Pranav Quasi Gamma Distribution: Properties and Applications

Sameer Ahmad Wani; Anwar Hassan; Shaista Shafi; Sumeera Shafi

doi:10.2991/jsta.d.201202.001

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Volume 19, Issue 4, December 2020, Pages 506 - 517

Pranav Quasi Gamma Distribution: Properties and Applications

Authors

Sameer Ahmad Wani¹, Anwar Hassan¹, Shaista Shafi¹, Sumeera Shafi²^{, *}

¹Department of Statistics, University of Kashmir, Srinagar, 190006, India

²Department of Mathematics, University of Kashmir, Srinagar, 190006, India

^*Corresponding author. Email: sumeera.shafi@gmail.com

Corresponding Author

Sumeera Shafi

Received 28 June 2020, Accepted 24 November 2020, Available Online 7 December 2020.

DOI: 10.2991/jsta.d.201202.001 How to use a DOI?
Keywords: Quasi gamma distribution; Pranav distribution; Mixture models; Simulation study; Mixing parameter; Structural properties and maximum likelihood estimation
Abstract: We have developed Pranav Quasi Gamma Distribution (PQGD) as a mixture of Pranav distribution (θ) and Quasi Gamma distribution (2,θ). We obtained various necessary statistical characteristics of PQGD. The flexibility of proposed model is clear from graph of hazard function. The reliability measures of proposed model are also obtained. Sample estimates of unknown parameters are obtained by making use of maximum likelihood estimation method. We have also carried out the simulation study for comparing our model with its related models. We then tested the significance of mixing parameter. Finally, applications to real-life data sets is presented to examine the significance of newly introduced model.
Copyright: © 2020 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

Continuous efforts have been made by researchers for many years to bring more and more flexibility in fitting probability models to real-life data. Flexibility can be introduced by generalizing the classical probability models or by mixing the two probability models. Need of mixture models arise when the population or distribution from which the data is obtained is a genuine mixture of k distinct populations or distributions and our aim is to estimate the proportions p1,p2,…pk in which these k distinct populations in which these occur. As we deal mostly with the data obtained from two or more populations mixed in different proportions, so mixture models find greater applicability in fitting models to data. Mixture models also extract more variation from the data. Data analysts use mixture models to the complex data for better interpretation of results. Stacy [1] obtained generalized form of gamma model using power transformation of gamma distribution. Nadarajah et al. [2] obtained another generalized form of gamma model and applied it to various real-life situations. Shukla [3] obtained Pranav distribution by mixing gamma and exponential models in appropriate proportions and obtained its properties. Ghitany et al. [4] formulated Lindley distribution by mixture technique and studied its important properties. Shanker et al. [5] introduced a Quasi Gamma distribution and obtained its vital properties. Shanker and Shukla [6] obtained Ishita distribution by using mixture technique. Hassan et al. [7] obtained Lindley-Quasi Xgamma distribution and studied its important properties. Hassan, Wani and Shafi [8] introduced Poisson Pranav distribution and obtained its various mathematical properties along with obtaining applications of the proposed model. Hassan, Wani and Para [9] formulated three parameter Quasi Lindley distribution by using weighting technique and obtained various properties of that model. Shafi et al. [10] obtained properties and applications of Sanna distribution.

A continuous r. v X will have a mixture distribution if its p.d.f f(x) is obtained as a mixture of k distinct populations having density functions f1(x),f2(x),…fk(x) and with mixing proportions p1,p2,…pk respectively. Mathematically

f(x)=p1f1(x)+p2f2(x)+…+pkfk(x)

where

0≤pi≤1

∑i=1kpi=1

We have used mixture technique to obtain Pranav Quasi Gamma distribution (PQGD) in this paper.

2. PRANAV QUASI GAMMA DISTRIBUTION

A nonnegative r.v X would follow a PQGD if it will have p.d.f f(x) which can be obtained as a mixture of Pranav (θ) having p.d.f f1(x) & Quasi Gamma distribution (2,θ) having p.d.f f2(x), in which θ is a scale parameter. Mathematically

f(x)=(1−p)f2(x)+pf1(x)(1)

In Equation (1) p is a mixing parameter and

f1(x)=θ4θ4+6θ+x3e−θx x>0,θ>0(2)

(2) is a p.d.f of Pranav (θ) distribution with the corresponding c.d.f F1(x) given below

F1(x)=1−3θx+6+θ2x2θxθ4+6+1e−θx(3)

And p.d.f of Quasi Gamma distribution (2,θ) is given in (4)

f2(x)=2θ2e−θx2x3 θ>0,x>0(4)

And corresponding c.d.f F2(x) of QGD is given below

F2(x)=−Γ2,θx2+1(5)

Substituting values of f1(x) & f2(x) in (1) we obtain p.d.f f(x) of PQGD as given below

f(x)=pθ4θ4+6θ+x3e−θx+2(1−p)θ2e−θx2x3 x>0,θ>0,0≤p≤1(6)

The graphs of p.d.f of PQGD are given in Figure 1a and 1b. These graphs show that for different parameter values indicating positively skewed nature of proposed model.

And the c.d.f of PQGD is found by using (3) and (5) and is given below

F(x)=p1−θ2x2+3θx+6θxθ4+6+1e−θx+(1−p)1−Γ2,θx2(7)

The above graphs represents the cumulative distribution function of PQGD.

3. RELIABILITY ANALYSIS

This different reliability measures are obtained in this particular area of paper. Expressions for survival function, failure rate and reverse failure rate of proposed PQGD are obtained in (8), (9), (10) respectively.

Rx=1−p1−3θx+6+θ2x2θxθ4+6+1e−θx+(1−p)1−Γ2,θx2(8)

h(x)=pθ4θ+x3e−θx+2θ4+61−pθ2e−θx2x3θ4+6−pθ4+6−θ4+6+θxθ2x2+3θx+6e−θx+(1−p)θ4+61−Γ2,θx2(9)

R.H.R=hr(x)=pθ4θ+x3e−θx+2(θ4+6)1−pθ2e−θx2x3pθ4+6−θ4+6+θxθ2x2+3θx+6e−θx+(1−p)θ4+61−Γ2,θx2(10)

Figure 3a and 3b represent the survival function of PQGD. Figure 4 represents the hazard rate of PQGD which shows the flexibility of proposed model as the graph is inverted bathtub shaped. The hazard rate is of monotonic increasing as well as monotonic decreasing nature which shows the flexibility of proposed model.

4. STATISTICAL PROPERTIES

Moments, mean deviation about mean, median characterize probability model among other properties. Here we have obtained these statistical properties for our proposed Pranav Quasi Gamma model.

4.1. Moments

Assuming X being a r.v having PQGD (θ,p). We now know that kth moment about origin of PQGD is given as below

μk′=EXk=∫0∞xkfx,θ,pdx =∫0∞xkpθ4θ4+6θ+x3e−θx+2(1−p)θ2e−θx2x3dx

μk′=k!θk(2+θα)2pk!(k+3)(θ4+(k+1)(k+2)θk(θ4+6)+(1−p)Γ4+k2θk2(11)

Put k=1 in Equation (11) we get

μ1′=pθ4+24θθ4+6+(1−p)Γ52θ12

which is mean of the PQGD

Put k=2 in Equation (11) we get

μ2′=2pθ4+60θ2θ4+6+(1−p)Γ(3)θ

And variance of PQGD is

V(x)=pθ8+84θ4+144θ2θ4+62+(1−p)2−Γ522θ

4.2. Average Deviation About Median and Arithmetic Mean of PQGD

We have derived the expressions for average deviation about median & arithmetic mean of PQGD in this area of paper.

Theorem 1:

If r.v X follows PQGD θ,p, then average deviation about median δ2(x) and arithmetic mean δ1(x)

δ1(X)=2μp1−1+θμθμ2+3θμ+6θ4+6e−θμ+(1−p)1−(Γ(2,θμ2)−2pθθ4+6θ4γ(2,μ)+γ(5,μ)+(1−p)θ12γ52,μ

And

δ2(X)=μ−2pθθ4+6θ4γ(2,M)+γ(5,M)+(1−p)θ12γ52,M

respectively.

Proof:

Average deviation about median δ2(x) & arithmetic mean δ1(x) are well defined as

δ1(X)=∫0∞|x−μ|f(x)dx

& δ2(X)=∫0∞|x−M|f(x)dx

respectively. where μ and M are arithmetic mean and median of PQGD respectively. The measures δ1(X) & δ2(X) are obtained by making use of the simplified relations given below.

δ1(X)=∫0μ(μ−x)f(x)dx+∫μ∞(x−μ)f(x)dx

δ1(X)=2μF(μ)−2∫0μxf(x)dx(12)

and

δ2(X)=∫M∞(x−M)f(x)dx+∫0M(M−x)f(x)dx

δ2(X)=μ−2∫0Mxf(x)dx(13)

Using (6) we obtain

∫0μxf(x)dx=pθθ4+6θ4γ(2,μ)+γ(5,μ)+(1−p)θ12γ52,μ(14)

∫0Mxf(x)dx=pθθ4+6θ4γ(2,M)+γ(5,M)+(1−p)θ12γ52,M(15)

where γs,x=∫0xts−1e−tdt is an incomplete gamma function.

γ2,μ=∫0μt2−1e−tdt

γ52,μ=∫0μt52−1e−tdt

and γ5,μ=∫0μt5−1e−tdt

Using expressions (12), (13), (14), and (15) and expression for c.d.f (7) we obtain average deviation about median δ2(x)&and arithmetic mean δ1(x)

δ1(X)=2μp1−1+θμ((θμ)2+3θμ+6)θ4+6e−θμ+(1−p)(1−Γ2,θμ2−2pθθ4+6θ4γ(2,μ)+γ(5,μ)+(1−p)θ12γ52,μ

δ2(X)=μ−2pθθ4+6θ4γ(2,M)+γ(5,M)+(1−p)θ12γ52,M

5. ORDER STATISTICS OF PQGD

Assuming X1,X2,X3‥‥,Xn being an order statistics for the random sample x1,x2,x3,‥‥xn obtained from PQGD having c.d.f Fx,θ,p and p.d.f fx,θ,p, then the p.d.f of vth order statistics Xv is given by: fv(x,θ,p)=n!(v−1)!(n−v)!f(x,θ,p)[F(x,θ,p)]v−1[1−F(x,θ,p)]n−v v=1,2,…n

Using the Equations (6) and (7), the probability density function of vth order statistics of PQGD is specified as

fvx,θ,p=n!v−1!n−v!pθ4θ4+6θ+x3e−θx+2(1−p)θ2e−θx2x3p1−1+θθ2x2+3θx+6(θ4+6)e−θx+(1−p)1−Γ2,θx2v−11−p1−1+θθ2x2+3θx+6(θ4+6)e−θx+(1−p)1−Γ2,θx2n−v.

Then, the p.d.f of first order statistic X1 of of PQGD is specified as

f1x,θ,p=npθ4θ4+6θ+x3e−θx+2(1−p)θ2e−θx2x31−p1−1+θθ2x2+3θx+6θ4+6e−θx+(1−p)1−Γ2,θx2n−1.

and the p.d.f of nth order statistic Xn of of PQGD is specified as

fnx,θ,p=npθ4θ4+6θ+x3e−θx+2(1−p)θ2e−θx2x3p1−1+θθ2x2+3θx+6θ4+6e−θx+(1−p)1−Γ2,θx2n−1.

6. ESTIMATION OF PARAMETERS OF PQGD

Assuming X1,X2,X3‥…Xn being a randomly selected sample of size n obtained from PQGD having density function given by (2.6), then the likelihood function of PQGD is given as

Lx|θ,p=∏i=1npθ4θ4+6θ+xi3e−θxi+2(1−p)θ2e−θxi2xi3

The log-likelihood function becomes

logL=2nlogθ−nlogθ4+6+∑i=1nlogpθ2θ+xi3e−θxi+2θ4+6(1−p)e−θxi2xi3(16)

By partially differentiating (16) w. r. to θ and p and then equating the outcome to zero, we get the resulting normal equations specified below

∂logL∂θ=2nθ−4nθ3θ4+6+∑i=1np3θ2+2θxi3e−θxi−θ3+θ2xi3xie−θxi+2(1−p)xi34θ3e−θxi2−xi 2e−θxi2θ4+6pθ2θ+xi3e−θxi+2θ4+6(1−p)e−θxi2xi3=0(17)

∂logL∂p=∑i=1nθ2θ+xi3e−θxi−2θ4+6e−θxi2xi3pθ2θ+xi3e−θxi+2θ4+6(1−p)e−θxi2xi3=0(18)

MLEs of θ,p cannot be obtained by solving above complex equations as these equations are not in closed form. So we solve above equations by using iteration method through R software.

7. SIMULATION STUDY

We have generated a data of 50 observations through R software by using inverse c.d.f technique from proposed model and we have obtained loss of information values AIC, BIC, AICC, and HQIC values for our model and its related models. We have also obtained the Shannon's entropy of our model and its related models. For testing the significance of mixing parameter p we used likelihood ratio (LR) statistic. In Table 1 estimates of parameters of fitted models along with model functions are given.

Distribution	Parameter Estimates	Model Function
Quasi Gamma (QGD)	θ^=3.176(0.317)	2θ2e−θx2x3
Pranav (PD)	θ^=2.217(0.156)	θ4θ4+6θ+x3e−θx
Pranav Quasi Gamma (PQGD)	θ^=2.978(0.309)p^=0.1222(0.068)	pθ4θ4+6θ+x3e−θx+2(1−p)θ2e−θx2x3

Table 1

ML estimates with standard errors in parenthesis, model function of proposed model, and its related models for simulated data of 50 observations.

In order to test the statistical significance of mixing parameter p for proposed PQGD we computed LR statistic by testing H0:p=0 against H1:p≠0 using LR statistic ω=2LΘ^−LΘ^0=23.01, where Θ^ and Θ^0 are MLEs under H1 and H0. LR statistic ω∼χ12(α=0.01)=6.635 as n→∞, d.f being the difference of dimensionality. From Table 2 ω=23.01>6.635 at 1% significance level, hence we rejected H0 and conclude that mixing parameter p plays statistically a significant role.

Distribution	−log L	AIC	BIC	AICC	HQIC	Shanon Entropy H(X)	Likelihood Ratio
PQGD	11.67180	27.3436	31.167	27.598	28.799	0.233	23.01
QGD	23.1788	48.3576	50.2696	48.440	48.44094	0.46	23.01
PD	41.09660	84.1932	86.105	84.276	84.921	0.82

Table 2

Model comparison and likelihood ratio statistic of proposed model and its related models.

Loss of information criteria's like AIC, BIC, AICC, and HQIC are computed along with measure of average uncertainty that is Shannon entropy H(X) for comparison of models fitted to data.

AIC=2v−2logL AICC=AIC+2v(v+1)f−v−1BIC=vlogf−2logL HQIC=2v loglogf+2logLH(X)=−logLf

where v gives count of parameters in the statistical model, f represents the sample size, and −2logL shows the maximized value of the log-likelihood function. From Table 2, it is observed that the PQGD possesses the lesser AIC, AICC, BIC, and HQIC and H(X) values as compared to QGD and PD for simulated data. Hence we can conclude that the PQGD gives much better fit as compared to QGD and PD for simulated data.

8. SPECIAL CASES

Case I: By putting p=0, then PQGD (6) reduces to Quasi Gamma distribution with p.d.f as

f2(x)=2θ2e−θx2x3 x>0,θ>0

Case II: By putting p=1, PQGD (6) reduces to Pranav distribution with p.d.f as

.f1(x)=θ4θ4+6θ+x3e−θx x>0,θ>0

9. APPLICATIONS OF PQGD

We fitted PQGD and its related distributions to two real-life data sets and showed that our proposed model fits well to these data sets as compared to its related models.

Data Set 1: The data set given in Table 3 has been taken from Kotz and Johnson [11] and represents the survival times (in years) after diagnosis of 43 patients with some kind of leukemia.

0.019	0.129	0.159	0.203	0.485	0.636	0.748	0.781
0.869	1.175	1.206	1.219	1.219	1.282	1.356	1.362
1.458	1.564	1.586	1.592	1.781	1.923	1.959	2.134
2.413	2.466	2.548	2.652	2.951	3.038	3.6	3.655
3.745	4.203	4.690	4.888	5.143	5.167	5.603	5.633
6.192	6.655	6.874

Table 3

Survival times (in years) after diagnosis of 43 patients with a certain kind of leukemia.

Data set 2: This data set given in Table 4 represents the Kevlar 49/epoxy strands failure times data (pressure at 90%) is taken from Makubate et al. [12].

0.01	0.01	0.02	0.02	0.02	0.03	0.03	0.04	0.05	0.06	0.07
0.07	0.08	0.09	0.09	0.10	0.10	0.11	0.11	0.12	0.13	0.18
0.19	0.20	0.23	0.24	0.24	0.29	0.34	0.35	0.36	0.38	0.40
0.42	0.43	0.52	0.54	0.56	0.60	0.60	0.63	0.65	0.67	0.68
0.72	0.72	0.72	0.73	0.79	0.79	0.80	0.80	0.83	0.85	0.90
0.92	0.95	0.99	1.00	1.01	1.02	1.03	1.05	1.10	1.10	1.11
1.15	1.18	1.20	1.29	1.31	1.33	1.34	1.40	1.43	1.45	1.50
1.51	1.52	1.53	1.54	1.54	1.55	1.58	1.60	1.63	1.64	1.80
1.80	1.81	2.02	2.05	2.14	2.17	2.33	3.03	3.03	3.34	4.20
4.69	7.89

Table 4

Kevlar 49/epoxy strands failure times data (pressure at 90%).

R software version 3.5.3 [13] is used for analyzing the data. We have fitted QGD, PD, GD, WD, and PQGD to the data sets 1 and 2. The summary statistics of data sets 1 and 2 are displayed in Tables 5 and 6, MLEs of the parameters with standard errors in parenthesis, model functions are displayed in Table 7 and log-likelihood values, LR statistic, AIC, AICC, BIC, HQIC, and Shannon's entropy are displayed in Tables 8 and 9 respectively.

Number of Observations	Minimum	First Quartile	Median	Mean	Third Quartile	Maximum
43	0.019	1.212	1.923	2.534	3.700	6.874

Table 5

Summary statistics of data set 1.

Number of Observations	Minimum	First Quartile	Median	Mean	Third Quartile	Maximum
101	0.010	0.240	0.800	1.025	1.450	7.890

Table 6

Summary statistics of data set 2.

Distribution	Parameter Estimates		Model Function
	Data Set 1	Data Set 2
Quasi Gamma Distribution (QGD)	θ^=0.199032(0.021465)	θ^=0.8730(0.0614)	2θ2e−θx2x3
Pranav Distribution (PD)	θ^=1.244312(0.075493)	θ^=1.8976(0.0821)	θ4θ4+6θ+x3e−θx
Pranav Quasi Gamma Distribution (PQGD)	θ^=1.143384(0.093793)p^=0.76297(0.11982019)	θ^=1.8784(0.0878)p^=0.7782(0.0870)	pθ4θ4+6θ+x3e−θx+2(1−p)θ2e−θx2x3
Gamma Distribution (GD)	α^=1.3017130(0.2527684)β^=1.946648(0.4588967)	α^=0.8718(0.10669)β^=1.17547(0.19074)	xα−1e−x/ββαΓα
Weibull distribution (WD)	λ^=2.702256(0.3482947)β^=1.2397431(0.1532526)	λ^=0.98994(0.111770)β^=0.92588(0.072598)	βλxλβ−1e−(x/λ)β

Table 7

ML estimates with standard errors in parenthesis, model function of proposed model, and its related models for data set 1 and 2.

Distribution	−log L	AIC	BIC	AICC	HQIC	Shanon Entropy H(X)	Likelihood Ratio
PQGD	80.07714	164.1543	167.6767	164.4543	165.4532	1.862	101.188
QGD	130.6714	263.3428	265.104	263.4404	263.9923	3.038	101.188
PD	82.17808	166.3562	168.1174	166.4537	167.0056	1.911
GD	82.12227	168.2445	171.7669	168.5445	169.5435	1.909
WD	81.61015	167.2203	170.7427	167.5203	168.5192	1.897

Table 8

Model comparison and Likelihood ratio statistic of proposed model and its related models for data set 1.

Distribution	−log L	AIC	BIC	AICC	HQIC	Shanon Entropy H(X)	Likelihood Ratio
PQGD	102.8728	209.7456	214.9758	209.868	211.8629	1.018	506.18
QGD	355.9660	713.9321	716.5472	713.9725	714.9907	3.52	506.18
PD	106.3711	214.7422	217.3573	214.7826	215.8008	1.05
GD	102.9827	209.9655	215.2048	210.2421	212.2049	1.020
WD	102.9768	209.9536	215.1839	210.0761	212.1071	1.019

Table 9

Model comparison and Likelihood ratio statistic of proposed model and its related models for data set 2.

In order to test the statistical significance of mixing parameter p for proposed PQGD we computed LR statistic by testing H0:p=0 against H1:p≠0 using LR statistic ω1=2LΘ^−LΘ^0=101.188 for data set 1 and ω2=2LΘ^−LΘ^0=506.18 for data set 2 where Θ^ and Θ^0 are MLEs under H1 and H0. LR statistic ω∼χ12(α=0.01)=6.635 as n→∞, d.f being the difference of dimensionality. From Table 8 ω1=101.188>6.635 and from Table 9 ω2=506.18>6.635 at 1% significance level, hence we rejected H0 and conclude that mixing parameter p plays statistically a significant role for both the data sets.

Loss of information criteria's like AIC, BIC, AICC, and HQIC are computed along with measure of average uncertainty that is Shannon entropy H(X) for comparison of models fitted to data.

AIC=2v−2logL AICC=AIC+2v(v+1)f−v−1BIC=vlogf−2logL HQIC=2v loglogf+2logLH(X)=−logLf

where v gives count of parameters in the statistical model, f represents the sample size, and −2logL shows the maximized value of the log-likelihood function. From Tables 8 and 9, it is observed that the PQGD possesses the lesser AIC, AICC, BIC, HQIC, and H(X) values as compared to QGD, GD, WD, and PD for both the data sets 1 and 2. Hence we can conclude that the PQGD leads to a better fit than the QGD, GD, WD, and PD for data sets 1 and 2.

10. CONCLUSION

We incorporated PQGD as a mixture of Pranav distribution and Quasi Gamma distribution. We obtained crucial properties of our proposed model. We carried out the simulation study and showed superiority of our model over its related models. We also obtained the estimates of our proposed model by using maximum likelihood method of estimation. Significance of mixing parameter has been tested. Finally we fitted our model and its related models to two real-life data sets and concluded that our model gives better fit to these data sets as compared to its related models.

CONFLICTS OF INTEREST

All the authors have no conflict of interest.

AUTHORS' CONTRIBUTIONS

All the authors contributed equally.

ACKNOWLEDGEMENT

We are highly thankful to reviewers for their valuable suggestions and we are also highly thankful to journal.

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Journal: Journal of Statistical Theory and Applications
Volume-Issue: 19 - 4
Pages: 506 - 517
Publication Date: 2020/12/07
ISSN (Online): 2214-1766
ISSN (Print): 1538-7887
DOI: 10.2991/jsta.d.201202.001 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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AU  - Sameer Ahmad Wani
AU  - Anwar Hassan
AU  - Shaista Shafi
AU  - Sumeera Shafi
PY  - 2020
DA  - 2020/12/07
TI  - Pranav Quasi Gamma Distribution: Properties and Applications
JO  - Journal of Statistical Theory and Applications
SP  - 506
EP  - 517
VL  - 19
IS  - 4
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.201202.001
DO  - 10.2991/jsta.d.201202.001
ID  - Wani2020
ER  -

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