdf of the DWD have been obtained considering weight as
w(x) = x and
characterize the distribution of a random variable X of the
DWD three functions have been introduced; namely the Reliability function, the
probability density function and the cumulative function. The moments, the
coefficient of variation, the coefficient of skewness and the coefficient of Kurtosis of
the DWD have been derived. For estimating the parameters of the DWD maximum
likelihood method have been used. The DWD have been fitted to 34 lifetimes of
electrical bulb from electrical manufacturing company in baghdad.DWD suggested a
good fit of the data and reliability function.
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