Parameter Estimation of the Weighted Generalized Inverse Weibull Distribution

Sofi Mudasir; S.P. Ahmad

doi:10.2991/jsta.d.210607.002

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Volume 20, Issue 2, June 2021, Pages 395 - 406

Parameter Estimation of the Weighted Generalized Inverse Weibull Distribution

Authors

Sofi Mudasir^*, S.P. Ahmad

Department of Statistics, University of Kashmir, Srinagar 190 006, India

^*Corresponding author. Email: sofimudasir3806@gmail.com

Corresponding Author

Sofi Mudasir

Received 2 August 2018, Accepted 28 November 2020, Available Online 10 July 2021.

DOI: 10.2991/jsta.d.210607.002 How to use a DOI?
Keywords: Generalized inverse Weibull distribution; Weighted generalized inverse Weibull distribution; Loss function; Bayesian estimation
Abstract: Weighted distributions are used widely in many fields of real life such as medicine, ecology, reliability, and so on. The idea of weighted distributions was given by Fisher and studied by Rao in a unified manner who pointed out that in many situations the recorded observations cannot be considered as a random sample from the original distribution. This can be due to nonobservability of some events, damage caused to the original observations or adoption of unequal probability sampling procedure. In this paper, we have proposed weighted version of generalized inverse Weibull distribution known as weighted generalized inverse Weibull distribution (WGIWD). Classical and Bayesian methods of estimation were proposed for estimating the parameters of the new model. The usefulness of the new model was demonstrated by applying it to a real-life data set.
Copyright: © 2021 The Authors. Published by Atlantis Press B.V.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

In many observational studies for wild life, human, fish population or insect, every unit in the population does not have the same chance of being included in the sample. In such cases, sampling frames are not well defined and recorded observations are biased. These observations don't follow the parent distribution and hence their modeling gives birth to the theory of weighted distributions. Fisher [1] and Rao [2] introduced and unified the concept of weighted distribution. Rao identified various situations that can be modeled by weighted distributions. These situations refer to instances where the recorded observations cannot be considered as a random sample from the original distributions. This may occur due to nonobservability of some events or damage caused to the original observation, or adoption of unequal probability sampling procedure. Weighted distributions were used frequently in research related to reliability, biomedicine, ecology and branching processes can be seen in Patil and Rao [3], Gupta and Kirmani [4], Gupta and Keating [5], Oluyede [6] and in references there in. There are many researchers for weighted distribution as Das and Roy [7] discussed the length-biased weighted generalized Rayleigh distribution with its properties, Sofi et al. [8] studied the structural properties of length-biased Nakagami distribution. For more important results of weighted distribution see Oluyede and George [9], Ghitany and Al-Mutairi [10], Ahmed, Reshi and Mir [11], Sofi et al. [12].

Suppose X is a nonnegative random variable with probability density function fx, then the probability density function of the weighted random variable is given by

fwx=wxfxμw, x>0,(1)

where wx be a nonnegative weight function and μw=Ewx<∞

The probability density function of generalized inverse Weibull distribution is given by

fx=λβαβxβ+1exp−λαxβ x>0;α,β,λ>0(2)

where α and β are called scale and shape parameters, respectively.

Let

wx=xθ, θ>0≤(3)

Now, μw=∫0∞wxfxdx

⇒ μw=αθλθβΓ1−θβ, θβ<1(4)

Substitute the value of Equations (2)–(4) in Equation (1), we get

fwx=βαβ−θλ1−θβxθ−β−1exp−λαxβΓ1−θβ,x>0;α,β,θ,λ>0 and θ<β(5)

The density function in Equation (5) is known as weighted generalized inverse Weibull distribution (WGIWD).

Also the cumulative distribution function (cdf) of weighted generalized inverse Weibull distribution (WGIWD) is

Fwx=Γ1−θβ,λαβxβΓ1−θβ

2. MAXIMUM LIKELIHOOD ESTIMATION

Let x1,x2,…,xn be a random sample from (5), then the likelihood function is given by

L=βαβ−θλ1−θβΓ1−θβn∏i=1nxiθ−β−1exp−λαβ∑i=1nxi−β

⇒ logL=nlogβ+n(β−θ)logα+n(1−θβ)logλ−nlogΓ1−θβ+θ−β−1∑i=1nlogxi−λαβ∑i=1nxi−β

Now differentiate the above equation with respect to α and equate to zero, we get

α^=nβ−θλβ∑i=1nxi−β1β

This is the required MLE of α

3. PARAMETER ESTIMATION UNDER SQUARED ERROR LOSS FUNCTION

In this section two different prior distributions namely Jeffrey's prior and extension of Jeffrey's prior are used for estimating the scale parameter of the WGIWD.

3.1. Using Jeffrey's Prior

Consider that the parameter α has Jeffrey's prior given by

π1α∝detIα, α>0

where Iα is the fisher information matrix obtained as

Iα=−nE∂2logfwx∂α2=nββ−θα2

Therefore, the Jeffrey's prior distribution is defined by

π1α∝1α(6)

The posterior distribution using Jeffrey's prior is obtained by using Bayes theorem given by

P1α|x∝Lα|xπ1α

⇒ P1α|x=kβλ1−θβΓ1−θβnηθ−β−1αnβ−θ−1exp−λαβT(7)

where η=∏i=1nxi, T=∑i=1nxi−β and k is the normalizing constant and is given by

k=Γn1−θβTn1−θβΓn1−θββn−1ηθ−β−1

With this value of k we get from (7) the posterior distribution as

P1α|x=βλTn1−θβΓn1−θβαnβ−θ−1exp−λαβT(8)

By using squared error loss function Lα^,α=cα^−α2, for some constant c, the risk function is given by

Rα^=cα^2+cΓnβ−θ+2βΓn1−θβλT2β−2cα^Γnβ−θ+1βΓn1−θβλT1β

The Bayes estimator α^ is the solution of the equation ∂Rα^∂α^=0, which results in

α^=Γnβ−θ+1βΓnβ−θβλT1β(9)

3.2. Using Extension of Jeffrey's Prior

The extension of Jeffrey's prior relating to the scale parameter α is given as

π2α∝detIαC1,C1∈R+

where Iα is same as in Jeffrey's prior.

⇒ π2α∝1α2C1

The posterior distribution using extension of Jeffrey's prior is obtained by using the same procedure as in case of Jeffrey's prior and is given by

P2α|x=βλTnβ−θ−2C1+1βΓnβ−θ−2C1+1βαnβ−θ−2C1exp−λαβT(10)

By using squared error loss function Lα^,α=cα^−α2, for some constant c, the risk function is given by

Rα^=cα^2+cΓnβ−θ−2C1+3βΓnβ−θ−2C1+1βλT2β−2cα^Γnβ−θ−2C1+2βΓnβ−θ−2C1+1βλT1β

The Bayes estimator α^ is the solution of the equation ∂Rα^∂α^=0, which results in

α^=Γnβ−θ−2C1+2βλT1βΓnβ−θ−2C1+1β(11)

Remark 1.

If C1=12 in (11), the same Bayes estimator is obtained as in (9) corresponding to Jeffrey's prior.

4. PARAMETER ESTIMATION UNDER QUADRATIC LOSS FUNCTION

In this section we use quadratic loss function to obtain Bayes estimators using Jeffrey's and extension of Jeffrey's prior information.

The quadratic loss function is defined as

Lα^,α=α−α^α2

4.1. Using Jeffrey's Prior

By using the quadratic loss function Lα^,α=α−α^α2, the risk function is given by

Rα^=1+α^2λT2βΓnβ−θ−2βΓn1−θβ−2α^λT1βΓnβ−θ−1βΓn1−θβ

The Bayes estimator α^ is the solution of the equation ∂Rα^∂α^=0, which results in

α^=Γnβ−θ−1βλT1βΓnβ−θ−2β(12)

4.2. Using Extension of Jeffrey's Prior

Taking the posterior distribution (11) and by using the quadratic loss function, the risk function is given by

Rα^=1+α^2λT2βΓnβ−θ−2C1−1βΓnβ−θ−2C1+1β−2α^λT1βΓnβ−θ−2C1βΓnβ−θ−2C1+1β

The Bayes estimator α^ is the solution of the equation ∂Rα^∂α^=0, which results in

α^=Γnβ−θ−2C1βλT1βΓnβ−θ−2C1−1β(13)

Remark 2.

Replacing C1=12 in (13), the same Bayes estimator is obtained as in (12) corresponding to Jeffrey's prior.

5. PARAMETER ESTIMATION UNDER NEW LOSS FUNCTION

In this section we obtain the Bayes estimators under new loss function introduced by Al-Bayyati [13] using Jeffrey's and extension of Jeffrey's prior

The Al-Bayyati's new loss function also called new loss function is of the form

Lα^,α=αC2α^−α2,C2∈R

Here we use this loss function to obtain the Bayes estimator of the scale parameter α of the WGIWD.

5.1. Using Jeffrey's Prior

By using the new loss function Lα^,α=αC2α^−α2, the risk function is given by

Rα^=α^2Γnβ−θ+C2βλTC2βΓn1−θβ+Γnβ−θ+C2+2βλTC2+2βΓn1−θβ−2α^Γnβ−θ+C2+1βλTC2+1βΓn1−θβ

The Bayes estimator α^ is the solution of the equation ∂Rα^∂α^=0, which results in

α^=Γnβ−θ+C2+1βλT1βΓnβ−θ+C2β(14)

Remark 3.

The Bayes estimator obtained under Jeffrey's prior given in (14) coincides with the Bayes estimator given in (9) if C2=0 in (14).

5.2. Using Extension of Jeffrey's Prior

By taking the posterior distribution (11) and using the new loss function, the risk function is given by

Rα^=α^2Γnβ−θ−2C1+C2+1βλTC2βΓnβ−θ−2C1+1β+Γn(β−θ)−2C1+C2+3βλTC2+2βΓn(β−θ)−2C1+1β−2α^Γnβ−θ−2C1+C2+2βλTC2+1βΓnβ−θ−2C1+1β

The Bayes estimator α^ is the solution of the equation ∂Rα^∂α^=0, which results in

α^=Γnβ−θ−2C1+C2+2βλT1βΓnβ−θ−2C1+C2+1β(15)

Remark 4.

If C1=12 in (15), the same Bayes estimator is obtained as in (14) corresponding to Jeffrey's prior and if C2=0 in (15) the Bayes estimator coincides with the Bayes estimator given in (11).

6. POSTERIOR MEAN AND POSTERIOR VARIANCE OF SCALE PARAMETER UNDER JEFFREY'S AND EXTENSION OF JEFFREY'S PRIORS

In this section, we calculate the posterior mean and posterior variance of the scale parameter α under Jeffrey's and extension of Jeffrey's Prior distribution.

6.1. Posterior Mean and Posterior Variance of Under Jeffrey's Prior

We have the posterior distribution under Jeffrey's prior as

P1α|x=βλTn1−θβΓn1−θβαnβ−θ−1exp−λαβT(16)

Now

Eαr=∫0∞αrP1α|xdα(17)

By using Equation (16) in Equation (17), we get

Eαr=Γnβ−θ+rβλTrβΓnβ−θβ(18)

If r = 1 in (18), we get

E(α)=Γnβ−θ+1βλT1βΓnβ−θβ

This is the posterior mean

If r = 2 in (18), we get

Eα2=Γnβ−θ+2βλT2βΓnβ−θβ

Thus the posterior variance is given by

vα=Γnβ−θβΓnβ−θ+2β−Γ2nβ−θ+1βλT2βΓ2nβ−θβ

6.2. Posterior Mean and Posterior Variance of Under Extension of Jeffrey's Prior

We have the posterior distribution under extension of Jeffrey's prior as

P2α|x=βλTnβ−θ−2C1+1βΓnβ−θ−2C1+1βαnβ−θ−2C1exp−λαβT(19)

Now

Eαr=∫0∞αrP2α|xdα(20)

By using Equation (19) in Equation (20), we get

Eαr=Γnβ−θ−2C1+r+1βλTrβΓnβ−θ−2C1+1β(21)

If r = 1 in (21), we get

Eα=Γnβ−θ−2C1+2βλT1βΓnβ−θ−2C1+1β

This is the posterior mean

If r = 2 in (21), we get

Eα2=Γnβ−θ−2C1+3βλT2βΓnβ−θ−2C1+1β

Thus the posterior variance is given by

⇒ vα=Γnβ−θ−2C1+1βΓnβ−θ−2C1+3β−Γ2nβ−θ−2C1+2βλT2βΓ2nβ−θ−2C1+1β

Remark 5.

If C1=12 then the posterior mean and posterior variance obtained under extension of Jeffrey's prior coincides with the posterior mean and posterior variance obtained under Jeffrey's prior.

7. DATA ANALYSIS

In this section we analyze real-life data set for illustration given by Lee and Wang [14] which represent remission times (in months) of a random sample of 128 bladder cancer patients (Table 1). A program has been developed in R language to obtain the Bayes estimates and posterior risks. The data are as follows: 0.08, 2.09, 3.48, 4.87, 6.94, 8.66, 13.11, 23.63, 0.20, 2.23, 3.52, 4.98, 6.97, 9.02, 13.29, 0.40, 2.26, 3.57, 5.06, 7.09, 9.22, 13.80, 25.74, 0.50, 2.46, 3.64, 5.09, 7.26, 9.47, 14.24, 25.82, 0.51, 2.54, 3.70, 5.17, 7.28, 9.74, 14.76, 26.31, 0.81, 2.62, 3.82, 5.32, 7.32, 10.06, 14.77, 32.15, 2.64, 3.88, 5.32, 7.39, 10.34, 14.83, 34.26, 0.90, 2.69, 4.18, 5.34, 7.59, 10.66, 15.96, 36.66, 1.05, 2.69, 4.23, 5.41, 7.62, 10.75, 16.62, 43.01, 1.19, 2.75, 4.26, 5.41, 7.63, 17.12, 46.12, 1.26, 2.83, 4.33, 5.49, 7.66, 11.25, 17.14, 79.05, 1.35, 2.87, 5.62, 7.87, 11.64, 17.36, 1.40, 3.02, 4.34, 5.71, 7.93, 11.79, 18.10, 1.46, 4.40, 5.85, 8.26, 11.98, 19.13, 1.76, 3.25, 4.50, 6.25, 8.37, 12.02, 2.02, 3.31, 4.51, 6.54, 8.53, 12.03, 20.28, 2.02, 3.36, 6.76, 12.07, 21.73, 2.07, 3.36, 6.93, 8.65, 12.63, 22.69.

Min.	1st Qu.	Median	Mean	3rd Qu.	Max.	Standard deviation	Skewness	Kurtosis
0.080	3.348	6.395	9.366	11.840	79.050	10.50833	3.286569	18.48308

Table 1

Descriptive statistics for the above real data set.

By using different loss functions, that is, SELF, QLF and NLF, the Bayes estimates and posterior risk through Jeffrey's and extension of Jeffrey's priors are presented in the tables below where posterior risk are in parentheses.

8. CONCLUSION

In this paper, we have primarily estimate the scale parameter of the new model known as WGID under Jeffrey's and extension of Jeffrey's prior distributions assuming different loss functions. For comparison, we use the real-life data set and the results are shown in the tables above.

Tables 2 and 3 show the estimates and the posterior risk in parentheses of the scale parameter α for different values of the parametersunder the Jeffrey's and extension of Jeffrey's priors. From the tables it is clear that the estimates and posterior risks obtained under extension of Jeffrey's prior coincides with the estimates obtained under Jeffrey's prior when the value of hyper-parameter C1=12. Also on comparing the posterior risks under different loss functions, it is observed that NLF has less value of posterior risk than other loss functions. Thus we conclude that in our case it is the NLF which is more preferable than other loss functions.

λ	β	θ	C	C2	MLE	SELF	QLF	NLF
1.05	1.5	0.5	1	0.5	0.7981785	0.7971406	0.7887990	0.7992192
	1.5	0.5	1	0.5	0.7981785	(0.003311692)	(0.005266604)	(0.0029587057)
	2.0	0.5	3	1.0	0.8956767	0.8945112	0.8898279	0.8968437
	2.0	0.5	3	1.0	0.8956767	(0.006259293)	(0.002628107)	(0.0018663487)
	2.5	1.0	4	1.5	0.8374464	0.8361384	0.8326298	0.8387554
	2.5	1.0	4	1.5	0.8374464	(0.005839690)	(0.002105796)	(0.0011153567)
	3.5	1.5	6	2.0	0.8687908	0.8675784	0.8656269	0.8695190
	3.5	1.5	6	2.0	0.8687908	(0.005057925)	(0.001127539)	(0.0006331024)
3.97	1.5	0.5	1	0.5	0.3288763	0.3284486	0.3250116	0.3293051
	1.5	0.5	1	0.5	0.3288763	(0.0005622312)	(0.005266605)	(0.0003224283)
	2.0	0.5	3	1.0	0.4606284	0.4600290	0.4576205	0.4612285
	2.0	0.5	3	1.0	0.4606284	(0.0016554805)	(0.002628793)	(0.0002538581)
	2.5	1.0	4	1.5	0.4919450	0.4911766	0.4891155	0.4927139
	2.5	1.0	4	1.5	0.4919450	(0.0020151567)	(0.002105852)	(0.0001732894)
	3.5	1.5	6	2.0	0.5941366	0.5933075	0.5919729	0.5946346
	3.5	1.5	6	2.0	0.5941366	(0.0023654540)	(0.001127542)	(0.0001384708)
4.12	1.5	0.5	1	0.5	0.3208446	0.3204274	0.3170743	0.3212629
	1.5	0.5	1	0.5	0.3208446	(0.0005351054)	(0.005266658)	(0.0003031018)
	2.0	0.5	3	1.0	0.4521654	0.4515770	0.4492128	0.4527545
	2.0	0.5	3	1.0	0.4521654	(0.0015952082)	(0.002628331)	(0.0002401214)
	2.5	1.0	4	1.5	0.4847009	0.4839439	0.4819132	0.4854585
	2.5	1.0	4	1.5	0.4847009	(0.0019562459)	(0.002105828)	(0.0001645215)
	3.5	1.5	6	2.0	0.5878742	0.5870538	0.5857333	0.5883669
	3.5	1.5	6	2.0	0.5878742	(0.0023158514)	(0.001127543)	(0.0001327244)

MLE = maximum likelihood estimator, SELF = square error loss function, QLF = quadratic loss function, NLF = new loss function.

Table 2

Estimates and (posterior risk) of α under Jeffrey's prior.

λ	β	θ	C	C1	C2	MLE	SELF	QLF	NLF
1.05	1.5	0.5	1	0.5	0.5	0.7981785	0.7971406	0.7887990	0.7992192
	1.5	0.5	1	0.5	0.5	0.7981785	(0.003311692)	(0.005266604)	(0.0029587057)
	2.0	0.5	3	1.0	1.0	0.8956767	0.8921727	0.8874770	0.8945112
	2.0	0.5	3	1.0	1.0	0.8956767	(0.006259250)	(0.002641994)	(0.0018614567)
	2.5	1.0	4	2.0	1.5	0.8374464	0.8308672	0.8273250	0.8335090
	2.5	1.0	4	2.0	1.5	0.8374464	(0.005858038)	(0.002139587)	(0.0011082842)
	3.5	1.5	6	2.5	2.0	0.8687908	0.8636644	0.8616906	0.8656269
	3.5	1.5	6	2.5	2.0	0.8687908	(0.005092232)	(0.001145624)	(0.0006316363)
3.97	1.5	0.5	1	0.5	0.5	0.3288763	0.3284486	0.3250116	0.3293051
	1.5	0.5	1	0.5	0.5	0.3288763	(0.0005622312)	(0.005266605)	(0.0003224283)
	2.0	0.5	3	1.0	1.0	0.4606284	0.4588263	0.4564114	0.4600290
	2.0	0.5	3	1.0	1.0	0.4606284	(0.0016554692)	(0.002642805)	(0.0002531927)
	2.5	1.0	4	2.0	1.5	0.4919450	0.4880801	0.4859993	0.4896320
	2.5	1.0	4	2.0	1.5	0.4919450	(0.0020214881)	(0.002143806)	(0.0001721906)
	3.5	1.5	6	2.5	2.0	0.5941366	0.5906308	0.5892810	0.5919729
	3.5	1.5	6	2.5	2.0	0.5941366	(0.0023814988)	(0.001145851)	(0.0001381502)
4.12	1.5	0.5	1	0.5	0.5	0.3208446	0.3204274	0.3170743	0.3212629
	1.5	0.5	1	0.5	0.5	0.3208446	(0.0005351054)	(0.005266658)	(0.0003031018)
	2.0	0.5	3	1.0	1.0	0.4521654	0.4503964	0.4480259	0.4515770
	2.0	0.5	3	1.0	1.0	0.4521654	(0.0015951972)	(0.002641997)	(0.0002394920)
	2.5	1.0	4	2.0	1.5	0.4847009	0.4808930	0.4788428	0.4824220
	2.5	1.0	4	2.0	1.5	0.4847009	(0.0019623923)	(0.002139695)	(0.0001634783)
	3.5	1.5	6	2.5	2.0	0.5878742	0.5844053	0.5830697	0.5857333
	3.5	1.5	6	2.5	2.0	0.5878742	(0.0023315597)	(0.001145638)	(0.0001324170)

MLE = maximum likelihood estimator, SELF = square error loss function, QLF = quadratic loss function, NLF = new loss function.

Table 3

Estimates and (posterior risk) of α under extension of Jeffrey's prior.

CONFLICTS OF INTEREST

The authors declare they have no conflicts of interest.

AUTHORS' CONTRIBUTIONS

Authors developed the new model and performed the analytical calculations. They also discussed the results and contributed to the final manuscript.

ACKNOWLEDGMENTS

The authors are very much thankful to the reviewers for their valuable inputs to bring this research paper to this form.

REFERENCES

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14.E.T. Lee and J.W. Wang, Statistical Methods for Survival Data Analysis, third, John Wiley and Sons, New York, USA, 2003.

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Journal: Journal of Statistical Theory and Applications
Volume-Issue: 20 - 2
Pages: 395 - 406
Publication Date: 2021/07/10
ISSN (Online): 2214-1766
ISSN (Print): 1538-7887
DOI: 10.2991/jsta.d.210607.002 How to use a DOI?
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - Sofi Mudasir
AU  - S.P. Ahmad
PY  - 2021
DA  - 2021/07/10
TI  - Parameter Estimation of the Weighted Generalized Inverse Weibull Distribution
JO  - Journal of Statistical Theory and Applications
SP  - 395
EP  - 406
VL  - 20
IS  - 2
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.210607.002
DO  - 10.2991/jsta.d.210607.002
ID  - Mudasir2021
ER  -

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