Weighted Analogue of Inverse Maxwell Distribution with Applications
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The document introduces a new statistical model called the weighted inverse Maxwell distribution (WIMD), detailing its statistical properties, reliability measures, and applications in various fields. It discusses characteristics such as moments, moment generating functions, and behavior of its probability density function, supported by graphs and analytical methods. The model's performance is illustrated through two datasets, concluding that WIMD provides a better fit compared to existing distributions.
![Weighted Analogue of Inverse Maxwell Distribution with Applications
Aijaz et al. 147
Making the substitution
1
𝑦2 = 𝑧 , so that
1
𝑦3 𝑑𝑦 = −
𝑑𝑧
2
and 𝑦 =
1
√𝑧
, we have
=
2
√𝜋
𝜃
3
2 ∫ 𝑦(
3−𝑘
2
)−1
∞
0
𝑒−
𝜃
𝑧 𝑑𝑧
𝐸(𝑦 𝑘) =
2
√𝜋
𝜃
𝑘
2Γ (
3 − 𝑘
2
) (1.3)
Weighted inverse Maxwell distribution (WIMD)
The weighted distributions have many practical applications to analyze real life data and have been used in various fields
such as biomedicine, ecology, reliability. Whenever existing statistical models does not providing a reliable results then
weighted analogue is used for the improvement of such distributions. Recently various authors have proposed and
analyzed different types of weighted as well as length biased distribution. Das et al introduced the Length biased weighted
Weibull distribution (2011). Abd El-Monsef et al (2015), proposed and studied various structural properties of the weighted
Kumaraswamy distribution. Afaq et al(2016) worked on the Length biased weighted Lomax distribution. Weighted inverse
Rayleigh distribution has been studied by Kawsar fatima et al (2017).Aijaz ahmad dar et al(2018), has introduced
characterization and estimation of weighted Maxwell distribution. Mudasir et al (2018), has introduced weighted version
of generalized inverse Weibull distribution and studied its several statistical properties. Hesham M et al (2017), has
established length biased Erlang distribution and discussed its various properties. In this paper, we create new distribution
which is a Weighted Inverse Maxwell distribution (WIMD) and discussing its several statistical properties.
Definition: suppose 𝑌 is a continuous random variate with p.d.f 𝑓(𝑦), then p.d.f of weighted variate 𝑌𝑤 is defined as.
𝑓𝑤(𝑦) =
𝑤(𝑦)𝑓(𝑦)
𝐸[𝑤(𝑦)]
, 𝑦 > 0 (2.1)
Where 𝑤(𝑦) is a non–negative weight function and 𝐸[𝑤(𝑦)] < ∞. Consider the weight function 𝑤(𝑦) = 𝑦 𝑘
for this
distribution.
The probability distribution function of weighted inverse Maxwell distribution is obtained by using equation (1.1),(1.3) and
(2.1), follows as.
𝑓𝑤(𝑦, 𝜃) =
2𝜃
3−𝑘
2
Γ(
3−k
2
)
𝑦 𝑘−4
𝑒
−
𝜃
𝑦2
, 𝑦 > 0, 𝑘, 𝜃 > 0 (2.2)
The cumulative distribution function (c.d.f) of weighted inverse Maxwell distribution is obtained as.
𝐹𝑤(𝑦, 𝜃) = ∫
2𝜃
3−𝑘
2
Γ (
3−k
2
)
𝑦 𝑘−4
𝑒
−
𝜃
𝑦2
𝑑𝑦
𝑦
0
⇒ 𝐹𝑤(𝑦, 𝜃) =
Γ (
3−k
2
,
θ
y2)
Γ (
3−k
2
)
(2.3)
Where Γ(𝜃, 𝑦) = ∫ 𝑦 𝜃−1
𝑒−𝑦
𝑑𝑦
∞
𝑦
is an upper incomplete gamma function.
Fig (1.1) and (1.2) illustrates the probability density function of Weighted Inverse Maxwell distribution.](https://image.slidesharecdn.com/pdfn-200806053226/85/Weighted-Analogue-of-Inverse-Maxwell-Distribution-with-Applications-2-320.jpg)
![Weighted Analogue of Inverse Maxwell Distribution with Applications
Int. J. Stat. Math. 148
Special Cases of Weighted Inverse Maxwell Distribution
Case 1: For = 0, then weighted inverse Maxwell distribution (2.2) reduces to inverse Maxwell distribution with p.d.f as .
𝑓(𝑦, 𝜃) =
4
√𝜋
𝜃
3
2
1
𝑦4
𝑒
−
𝜃
𝑦2
Case 2: For 𝑘 = 1, then weighted inverse Maxwell distribution (2.2) reduces to length biased inverse Maxwell distribution
with p.d.f as .
𝑓(𝑦, 𝜃) = 2𝜃
1
𝑦3
𝑒
−
𝜃
𝑦2
Case 3: For = 2,then weighted inverse Maxwell distribution (2.2) reduces to area biased inverse Maxwell distribution with
p.d.f as .
𝑓(𝑦, 𝜃) = 2√
𝜃
𝜋
1
𝑦2
𝑒
−
𝜃
𝑦2
STATISTICAL PROPERTIES
Moments of weighted inverse Maxwell distribution
Let 𝑋 be a random variable from weighted inverse Maxwell distribution with parameter, then its 𝑟 𝑡ℎ
moment is given as.
𝜇 𝑟 = 𝐸(𝑌 𝑟) = ∫ 𝑦 𝑟
∞
0
𝑓𝑤(𝑦, 𝜃, 𝑘)𝑑𝑦
Now using (2.2), we get
= ∫ 𝑦 𝑟
∞
0
2𝜃
3−𝑘
2
Γ (
3−k
2
)
𝑦 𝑘−4
𝑒
−
𝜃
𝑦2
𝑑𝑦
=
2𝜃
3−𝑘
2
Γ (
3−k
2
)
∫ 𝑦 𝑟+𝑘−1
1
𝑦3
𝑒
−
𝜃
𝑦2
𝑑𝑦
∞
0
By making the substitution
𝜃
𝑦2 = 𝑡 sothat
1
𝑦3 𝑑𝑦 = −
1
2𝜃
𝑑𝑡 and 𝑦 = √
𝜃
𝑡
, then the integration yields as.
=
𝜃
𝑟
2
Γ (
3−k
2
)
∫ 𝑡
[1−(𝑟+𝑘)]
2
∞
0
𝑒−𝑡
𝑑𝑡
=
𝜃
𝑟
2
Γ (
3−k
2
)
∫ 𝑡[
3−(𝑟+𝑘)
2
]−1
𝑒−𝑡
𝑑𝑡
∞
0
Therefore
𝜇 𝑟 = 𝐸(𝑌 𝑟) =
𝜃
𝑟
2
Γ (
3−k
2
)
Γ (
3 − (𝑟 + 𝑘)
2
)
Substituting 𝑟 = 1,2 we get
𝑴𝒆𝒂𝒏 = 𝐸(𝑌) =
𝜃
1
2
Γ (
3−k
2
)
Γ (
2 − 𝑘
2
)
𝐸(𝑌2) =
𝜃
Γ (
3−k
2
)
Γ (
1 − 𝑘
2
)
Variance = 𝐸(𝑌2) − [𝐸(𝑌)]2
=
𝜃
Γ(
3−k
2
)
Γ (
1−𝑘
2
) − [
𝜃
1
2
Γ(
3−k
2
)
Γ (
2−𝑘
2
)]
2
Moment generating function of weighted inverse Maxwell distribution
Let y be a random variable from weighted inverse Maxwell distribution, then the moment generating function of ydenoted
by 𝑀 𝑌(𝑡) is given as.](https://image.slidesharecdn.com/pdfn-200806053226/85/Weighted-Analogue-of-Inverse-Maxwell-Distribution-with-Applications-3-320.jpg)
![Weighted Analogue of Inverse Maxwell Distribution with Applications
Int. J. Stat. Math. 150
Reliability measures
Suppose Y be a continuous random variable with c.d.f 𝐹𝑤(𝑦) , 𝑦 ≥ 0 .then its reliability function which is also called survival
function is defined as
𝑆 𝑤(𝑦) = 𝑝𝑟(𝑌 > 𝑦) = ∫ 𝑓𝑤(𝑦)
∞
0
𝑑𝑦 = 1 − 𝐹𝑤(𝑦)
The survival function of weighted inverse Maxwell distribution is given as
𝑆 𝑤(𝑦) = 1 −
Γ (
3−k
2
,
θ
y2)
Γ (
3−k
2
)
=
𝛾 (
3−k
2
,
θ
y2)
Γ (
3−k
2
)
(4.1)
The hazard rate function of the random variable y is given as
𝐻 𝑤(𝑦) =
𝑓𝑤(𝑦)
𝑆 𝑤(𝑦)
( 4.2 )
Substituting (2.2) and(4.1),into (4.2), we get.
𝐻 𝑤(𝑦) =
2𝜃
3−𝑘
2 𝑦 𝑘−4
𝑒
−
𝜃
𝑦2
𝛾 (
3−k
2
,
θ
y2)
Shannon’s Entropy of weighted inverse Maxwell distribution
The concept of information entropy was introduced by Shanon in 1948. The entropy can be interpreted as the average
rate at which information is produced by a random source of data and is given by
𝐻 𝑤(𝑥, 𝜃) = −𝐸[log 𝑓𝑤(𝑥, 𝜃)]
= −𝐸 [log (
2𝜃
3−𝑘
2
Γ (
3−k
2
)
𝑦 𝑘−4
𝑒
−
𝜃
𝑦2
)]
= − log (
2𝜃
3−𝑘
2
Γ (
3−k
2
)
) − (𝑘 − 4)𝐸(log 𝑦) + 𝐸 (
𝜃
𝑦2
) (5.1)
Now
𝐸(log 𝑦) = ∫ log 𝑦 𝑓𝑤(𝑦)
∞
0
𝑑𝑦
= ∫ log 𝑦
∞
0
2𝜃
3−𝑘
2
Γ (
3−k
2
)
𝑦 𝑘−4
𝑒
−
𝜃
𝑦2
𝑑𝑦
=
2𝜃
3−𝑘
2
Γ (
3−k
2
)
∫ log 𝑦 𝑦 𝑘−4
𝑒
−
𝜃
𝑦2
𝑑𝑦
∞
0
Making the substitution
𝜃
𝑦2 = 𝑡 so that
1
𝑦3 𝑑𝑦 = −
1
2𝜃
𝑑𝑡, and 𝑦 = (
𝜃
𝑡
)
1
2
=
1
Γ (
3−k
2
)
∫ log (
𝜃
𝑡
)
∞
0
1
2
𝑡
1−𝑘
2 𝑒−𝑡
𝑑𝑡
After solving the integral, we get
=
1
2
[log 𝜃 − 𝜓 (
3 − 𝑘
2
)] (5.2)
Also 𝐸 (
𝜃
𝑦2) = 𝜃 ∫
1
𝑦2
∞
0
𝑓𝑤(𝑦 , 𝜃)𝑑𝑦
=
2𝜃
3−𝑘
2 𝜃
Γ (
3−k
2
)
∫
1
𝑦2
𝑦 𝑘−4
𝑒
−
𝜃
𝑦2
𝑑𝑦
∞
0](https://image.slidesharecdn.com/pdfn-200806053226/85/Weighted-Analogue-of-Inverse-Maxwell-Distribution-with-Applications-5-320.jpg)
![Weighted Analogue of Inverse Maxwell Distribution with Applications
Aijaz et al. 151
Making substitution
𝜃
𝑦2 = 𝑡 , we get
=
1
Γ (
3−k
2
)
∫ 𝑡
3−𝑘
2
∞
0
𝑒−𝑡
𝑑𝑡
Therefore 𝐸 (
𝜃
𝑦2) =
3−𝑘
2
(5.3)
Substituting the values (5.2), (5.3) in (5.1), we get
𝐻 𝑤(𝑥, 𝜃) = − log
2𝜃
3−𝑘
2 𝜃
Γ (
3−k
2
)
−
𝑘 − 2
2
[log 𝜃 − 𝜓 (
3 − 𝑘
2
)] + (
3 − 𝑘
2
)
Where 𝜓(. ) denotes the digamma function.
Order statistics of weighted inverse Maxwell distribution
Let us suppose 𝑌1, 𝑌2, 𝑌3 … , 𝑌𝑛 be random samples of size n from weighted inverse Maxwell distribution with p.d.f 𝑓(𝑦) and
c.d.f 𝐹(𝑦). Then the probability density function of 𝑘 𝑡ℎ
order statistics is given as.
𝑓𝑋(𝑘)
(𝑦, 𝜃) =
𝑛!
(𝑘 − 1)! (𝑛 − 𝑘)!
[𝐹(𝑦)] 𝑘−1[1 − 𝐹(𝑦)] 𝑛−𝑘
𝑓(𝑥) . 𝑘 = 1,2,3, … . , 𝑛 (6.1)
Now using the equation (2.2) and (2.3) in (6.1). the probability of 𝑘 𝑡ℎ
order statistics of weighted inverse Maxwell
distribution is given as
𝑓𝑤(𝑘)(𝑦, 𝜃) =
2 𝑛! 𝜃
3−𝑘
2 𝑦 𝑘−4
(𝑘 − 1)! (𝑛 − 𝑘)!
𝑒
−
𝜃
𝑦2
[Γ (
3−𝑘
2
)]
𝑛 [𝛾 (
3 − k
2
,
θ
y2
)]
𝑘−1
[Γ (
3 − 𝑘
2
,
𝜃
𝑦2
)]
𝑛−𝑘
Then, the p.d.f of first order 𝑌1 weighted inverse Maxwell distribution is given as
𝑓1 𝑤
(𝑦, 𝜃) =
2𝑛𝜃
3−𝑘
2 𝑦 𝑘−4
𝑒
−
𝜃
𝑦2
[Γ (
3−𝑘
2
)]
𝑛 [Γ (
3 − 𝑘
2
,
𝜃
𝑦2
)]
𝑛−1
And the p.d.f of nth order 𝑌𝑛 weighted inverse Maxwell distribution is given as
𝑓𝑛 𝑤
(𝑦, 𝜃) =
2𝑛𝜃
3−𝑘
2 𝑦 𝑘−4
𝑒
−
𝜃
𝑦2
[Γ (
3−𝑘
2
)]
𝑛 [𝛾 (
3 − k
2
,
θ
y2
)]
𝑛−1
ESTIMATION OF PARAMETER
Method of Moments
In order to obtain the sample moments of weighted inverse Maxwell distribution, we equate population moments with
sample moments
𝜇1 =
1
𝑛
∑ 𝑦𝑖
𝑛
𝑖=1
𝑌̅ =
𝜃
1
2
Γ (
3−k
2
)
Γ (
2 − 𝑘
2
)
𝜃̂ = [𝑌̅
Γ (
3−k
2
)
Γ (
2−𝑘
2
)
]
1
2
Method of maximum likelihood Estimation:
The estimation of parameters of weighted inverse Maxwell distribution is doing by using the method of maximum likelihood
estimation. Suppose 𝑌1, 𝑌2, 𝑌3 … 𝑌𝑛 be random samples of size n from weighted inverse Maxwell distribution. Then the
likelihood function of weighted inverse Maxwell distribution is given as.
𝑙 = ∏ 𝑓(𝑦𝑖, 𝜃)
𝑛
𝑖=1
= ∏
2 𝜃
3−𝑘
2
Γ (
3−k
2
)
𝑦𝑖
𝑘−4
𝑒
−𝜃 ∑
1
𝑦 𝑖
∞
𝑖=0
𝑛
𝑖=1
(7.1)](https://image.slidesharecdn.com/pdfn-200806053226/85/Weighted-Analogue-of-Inverse-Maxwell-Distribution-with-Applications-6-320.jpg)
Weighted Analogue of Inverse Maxwell Distribution with Applications
- 1. Weighted Analogue of Inverse Maxwell Distribution with Applications IJSM Weighted Analogue of Inverse Maxwell Distribution with Applications *1Ahmad Aijaz,2Afaq Ahmad and 1Rajnee Tripathi 1Department of Mathematics, Bhagwant university, Ajmer, India 2Department of Mathematical Sciences, Islamic University of Science Technology, Awantipora, Kashmir *Corresponding Author: Ahmad Aijaz, Department of Mathematics, Bhagwant university, Ajmer, India Email: ahmadaijaz@gmail.com In the present study, we established a new statistical model named as weighted inverse Maxwell distribution (WIMD). Its several statistical properties including moments, moment generating function, characteristics function, order statistics, shanon entropy has been discussed. The expression for reliability, mode, harmonic mean, hazard rate function has been derived. In addition, it also contains some special cases that are well known. Moreover, the behavior of probability density function (p.d.f) has been shown through graphs by choosing different values of parameters. Finally, the performance of the proposed model is explained through two data sets. By which we conclude that the established distribution provides better fit. Key words: Weighted distribution, inverse Maxwell distribution, Reliability, mode, maximum likelihood estimation. Mathematics Subject Classification: 60E05, 62E15. INTRODUCTION James clark Maxwell and Ludwing Boltzman were known for their work in physics. They have proposed a continuous statistical probability distribution which depicts average distribution matter particles that do not interact with each other from one energy state to another state in thermal equilibrium This famous distribution is commonly known as Maxwell distribution. It has wide applications in physics, chemistry, statistics and mechanics. Let 𝑋 follows the probability distribution function of Maxwell distribution, then 𝑋 = 1 𝑌 is said to follow inverse of Maxwell distribution having p.d.f as. 𝑓(𝑦, 𝜃) = 4 √𝜋 𝜃 3 2 1 𝑦4 𝑒 − 𝜃 𝑦2 𝑦 > 0 , 𝜃 > 0 (1.1) Where 𝜃 is the scale parameter. The corresponding cumulative distribution function ( c.d.f) of (1.1), is given as. 𝐹(𝑦, 𝜃) = 2𝜃 √𝜋 Γ ( 3 2 , θ y2 ) y > 0, 𝜃 > 0 (1.2) Where Γ(𝜃, 𝑦) = ∫ 𝑦 𝜃−1 𝑒−𝑦 𝑑𝑦 ∞ 𝑦 is incomplete gamma function. The kth moment of (1.1), is determined as. 𝐸(𝑦 𝑘) = ∫ 𝑦 𝑘 𝑓(𝑦, 𝜃)𝑑𝑦 ∞ 0 = 4 √ 𝜋 𝜃 3 2 ∫ 𝑦 𝑘 1 𝑦4 𝑒 − 𝜃 𝑦2 𝑑𝑦 ∞ 0 = 4 √ 𝜋 𝜃 3 2 ∫ 𝑦 𝑘−1 ∞ 0 1 𝑦3 𝑒 − 𝜃 𝑦2 𝑑𝑦 Research Article Vol. 7(1), pp. 146-153, July, 2020. © www.premierpublishers.org. ISSN: 2375-0499 International Journal of Statistics and Mathematics
- 2. Weighted Analogue of Inverse Maxwell Distribution with Applications Aijaz et al. 147 Making the substitution 1 𝑦2 = 𝑧 , so that 1 𝑦3 𝑑𝑦 = − 𝑑𝑧 2 and 𝑦 = 1 √𝑧 , we have = 2 √𝜋 𝜃 3 2 ∫ 𝑦( 3−𝑘 2 )−1 ∞ 0 𝑒− 𝜃 𝑧 𝑑𝑧 𝐸(𝑦 𝑘) = 2 √𝜋 𝜃 𝑘 2Γ ( 3 − 𝑘 2 ) (1.3) Weighted inverse Maxwell distribution (WIMD) The weighted distributions have many practical applications to analyze real life data and have been used in various fields such as biomedicine, ecology, reliability. Whenever existing statistical models does not providing a reliable results then weighted analogue is used for the improvement of such distributions. Recently various authors have proposed and analyzed different types of weighted as well as length biased distribution. Das et al introduced the Length biased weighted Weibull distribution (2011). Abd El-Monsef et al (2015), proposed and studied various structural properties of the weighted Kumaraswamy distribution. Afaq et al(2016) worked on the Length biased weighted Lomax distribution. Weighted inverse Rayleigh distribution has been studied by Kawsar fatima et al (2017).Aijaz ahmad dar et al(2018), has introduced characterization and estimation of weighted Maxwell distribution. Mudasir et al (2018), has introduced weighted version of generalized inverse Weibull distribution and studied its several statistical properties. Hesham M et al (2017), has established length biased Erlang distribution and discussed its various properties. In this paper, we create new distribution which is a Weighted Inverse Maxwell distribution (WIMD) and discussing its several statistical properties. Definition: suppose 𝑌 is a continuous random variate with p.d.f 𝑓(𝑦), then p.d.f of weighted variate 𝑌𝑤 is defined as. 𝑓𝑤(𝑦) = 𝑤(𝑦)𝑓(𝑦) 𝐸[𝑤(𝑦)] , 𝑦 > 0 (2.1) Where 𝑤(𝑦) is a non–negative weight function and 𝐸[𝑤(𝑦)] < ∞. Consider the weight function 𝑤(𝑦) = 𝑦 𝑘 for this distribution. The probability distribution function of weighted inverse Maxwell distribution is obtained by using equation (1.1),(1.3) and (2.1), follows as. 𝑓𝑤(𝑦, 𝜃) = 2𝜃 3−𝑘 2 Γ( 3−k 2 ) 𝑦 𝑘−4 𝑒 − 𝜃 𝑦2 , 𝑦 > 0, 𝑘, 𝜃 > 0 (2.2) The cumulative distribution function (c.d.f) of weighted inverse Maxwell distribution is obtained as. 𝐹𝑤(𝑦, 𝜃) = ∫ 2𝜃 3−𝑘 2 Γ ( 3−k 2 ) 𝑦 𝑘−4 𝑒 − 𝜃 𝑦2 𝑑𝑦 𝑦 0 ⇒ 𝐹𝑤(𝑦, 𝜃) = Γ ( 3−k 2 , θ y2) Γ ( 3−k 2 ) (2.3) Where Γ(𝜃, 𝑦) = ∫ 𝑦 𝜃−1 𝑒−𝑦 𝑑𝑦 ∞ 𝑦 is an upper incomplete gamma function. Fig (1.1) and (1.2) illustrates the probability density function of Weighted Inverse Maxwell distribution.
- 3. Weighted Analogue of Inverse Maxwell Distribution with Applications Int. J. Stat. Math. 148 Special Cases of Weighted Inverse Maxwell Distribution Case 1: For = 0, then weighted inverse Maxwell distribution (2.2) reduces to inverse Maxwell distribution with p.d.f as . 𝑓(𝑦, 𝜃) = 4 √𝜋 𝜃 3 2 1 𝑦4 𝑒 − 𝜃 𝑦2 Case 2: For 𝑘 = 1, then weighted inverse Maxwell distribution (2.2) reduces to length biased inverse Maxwell distribution with p.d.f as . 𝑓(𝑦, 𝜃) = 2𝜃 1 𝑦3 𝑒 − 𝜃 𝑦2 Case 3: For = 2,then weighted inverse Maxwell distribution (2.2) reduces to area biased inverse Maxwell distribution with p.d.f as . 𝑓(𝑦, 𝜃) = 2√ 𝜃 𝜋 1 𝑦2 𝑒 − 𝜃 𝑦2 STATISTICAL PROPERTIES Moments of weighted inverse Maxwell distribution Let 𝑋 be a random variable from weighted inverse Maxwell distribution with parameter, then its 𝑟 𝑡ℎ moment is given as. 𝜇 𝑟 = 𝐸(𝑌 𝑟) = ∫ 𝑦 𝑟 ∞ 0 𝑓𝑤(𝑦, 𝜃, 𝑘)𝑑𝑦 Now using (2.2), we get = ∫ 𝑦 𝑟 ∞ 0 2𝜃 3−𝑘 2 Γ ( 3−k 2 ) 𝑦 𝑘−4 𝑒 − 𝜃 𝑦2 𝑑𝑦 = 2𝜃 3−𝑘 2 Γ ( 3−k 2 ) ∫ 𝑦 𝑟+𝑘−1 1 𝑦3 𝑒 − 𝜃 𝑦2 𝑑𝑦 ∞ 0 By making the substitution 𝜃 𝑦2 = 𝑡 sothat 1 𝑦3 𝑑𝑦 = − 1 2𝜃 𝑑𝑡 and 𝑦 = √ 𝜃 𝑡 , then the integration yields as. = 𝜃 𝑟 2 Γ ( 3−k 2 ) ∫ 𝑡 [1−(𝑟+𝑘)] 2 ∞ 0 𝑒−𝑡 𝑑𝑡 = 𝜃 𝑟 2 Γ ( 3−k 2 ) ∫ 𝑡[ 3−(𝑟+𝑘) 2 ]−1 𝑒−𝑡 𝑑𝑡 ∞ 0 Therefore 𝜇 𝑟 = 𝐸(𝑌 𝑟) = 𝜃 𝑟 2 Γ ( 3−k 2 ) Γ ( 3 − (𝑟 + 𝑘) 2 ) Substituting 𝑟 = 1,2 we get 𝑴𝒆𝒂𝒏 = 𝐸(𝑌) = 𝜃 1 2 Γ ( 3−k 2 ) Γ ( 2 − 𝑘 2 ) 𝐸(𝑌2) = 𝜃 Γ ( 3−k 2 ) Γ ( 1 − 𝑘 2 ) Variance = 𝐸(𝑌2) − [𝐸(𝑌)]2 = 𝜃 Γ( 3−k 2 ) Γ ( 1−𝑘 2 ) − [ 𝜃 1 2 Γ( 3−k 2 ) Γ ( 2−𝑘 2 )] 2 Moment generating function of weighted inverse Maxwell distribution Let y be a random variable from weighted inverse Maxwell distribution, then the moment generating function of ydenoted by 𝑀 𝑌(𝑡) is given as.
- 4. Weighted Analogue of Inverse Maxwell Distribution with Applications Aijaz et al. 149 𝑀 𝑌(𝑡) = 𝐸(𝑒 𝑡𝑦) = ∫ 𝑒 𝑡𝑦 𝑓𝑤(𝑦)𝑑𝑦 ∞ 0 = ∫ (1 + 𝑡𝑦 + (𝑡𝑦)2 2! + ⋯ ) 𝑓𝑤(𝑦)𝑑𝑦 ∞ 0 = ∫ ∑ 𝑡 𝑟 𝑟! ∞ 𝑟=0 ∞ 0 𝑦 𝑟 𝑓𝑤(𝑦)𝑑𝑦 = ∑ 𝑡 𝑟 𝑟! ∞ 𝑟=0 ∫ 𝑦 𝑟 𝑓𝑤(𝑦)𝑑𝑦 ∞ 0 = ∑ 𝑡 𝑟 𝑟! ∞ 𝑟=0 𝜃 𝑟 2 Γ ( 3−k 2 ) Γ ( 3 − (𝑟 + 𝑘) 2 ) Characteristics function of weighted inverse Maxwell distribution Let y be a random variable from weighted inverse Maxwell distribution, then the characteristics function of y denoted by 𝜙 𝑌(𝑡) is given as. 𝜙 𝑌(𝑡) = 𝐸(𝑒 𝑖𝑡𝑦) = ∫ 𝑒 𝑖𝑡𝑦 𝑓𝑤(𝑦) ∞ 0 𝑑𝑦 = ∫ (1 + 𝑖𝑡𝑦 + (𝑖𝑡𝑦)2 2! + ⋯ ) 𝑓𝑤(𝑦)𝑑𝑦 ∞ 0 = ∫ ∑ (𝑖𝑡) 𝑟 𝑟! ∞ 𝑟=0 ∞ 0 𝑦 𝑟 𝑓𝑤(𝑦)𝑑𝑦 = ∑ (𝑖𝑡) 𝑟 𝑟! ∞ 𝑟=0 ∫ 𝑦 𝑟 𝑓𝑤(𝑦)𝑑𝑦 ∞ 0 = ∑ (𝑖𝑡) 𝑟 𝑟! ∞ 0 𝜃 𝑟 2 Γ ( 3−k 2 ) Γ ( 3 − (𝑟 + 𝑘) 2 ) Harmonic mean of weighted inverse Maxwell distribution The harmonic mean (H) is given as: 1 𝐻 = 𝐸 ( 1 𝑌 ) = ∫ 1 𝑦 𝑓𝑤(𝑦) ∞ 0 𝑑𝑦 1 𝐻 = ∫ 1 𝑦 ∞ 0 2𝜃 3−𝑘 2 Γ ( 3−k 2 ) 𝑦 𝑘−4 𝑒 − 𝜃 𝑦2 𝑑𝑦 = 2𝜃 3−𝑘 2 Γ ( 3−k 2 ) ∫ 𝑦 𝑘−5 𝑒 − 𝜃 𝑦2 𝑑𝑦 ∞ 0 Making the substitution 𝜃 𝑦2 = 𝑧, after solving the integral we get. 1 𝐻 = 1 𝜃 1 2 Γ( 3−𝑘 2 ) Γ ( 4 − 𝑘 2 ) Mode of weighted inverse Maxwell distribution Taking the log of weighted inverse Maxwell distribution, we get ln 𝑓𝑤(𝑦, 𝜃) = ln 2𝜃 3−𝑘 2 − ln Γ ( 3 − 𝑘 2 ) + (𝑘 − 4) ln 𝑦 − 𝜃 𝑦 (4.1) Differentiate (4.1),w.r.t y we get 𝜕 ln 𝑓𝑤(𝑦, 𝜃) 𝜕𝑦 = 𝑘 − 4 𝑦 + 2𝜃 𝑦3 (3.1) Equating (3.1) to zero and solve for y, we obtain 𝑦0 = √( 2𝜃 4 − 𝑘 )
- 5. Weighted Analogue of Inverse Maxwell Distribution with Applications Int. J. Stat. Math. 150 Reliability measures Suppose Y be a continuous random variable with c.d.f 𝐹𝑤(𝑦) , 𝑦 ≥ 0 .then its reliability function which is also called survival function is defined as 𝑆 𝑤(𝑦) = 𝑝𝑟(𝑌 > 𝑦) = ∫ 𝑓𝑤(𝑦) ∞ 0 𝑑𝑦 = 1 − 𝐹𝑤(𝑦) The survival function of weighted inverse Maxwell distribution is given as 𝑆 𝑤(𝑦) = 1 − Γ ( 3−k 2 , θ y2) Γ ( 3−k 2 ) = 𝛾 ( 3−k 2 , θ y2) Γ ( 3−k 2 ) (4.1) The hazard rate function of the random variable y is given as 𝐻 𝑤(𝑦) = 𝑓𝑤(𝑦) 𝑆 𝑤(𝑦) ( 4.2 ) Substituting (2.2) and(4.1),into (4.2), we get. 𝐻 𝑤(𝑦) = 2𝜃 3−𝑘 2 𝑦 𝑘−4 𝑒 − 𝜃 𝑦2 𝛾 ( 3−k 2 , θ y2) Shannon’s Entropy of weighted inverse Maxwell distribution The concept of information entropy was introduced by Shanon in 1948. The entropy can be interpreted as the average rate at which information is produced by a random source of data and is given by 𝐻 𝑤(𝑥, 𝜃) = −𝐸[log 𝑓𝑤(𝑥, 𝜃)] = −𝐸 [log ( 2𝜃 3−𝑘 2 Γ ( 3−k 2 ) 𝑦 𝑘−4 𝑒 − 𝜃 𝑦2 )] = − log ( 2𝜃 3−𝑘 2 Γ ( 3−k 2 ) ) − (𝑘 − 4)𝐸(log 𝑦) + 𝐸 ( 𝜃 𝑦2 ) (5.1) Now 𝐸(log 𝑦) = ∫ log 𝑦 𝑓𝑤(𝑦) ∞ 0 𝑑𝑦 = ∫ log 𝑦 ∞ 0 2𝜃 3−𝑘 2 Γ ( 3−k 2 ) 𝑦 𝑘−4 𝑒 − 𝜃 𝑦2 𝑑𝑦 = 2𝜃 3−𝑘 2 Γ ( 3−k 2 ) ∫ log 𝑦 𝑦 𝑘−4 𝑒 − 𝜃 𝑦2 𝑑𝑦 ∞ 0 Making the substitution 𝜃 𝑦2 = 𝑡 so that 1 𝑦3 𝑑𝑦 = − 1 2𝜃 𝑑𝑡, and 𝑦 = ( 𝜃 𝑡 ) 1 2 = 1 Γ ( 3−k 2 ) ∫ log ( 𝜃 𝑡 ) ∞ 0 1 2 𝑡 1−𝑘 2 𝑒−𝑡 𝑑𝑡 After solving the integral, we get = 1 2 [log 𝜃 − 𝜓 ( 3 − 𝑘 2 )] (5.2) Also 𝐸 ( 𝜃 𝑦2) = 𝜃 ∫ 1 𝑦2 ∞ 0 𝑓𝑤(𝑦 , 𝜃)𝑑𝑦 = 2𝜃 3−𝑘 2 𝜃 Γ ( 3−k 2 ) ∫ 1 𝑦2 𝑦 𝑘−4 𝑒 − 𝜃 𝑦2 𝑑𝑦 ∞ 0
- 6. Weighted Analogue of Inverse Maxwell Distribution with Applications Aijaz et al. 151 Making substitution 𝜃 𝑦2 = 𝑡 , we get = 1 Γ ( 3−k 2 ) ∫ 𝑡 3−𝑘 2 ∞ 0 𝑒−𝑡 𝑑𝑡 Therefore 𝐸 ( 𝜃 𝑦2) = 3−𝑘 2 (5.3) Substituting the values (5.2), (5.3) in (5.1), we get 𝐻 𝑤(𝑥, 𝜃) = − log 2𝜃 3−𝑘 2 𝜃 Γ ( 3−k 2 ) − 𝑘 − 2 2 [log 𝜃 − 𝜓 ( 3 − 𝑘 2 )] + ( 3 − 𝑘 2 ) Where 𝜓(. ) denotes the digamma function. Order statistics of weighted inverse Maxwell distribution Let us suppose 𝑌1, 𝑌2, 𝑌3 … , 𝑌𝑛 be random samples of size n from weighted inverse Maxwell distribution with p.d.f 𝑓(𝑦) and c.d.f 𝐹(𝑦). Then the probability density function of 𝑘 𝑡ℎ order statistics is given as. 𝑓𝑋(𝑘) (𝑦, 𝜃) = 𝑛! (𝑘 − 1)! (𝑛 − 𝑘)! [𝐹(𝑦)] 𝑘−1[1 − 𝐹(𝑦)] 𝑛−𝑘 𝑓(𝑥) . 𝑘 = 1,2,3, … . , 𝑛 (6.1) Now using the equation (2.2) and (2.3) in (6.1). the probability of 𝑘 𝑡ℎ order statistics of weighted inverse Maxwell distribution is given as 𝑓𝑤(𝑘)(𝑦, 𝜃) = 2 𝑛! 𝜃 3−𝑘 2 𝑦 𝑘−4 (𝑘 − 1)! (𝑛 − 𝑘)! 𝑒 − 𝜃 𝑦2 [Γ ( 3−𝑘 2 )] 𝑛 [𝛾 ( 3 − k 2 , θ y2 )] 𝑘−1 [Γ ( 3 − 𝑘 2 , 𝜃 𝑦2 )] 𝑛−𝑘 Then, the p.d.f of first order 𝑌1 weighted inverse Maxwell distribution is given as 𝑓1 𝑤 (𝑦, 𝜃) = 2𝑛𝜃 3−𝑘 2 𝑦 𝑘−4 𝑒 − 𝜃 𝑦2 [Γ ( 3−𝑘 2 )] 𝑛 [Γ ( 3 − 𝑘 2 , 𝜃 𝑦2 )] 𝑛−1 And the p.d.f of nth order 𝑌𝑛 weighted inverse Maxwell distribution is given as 𝑓𝑛 𝑤 (𝑦, 𝜃) = 2𝑛𝜃 3−𝑘 2 𝑦 𝑘−4 𝑒 − 𝜃 𝑦2 [Γ ( 3−𝑘 2 )] 𝑛 [𝛾 ( 3 − k 2 , θ y2 )] 𝑛−1 ESTIMATION OF PARAMETER Method of Moments In order to obtain the sample moments of weighted inverse Maxwell distribution, we equate population moments with sample moments 𝜇1 = 1 𝑛 ∑ 𝑦𝑖 𝑛 𝑖=1 𝑌̅ = 𝜃 1 2 Γ ( 3−k 2 ) Γ ( 2 − 𝑘 2 ) 𝜃̂ = [𝑌̅ Γ ( 3−k 2 ) Γ ( 2−𝑘 2 ) ] 1 2 Method of maximum likelihood Estimation: The estimation of parameters of weighted inverse Maxwell distribution is doing by using the method of maximum likelihood estimation. Suppose 𝑌1, 𝑌2, 𝑌3 … 𝑌𝑛 be random samples of size n from weighted inverse Maxwell distribution. Then the likelihood function of weighted inverse Maxwell distribution is given as. 𝑙 = ∏ 𝑓(𝑦𝑖, 𝜃) 𝑛 𝑖=1 = ∏ 2 𝜃 3−𝑘 2 Γ ( 3−k 2 ) 𝑦𝑖 𝑘−4 𝑒 −𝜃 ∑ 1 𝑦 𝑖 ∞ 𝑖=0 𝑛 𝑖=1 (7.1)
- 7. Weighted Analogue of Inverse Maxwell Distribution with Applications Int. J. Stat. Math. 152 The log likelihood function of (7.1), is given as ln 𝑙 = ln ( 2 𝜃 3−𝑘 2 Γ ( 3−k 2 ) ) 𝑛 + ∑ ln 𝑦𝑖 𝑘−4 − ∑ 𝜃 𝑦𝑖 2 𝑛 𝑖=1 𝑛 𝑖=1 = 𝑛 ln 2 + 𝑛 (𝜃 3−𝑘 2 ) ln 𝜃 − 𝑛 ln Γ ( 3 − k 2 ) + (𝑘 − 4) ∑ ln 𝑦𝑖 − 𝜃 ∑ 1 𝑦𝑖 2 𝑛 𝑖=1 𝑛 𝑖=1 Differentiate w.r.t 𝜃 , we get 𝜕 ln 𝑙 𝜕𝜃 = 𝑛(3 − 𝑘) 2𝜃 − ∑ 1 𝑦𝑖 2 𝑛 𝑖=1 (7.2) Now equating (7.2), to zero, we get ⇒ 𝑛(3 − 𝑘) 2𝜃 − ∑ 1 𝑦𝑖 2 𝑛 𝑖=1 = 0 𝜃̂ = 𝑛(3 − 𝑘) 2 ∑ 𝑦𝑖 2 𝑛 𝑖 DATA ANALYSIS Data 1: In this section we provide an application which explains the performance of the newly developed distribution. The data set has been taken from Gross and Clark (1975), which signifies the relief times of 20 patients getting an analgesic. We use previous data to associate the fit of the newly developed model with inverse Maxwell distribution. The data are follows. 1.1,1.4,1.3,1.7,1.9,1.8,1.6,2.2,1.7,2.7,4.1,1.8,1.5,1.2,1.4,3.0,1.7,2.3,1.6,2.0. In order to compare the two distribution models, we consider the criteria like AIC (Akaike information criterion, AICC (corrected Akaike information criterion) and BIC (Bayesian information criterion. The better distribution corresponds to lesser AIC, AICC and BIC values. 𝐴𝐼𝐶 = −2𝑙𝑛𝐿 + 2𝑘, 𝐴𝐼𝐶𝐶 = 𝐴𝐼𝐶 + 2𝑘(𝑘+1) (𝑛−𝑘−1) ,𝐵𝐼𝐶 = −2𝑙𝑛𝐿 + 𝑘𝑙𝑛 Table 1: ML estimates and Criteria for Comparison Distribution Estimates Standard Error -2logL AIC AICC BIC Weighted inverse Maxwell distribution 0.09297414 1.65287637 0.1146933 0.2608437 43.90867 45.90867 46.1308922 45.2097 Inverse Maxwell distribution 0.2070503 0.1690519 69.44787 71.44787 71.6700922 70.7489 Data 2: The data set is on the breaking stress of carbon fibres of 50 mm length (GPa). The data has been previously used by Cordeiro and Lemonte (2011) and Al-Aqtash et al.(2014). The data is as follows: 0.39, 0.85, 1.08, 1.25, 1.47, 1.57, 1.61, 1.61, 1.69, 1.80, 1.84, 1.87, 1.89, 2.03, 2.03, 2.05, 2.12, 2.35, 2.41, 2.43, 2.48, 2.50, 2.53, 2.55, 2.55, 2.56, 2.59, 2.67, 2.73, 2.74, 2.79, 2.81, 2.82, 2.85, 2.87, 2.88, 2.93, 2.95, 2.96, 2.97, 3.09, 3.11, 3.11, 3.15, 3.15, 3.19, 3.22, 3.22, 3.27, 3.28, 3.31, 3.31, 3.33, 3.39, 3.39, 3.56, 3.60, 3.65, 3.68, 3.70, 3.75, 4.20, 4.38, 4.42, 4.70, 4.90. Table 2:ML estimates and Criteria for Comparison Distribution Estimates Standard Error -2logL AIC AICC BIC Weighted Inverse Distribution 0.02658982 1.99391471 0.03755243 0.11173997 238.6447 240.6447 240.866922 240.464244 Inverse Maxwell Distribution 0.07914596 0.06461209 403.0806 405.0806 405.302822 404.900144 From Table 1 and 2, it has been observed that the Weighted inverse Maxwell model have the lesser AIC, AICC, - 2logL and BIC values as compared to inverse Maxwell distribution. Hence, we can conclude that Weighted inverse Maxwell distribution leads to a better fit as compared to inverse Maxwell model.
- 8. Weighted Analogue of Inverse Maxwell Distribution with Applications Aijaz et al. 153 CONCLUSION In this paper the weighted inverse Maxwell distribution has been established. Many times, classical distributions do not provide better results for fitting different data taken from various fields.The weighted distributions in such conditions provide an adequate result for analyzing data. Some statistical properties including moments, moment generating function, characteristics function, mode, shanon entropy, survival function, hazard rate function has been discussed. The estimation of the parameters of the established distribution has been estimated by the method of moments and maximum likelihood estimator. Finally, the performance of the established distribution has been examined through two data sets. The established distribution leads the compared one. REFERENCES Afaq. A . Ahmad S.P, Ahmad A.(2016), Length-Biased Weighted Lomax Distribution Statistical Properties and Application. Pak .j .stat.oper.res, vol XII No.2, pp245-255. Abdl .M.M.E El-Monsef and Ghonien .A.E (2015).The Weighted Kumaraswamy Distribution.International Information Institute. Volume 18 , No. 8, pp. 3289-3300. Aqtash, A., Lee, C and Famoye, F, “Gumbel-weibull distribution: Properties and applications”, Journal of Modern Applied Statistical Methods, vol.13, pp.201-225, (2014). Aijaz. A. D, Ahmed.A and Reshi.J.A, (2018) , Applied Mathematics and Information Sciences An International Journal. Characterization and Estimation of Weighted Maxwell-Boltzmann Distribution,12 nNo.1,193-203. Bekker, A., Roux, J.J.J., 2005. Reliability characteristics of the Maxwell distribution: a bayes estimation study. Communication in Statistics- Theory and Methods 34, 2169–2178. Cordeiro, G.M and Lemonte, A.J. “The β-Birnbaum-Saunders distribution: An improved distribution for fatigue life modeling”, Computational Statistics and Data Analysis, vol.55, pp.1445-1461, (2011). Das.K.K, Roy.T.D,(2011). On some Length –biased Weighted Weibull distribution , Advances in Applied science Research ,2,No.5,465-475. Gross, A. J., Clark, V. A., Survival Distributions: Reliability Applications in the Biometrical Sciences, John Wiley, New York, (1975). Kawsar.Fand Ahmad.S.P (2017).Weighted Inverse Rayleigh Distribution. International Journal of Statistics and System, Vol 12 No.1, 119-137. Mudasir. S, Ahmad.S.P (2018). Weighted Version Of Generalized Inverse Weibull Distribution. Journal Of Modern Applied Statistical Methods, Vol 17(2) ep2691. Sharma, V.K., Dey S., Singh, S.K &Manzoor. U (2017), “On Length and Area biased Maxwell distributions”, Communications in Statistics - Simulation andComputation. Hesham. M.R, Soha.A.O and Alaaed. A.M (2017). The Length Biased Erlang Distribution. Asian Research Journal Of Mathematics,6(3); 1-15. Accepted 21 July 2020 Citation: Aijaz A, Ahmad A, Tripathi R(2020). Weighted Analogue of Inverse Maxwell Distribution with Applications. International Journal of Statistics and Mathematics, 7(1): 146-153. Copyright: © 2020Aijaz et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are cited.