On a family that unifies Generalized Marshall-Olkin and Poisson-G family of distribution
11 On a family that unifies Generalized Marshall-Olkin and Poisson-G family of distribution Laba Handique, Farrukh Jamal and Subrata Chakraborty
1, 3
Department of Statistics, Dibrugarh University, Dibrugarh-786004, India Department of Statistics, Government S.A. Post-Graduate College, Dera Nawab Sahib, Pakistan.
Abstract
Unifying the generalized Marshall-Olkin (GMO) and Poisson-G (P-G) a new family of distribution is proposed. Density and the survival function are expressed as infinite mixtures of P-G family. The quantile function, asymptotes, shapes, stochastic ordering, moment generating function, order statistics, probability weighted moments and Rényi entropy are derived. Maximum likelihood estimation with large sample properties is presented. A Monte Carlo simulation is used to examine the pattern of the bias and the mean square error of the maximum likelihood estimators.
An illustration of comparison with some of the important sub models of the family in modeling a real data reveals the utility of the proposed family.
Keywords:
GMO family, Poisson-G family, MLE, AIC, A, W ___________________________________________
Preprint (June 06, 2020) * Corresponding author: Email: [email protected] Introduction
With the basic motivation to bring in more flexibility in the modelling different type of data, a preferred area of research in the field of probability distribution is that of generating new distributions starting with a base line distribution by inducing one or more additional parameters through various methodologies. A number of useful continuous univariate-G families have been added in the literature in recent times. Notable families introduced since 2017 are Poisson-G family (Abouelmagd et al ., 2017), Beta-G Poisson family (Gokarna et al ., 2018), Marshall-Olkin Kumaraswamy-G family (Handique et al ., 2017), Generalized Marshall-Olkin Kumaraswamy-G family (Chakraborty and Handique 2017), Exponentiated generalized-G Poisson family (Gokarna and Haitham, 2017), Beta Kumaraswamy-G family (Handique et al ., 2017), Beta generated Kumaraswamy Marshall-Olkin-G family (Handique and Chakraborty, 2017a), Beta generalized Marshall-Olkin Kumaraswamy-G family (Handique and Chakraborty, 2017b), Beta generated Marshall-Olkin-Kumaraswamy-G (Chakraborty et al ., 2018), Exponentiated generalized Marshall-Olkin-G family by (Handique et al ., 2019), Kumaraswamy generalized Marshall-Olkin-G family (Chakraborty and Handique, 2018), Odd modified exponential generalized family (Ahsan et al ., 2018), Zografos-Balakrishnan Burr XII family (Emrah et al., et al ., 2019), Extended generalized Gompertz family (Thiago et al ., 2019), Odd Half-Cauchy family (Chakraborty et al ., 2020), Generalized modified exponential-G family (Handique et al ., 2020), Beta Poisson-G family (Handique et al ., 2020) and Kumaraswamy Poisson-G (Chakraborty et al ., 2020) among others. In this current article, we propose a new family of continuous probability distribution to unify generalized Marshall-Olkin (GMO) of Jayakumar and Mathew (2008) and the Poisson-G (P-G) family of distribution (Tahir et al ., 2016) and call it the Generalized Marshall-Olkin Poisson-G ( ),,(GGMOP ). Now we briefly describe the GMO and P-G family and then introduce GMOP-G in the next section.
GMO : Jayakumar and Mathew (2008) proposed a generalization of the Marshall-Olkin family (Marshall and Olkin, 1997) called generalized Marshall-Olkin (GMO) family of distribution with baseline distribution having survival function (sf) )( tF and probability distribution function (pdf) )( tf . The sf and pdf of the GMO distribution are given respectively by ),;(
GMO tF )(1 )( tFtF and ),;( GMO tf )](1[ )()( tF tFtf , (1) where t ( ) and is an additional shape parameter. When ,1 );(),;( MOGMO tFtF and for ,1 )(),;( GMO tFtF . P-G : The Poisson-G (P-G) family of distribution with sf and cdf is given by (Chakraborty et al ., 2020) and1);( )(PG e eetG tG ,11);( )(PG eetG tG }0{ R ; ,...2,1 n (2) where )( tG is the baseline distribution. The corresponding pdf of (2) is given by )(1PG )()1();( tG etgetg , }0{ R ; . t (3) Rest of the article is arranged in 5 more sections. In Section 2 we introduce the proposed family along with its physical basis and list of some important sub models also defined some mathematical properties. In Section 3, linear representation of the sf and pdf of the proposed family also we discuss some statistical properties of the proposed family. In Section 4, maximum likelihood methods of estimation of parameters and simulation are presented. The data fitting applications is presented in Section 5. The article ends with a conclusion in Section 6. Generalized Marshall-Olkin Poisson-G
In this section we introduce the ),,(GGMOP family and also provide its special cases and a statistical genesis. The sf, cdf, pdf and hrf of this proposed distribution are respectively given by: ),,;(
GMOPG tF )()( tGtG ee ee , ),,;( GMOPG tF )()( tGtG ee ee , (4) ),,;( GMOPG tf )1( )()()1( tG tGtG ee eeetge , (5) and ),,;( GMOPG th )( 1)()( tG tGtG ee eeetge In particular, we get for (i) , the ),(G-MOP distribution. (ii) , the )(GP distribution. (iii) , the ),(GMO distribution. (iv) , the )(MO distribution. Proposition 1:
Let iNi2i1 ...,,,
TTT , ,,2,1 i be a sequence of N i.i.d. random variables from Poisson-G distribution and )...,,,min( iNi2i1i TTTW and )...,,,max( iNi2i1i TTTV . Then (i) i min i W follows ),,(GGMOP if ( ) Geometric (ii) i max i V follows ),,(GGMOP if (1 / ) Geometric . Proof: Case (i) When , considering N has a geometric distribution with parameter , we get ]},...,,[min{P θ21 tWWW ][P...][P][P θ21 tWtWtW θMOPGθ1i i ]),([][P t;FtW )()( tGtG ee ee . Case (ii) For , considering N has a geometric distribution with parameter /1 , we get ]},...,,[min{P θ21 tVVV ][P...][P][P θ21 tVtVtV θMOPG1 ]),([][ t;FtVP i i )()( tGtG ee ee In what follows we investigate some general properties, parameter estimation and real life applications.
Special models and shape of the density and hazard function
In this section we have plotted the pdf and hrf of the ),,(GGMOP taking G to be Weibull (W) and exponential (E) for some chosen values of the parameters to show the variety of shapes assumed by the family. The pdf and hrf of these distributions are obtained from ),,(GGMOP as follows: The -GMOP
Weibull (
W-GMOP ) distribution Considering the Weibull distribution (Weibull, 1951) with parameters and having pdf and cdf t ettg )( and t etG respectively we get the pdf and hrf of ),,,,(WGMOP distribution as ),,,,;( GMOPW tf )1( )()1( t tt e eet ee eeeete , and ),,,,;( GMOPW th )1( 1)1()1(1 t tt e eet ee eeeete . Taking in ),,,,(WGMOP we get the ),,,(E-GMOP with pdf and hrf is given by ),,,;( GMOPE tf )1( )()1( t tt e eet ee eeeee , and ),,,;( GMOPE th )1( 1)1()1( t tt e eet ee eeeee . Fig 1: pdf plots of the ),,,(E-GMOP distribution
Fig 2: hrf plots of the ),,,(E-GMOP distribution 2.2
Quantile and related measures
The th p quantile p t for ),,(GGMOP can be easily obtained by solving the equation ptF )( GMOPG as ))(1( ))(1()1(log1 tF tFeeGt p . A random number ‘ t ’ from ),,(GGMOP via an uniform random number ‘ u ’ can be generated by using the formula )1( )1()1(log1 u ueeGt . (6) For example, when we consider the exponential distribution having pdf and cdf as xexg x ,0 and ,1):( x exG the p th quantile, p t is given by )1( )1()1(log11log1 u ueet p . Here the flexibility of skewness and kurtosis of ),,(GGMOP is checked by plotting Galton skewness (S) that measures the degree of the long tail and Morris (1988) kurtosis (K) that measures the degree of tail heaviness. These are respectively defined by (2/8)Q-(6/8)Q (2/8)Q(4/8)2Q-(6/8)Q S and Q (7 / 8) - Q (5 / 8) Q (3 / 8) - Q (1 / 8) .Q (6 / 8) - Q (2 / 8) K be t a l a m bda Kurtosis
Fig 3:
Plots of the Galton skewness S and the Moor kurtosis K for the GMOP-E distribution with parameters Asymptotes and shapes
Two propositions regarding asymptotes of the proposed family are discussed here.
Proposition 2:
The asymptotes of pdf, cdf and hrf of ),,(GGMOP as t are given by GMOPG ( ; , , ) ~ ( ) (1 ) f t g t e , GMOPG tF and GMOPG ( ; , , ) ~ ( ) (1 ) h t g t e . Proposition 3:
The asymptotes of pdf, cdf and hrf of ),,(GGMOP as t are given by GMOPG ( ) 1 ( ; , , ) ~ ( ) ( ) / (1 )
G t f t e g t e e e , GMOPG ( ) ( ; , , ) ~ 1 ( ) / (1 )
G t
F t e e e and )()(~),,;( eetgeth tG . The shapes of the density and hazard rate function can be described analytically. The critical points of the ),,(GGMOP family density function are the roots of the equation
01 )()1()()1()()( )( )()()( )( tGtGtG tG ee tgeee tgetgtg tg . (7) The critical point of ),,(GGMOP family hazard rate are the roots of the equation
01 )()()()( )( )()()( )( tGtGtG tG ee tgeee tgetgtg tg . (8) There may be more than one root of (7) and (8). If tt is a root then it is a local maximum, a local minimum or a point of inflexion if tortt and for (8) if tortt where )]([log)()( tfdtdt and )]([log)()( thdtdt . Stochastic orderings
Let X and Y be two random variables with cdfs F and G, respectively, corresponding pdf’s f and g . Then X is said to be smaller than Y in the likelihood ratio order ( YX lr ) if )()( tgtf is decreasing in t . Here we present a result of likelihood ratio ordering. Theorem 1:
Let ),,(GMOPG~ X and ),,(GMOPG~ Y . If , then YX lr Proof: tG tG ee eetg tf ( ) ( )2 21 2 1 2 ( ) 21 1 [1 ] ( ) (1 )( ( ) / ( )) ( 1) ( ) ( ) .[1 ]
G t G tG t e e e g t ed f t g tdt e e
Now this is always less than 0 since . Hence, )(/)( tgtf is decreasing in t . That is YX lr Linear representation
Linear representation of sf and pdf, etc in terms of corresponding functions of known distributions is an important tool for further mathematical properties. In this section we present some important results for proposed family.
Expansions of the survival and density functions as infinite linear mixture
Here we express the sf and pdf of the ),,(GGMOP as infinite linear mixture of the corresponding functions of )(P G distribution. Consider the series representation jjjjk zjk kjzjk jkz !!)1( !)1(!)(Γ )(Γ)1( . (9) This is valid for z and ,0 k where (.) is the gamma function. Using equation (9) in equation (4), for )1,0( we obtain ),,;( GMOPG tF jjj tGjjtG )};({)1(!)!1( )!1()};({ PG0PG jj j tG )];([ PG0 (10) ))(;( j j jtG Differentiating in equation (10) with respect to ‘ t ’ we get ),,;( GMOPG tf
10 PGPG ]);([);( jj j tGtg (11) ]))(;([ j j jtGdtd (12) ))(;( j j jtg , where jjj jj )1(1)( , jjj j )()( . The moment generating function (mgf) of ),,(GGMOP family can be easily expressed in terms of those of the exponentiated )(G-P distribution using the results of Section 2.1. For example using equation (12) it can be seen that dtjtGdtdedttfeeEsM j jststsTT ]))(;([)(][)( )(]))(;([ PG0PG0 sMdtjtGe
Tj jstj j where )( PG sM T is the mgf of a )(G-P distribution. Table 1:
Mean, variance, skewness and kurtosis of the GMOP E distribution with different values of and,, Mean Variance Skewness Kurtosis 10 10 2 2 0.1572 0.0163 1.2096 4.6452 10 10 1 2 0.2224 0.0306 1.0761 4.0834 10 10 0.5 2 0.2648 0.0408 0.9749 3.7373 10 10 0.1 2 0.3033 0.0502 0.8840 3.4698 10 10 2 1 0.3145 0.0652 1.2096 4.6452 10 10 2 0.5 0.6291 0.2610 1.2096 4.6452 10 10 0.5 0.5 1.0592 0.6528 0.9749 3.7373 10 10 0.1 0.1 6.0675 20.1022 0.8840 3.4698 10 5 2 2 0.0918 0.0066 1.5144 6.0241 10 2 2 2 0.0419 0.0016 1.9171 8.4744 10 0.5 2 2 0.0115 0.0001 2.4163 12.8206 10 0.5 0.5 0.5 0.0834 0.0077 2.3537 12.1190 5 10 2 2 0.2692 0.0424 1.0978 4.3497 5 5 2 2 0.1676 0.0206 1.4448 5.8193 2 5 2 2 0.3615 0.0914 1.5105 6.3958 2 2 2 2 0.2049 0.0427 2.1968 10.7924 1 2 2 2 0.4180 0.19284 2.3136 11.2913 5 0.1 0.1 0.1 0.2309 0.0804 3.6320 30.5202 5 5 5 3 0.0489 0.0016 1.4027 5.7915 5 5 3 5 0.04882 0.0017 1.5071 6.2776 5 5 5 8 0.0183 0.0002 1.3967 5.7764 5 5 10 10 0.0069 0.00001 1.1168 2.1041
The entropy of a random variable is a measure of uncertainty variation and has been used in various situations in science and engineering. The Rényi entropy (see details, Song, 2001) is defined by dttfI R )(log)1()( , where and . Thus the Rényi entropy of ),,(GGMOP distribution can be obtained as dttGtGtgI jj jR )];([]);();([log)1()( PG1PGPG01 , where }!)]1([/{]})1([)1({)( jj jjj . Table 2:
Rényi entropy ),,,(EGMOP distribution with different values of and,,
Parameter
10 10
2 2 -0.2550 -0.6403 -0.9916 -1.0647 -1.1549 -1.2490 5 5 0.5 0.5 1.6816 1.3053 0.9722 0.9032 0.8176 0.7280 5 5 2 0.5 1.3469 0.8661 0.4519 0.3687 0.2669 0.1621 3 3 2 0.5 1.6090 1.0220 0.5252 0.42839 0.3113 0.1924 1.5 1.5 2 0.5 2.1288 1.3773 0.6945 0.5640 0.4097 0.2571 2 0.5 0.5 0.5 1.7182 0.8594 0.4390 -0.1133 -0.2982 -0.4789
As expected the Rényi entropy turns out to be non increasing with . Suppose n TTT ...,,, is a random sample from any ),,(GGMOP distribution. Let ni T : denote the t h i order statistics. The pdf of ni T : can be expressed as inini tFtFtfini ntf )()](1[)()!()!1( !)( GMOPG1GMOPGGMOPG: ilnil l tFlil itfini n )()!1(! )!1()1()()!()!1( ! GMOPG10GMOPG
Now using the general expansion of the pdf and sf of ),,(GGMOP distribution we get the pdf of the th i order statistics for of the ),,(GGMOP as ]]);([);([)!()!1( !)(
10 PGPG: jj jni tGtgini ntf ])];([[1)1( )(PG010 ilnkk kil l tGli , where j and k are defined earlier.
0, 1)1(PG,PG )];([);( kj ilnkjkj tGΦtg (13)
0, )1(PG, )];([]))1(/([ kj ilnkjkj tGdtdilnkjΦ
0, PG, )))]1((;( kj kj ilnkjtgΦ , where lilkjkj liinnΦ )1(111 and ))1(/( ,, ilnkjΦΦ kjkj . The probability weighted moments (PWM), first proposed by Greenwood et al . (1979), are expectations of certain functions of a random variable whose mean exists. The ( , , ) th p q r PWM of T is defined by dttftFtFt rqprqp )()](1[)( ,, . The th s moment of T can be written as dttgtGtTE jsj js );()];([)( j jsj where j define in Section 2.1 and dteetgeeeet tGrtGqtGprqp )()()(0,, is the PWM of )(G-P distribution. Therefore the moments of the ),,(GGMOP may be expressed in terms of the PWMs of )(G-P . Similarly proceeding we can derive th s moment of the th i order statistic , : ni T in a random sample of size n from ),,(GGMOP by using equations (13) as )( , nis TE
0, 1)1(,0,, kj ilnkjskj Φ where j and kj Φ , defined above. 4. Estimation
This section is devoted to the estimation of the ),,(GGMOP model parameters via the maximum likelihood (ML) method.
Let ),...,,( n tttT be a random sample of size n from ),,(GGMOP with parameter vector ),,,( ξρ , where ),...,,( q ξ is the parameter vector of G . Then the log-likelihood function for ρ is given by )(log)1(]),([]),([log)1(log)log()( ),(1 11 eetGtgenn i tGni niini i ξ ξξ ρ )1(log)1( ),(1 ξ i tGni ee . This log-likelihood function can’t be solved analytically because of its complex form, but it can be maximized numerically by employing global optimization methods available with the software’s R. By taking the partial derivatives of the log-likelihood function with respect to ξ and,, we obtain the components of the score vector ),,,( ξ UUUUU ρ . The asymptotic variance-covariance matrix of the MLEs of parameters can obtained by inverting the Fisher information matrix )(I ρ which can be derived using the second partial derivatives of the log-likelihood function with respect to each parameter. The th ji elements of )(I ρ n are given by ,])([I jiji lE ρ qji . The exact evaluation of the above expectations may be cumbersome. In practice one can estimate )(I ρ n by the observed Fisher’s information matrix )Iˆ()ˆ(Iˆ jin ρ defined as ,)(Iˆ ˆ2 ηη ρ jiji l qji . Using the general theory of MLEs under some regularity conditions on the parameters as n the asymptotic distribution of )ˆ( ρρ n is ),0( nk VN where )(I)( ρ njjn vV . The asymptotic behaviour remains valid if n V is replaced by )ˆ(Iˆˆ ρ n V . Using this result large sample standard errors of j th parameter j is given by jj v ˆ . In this section Monte Carlo simulation study is conducted to compare the performance of the different estimators of the unknown parameters for the ),,,(EGMOP distribution using R program. We generate N samples of size n from GMOP-E distribution with true parameters values , and calculate the bias and mean square error (MSE) of the MLEs empirically by Ni ihNi ih hhNMSEhhNBias )ˆ(1and)ˆ(1 respectively (for ,,, h ). Results of this simulation study are presented graphically in Figures 4 and 5 tells us that as the sample sizes increases the biases and MSE’s approach to 0 in all cases which is consistent with the theoretical properties of the MLEs. This fact supports that the asymptotic normal distribution provides an adequate approximation to the finite sample distribution of the MLEs. The simulation study shows that the maximum likelihood method is appropriate for estimating the ),,,(EGMOP distribution parameters. Fig 4:
The Biases for the parameter values for ),,,(EGMOP distribution.
Fig 5:
The MSEs for the parameter values for ),,,(EGMOP distribution.
5. Real life application
Here we consider modelling of the one failure time data set to illustrate the suitability of the ),,,(EGMOP distribution in comparison to some existing distributions by estimating the parameters by numerical maximization of log-likelihood functions. The data set comprises survival time of 72 guinea pigs infected with virulent tubercle bacilli, observed and reported by Bjerkedal 1960. The descriptive statistics about the data set shown in Table 3 reveal that the data set are positively skewed as expected from the nature of life time data and has higher kurtosis. Table 3:
Descriptive Statistics for the data set I
Data Sets n Min. Mean Median s.d. Skewness Kurtosis 1 st Qu. 3 rd Qu. Max. I 72 0.100 1.851 1.560 1.200 1.788 4.157 1.080 2.303 7.000
We have compared the ),,,(EGMOP distribution with some of its sub models namely, exponential (Exp), moment exponential (ME), Poisson exponential (P-E), Marshall-Olkin exponential (MO-E) (Marshall and Olkin, 1997), generalized Marshall-Olkin exponential (GMO-E) (Jayakumar and Mathew, 2008) and Marshall-Olkin Poisson exponential (MOP-E) models and also with other recently introduced models Kumaraswamy exponential (Kw-E) (Cordeiro and de Castro, 2011), Beta exponential (BE) (Eugene et al ., 2002), Marshall-Olkin Kumaraswamy exponential (MOKw-E) (Handique et al ., 2017) and Kumaraswamy Marshall-Olkin exponential (KwMO-E) (Alizadeh et al ., 2015), beta Poisson exponential (BP-E) (Handique et al ., 2020) and Kumaraswamy Poisson exponential (KwP-E) (Chakraborty et al ., 2020) distributions for failure time data set. Upon fitting the best model is chosen as the one having lowest AIC (Akaike Information Criterion), BIC (Bayesian Information Criterion), CAIC (Consistent Akaike Information Criterion), and HQIC (Hannan-Quinn Information Criterion) and also, we apply formal goodness-of-fit tests in order to verify which distribution fits better to this data by considering Anderson-Darling (A), Cram′er-von Mises (W) and Kolmogorov-Smirnov (K-S) statistics to compare the fitted models. The model with minimum values for these statistics and highest p -value of K-S statistics could be chosen as the best model to fit the data. We have also provided the asymptotic standard errors and confidence intervals of the mles of the parameters for each competing model. For visual comparison of the best fitted density and the fitted cdf are plotted with the corresponding observed histograms and ogives in Fig. 7. These plots indicate that the proposed distributions provide a good fit to this data set. To check the shape of the observed hazard function the total time on test (TTT) plot (see Aarset, 1987) is used. A straight diagonal line indicates constant hazard for the data set, where as a convex (concave) shape implies decreasing (increasing) hazard. The TTT plots for the data set Fig. 6 indicate that the data set has increasing hazard rate. We also provide the box plot of the data to summerise the minimum, first quartile, median, third quartile, and maximum where a box is shown from the first quartile to the third quartile with a a vertical line going through the box at the median. Fig: 6
TTT and Box plot for the failure time data set Table 4:
MLEs, standard errors (in parentheses) values for the guinea pigs survival time’s data set Models ˆ ˆ a ˆ b ˆ ˆ ˆ Exp --- --- --- --- --- 0.540 )( (0.063) ME --- --- --- --- --- 0.925 )( (0.077) P-E --- --- --- --- -5.268 1.236 ),( (1.069) (0.129) MO-E --- 8.778 --- --- --- 1.379 ),( (3.555) (0.193) GMO-E 0.179 47.635 --- --- --- 4.465 ),,( (0.070) (44.901) (1.327) MOP-E --- 2.036 --- --- -3.366 1.312 ),,( (1.071) (0.142) (0.093) Kw-E --- --- 3.304 1.100 --- 1.037 ),,( ba (1.106) (0.764) (0.614) B-E --- --- 0.807 3.461 --- 1.331 ),,( ba (0.696) (1.003) (0.855) MOKw-E --- 0.008 2.716 1.986 --- 0.099 ),,,( ba (0.002) (1.316) (0.784) (0.048) KwMO-E --- 0.373 3.478 3.306 --- 0.299 ),,,( ba (0.136) (0.861) (0.779) (1.112) BP-E --- --- 3.595 0.724 0.014 1.482 ),,,( ba (1.031) (1.590) (0.010) (0.516) KwP-E --- --- 3.265 2.658 4.001 0.177 ),,,( ba (0.991) (1.984) (5.670) (0.226) GMOP-E 0.333 12.584 --- --- 0.054 2.858 ),,,( (0.151) (7.696) (1.376) (0.959) Table 5:
Log-likelihood, AIC, BIC, CAIC, HQIC, A, W and KS ( p -value) values for the guinea pigs survival times data set Models AIC BIC CAIC HQIC A W KS ( p -value) Exp 234.63 236.91 234.68 235.54 6.53 1.25 0.27 )( (0.06) ME 210.40 212.68 210.45 211.30 1.52 0.25 0.14 )( (0.13) P-E 209.86 214.42 210.03 211.66 0.98 0.19 0.09 ),( (0.66) MO-E 210.36 214.92 210.53 212.16 1.18 0.17 0.10 ),( (0.43) GMO-E 210.54 217.38 210.89 213.24 1.02 0.16 0.09 ),,( (0.51) MOP-E 208.32 215.15 208.67 211.04 0.96 0.17 0.09 ),,( (0.56) Kw-E 209.42 216.24 209.77 212.12 0.74 0.11 0.08 ),,( ba (0.50) B-E 207.38 214.22 207.73 210.08 0.98 0.15 0.11 ),,( ba (0.34) MOKw-E 209.44 218.56 210.04 213.04 0.79 0.12 0.10 ),,,( ba (0.44) KwMO-E 207.82 216.94 208.42 211.42 0.61 0.11 0.08 ),,,( ba (0.73) BP-E 205.42 214.50 206.02 209.02 0.55 0.08 0.09 ),,,( ba (0.81) KwP-E 206.63 215.74 207.23 210.26 0.48 0.07 0.09 ),,,( ba (0.79) GMOP-E ),,,( ( ) In the Tables 4 and 5 the MLEs with standard errors of the parameters for all the fitted models along with their AIC, BIC, CAIC, HQIC, A, W and K-S statistic with p -value for the six sub models for the failure time data set are presented respectively. From these findings, it is evident that the E-GMOP distribution with lowest value of AIC, BIC, CAIC, HQIC, A, W and highest p -value of K-S statistics is not only a better model than all the sub models Exp, ME, P-E, MO-E, GMO-E, MOP-E but also the better than the most of the recently introduced three or four parameters models namely Kw-E, B-E, MOKw-E, KwMO-E, BP-E and KwP-E. These findings are further validated from the plots of fitted density with histogram of the observed data and fitted cdf with ogive of observed data in Figure 7. These plots clearly indicate that the proposed distribution provide closest fit to the data set considered here. Fig: 7
Plots of the observed histogram and estimated pdf on left and on right the observed ogive and estimated cdf for failure time data set for the GMOP-E model
6. Conclusions
In this work, we propose a new family of continuous distributions called the
Generalized Marshall-Olkin Poisson -G family of distributions. Several new models can be generated by considering special distributions for G. We demonstrate that the pdf of any GMOPG distribution can be expressed as a linear combination of exponentiated-G density functions, which result allowed us to derive some of its mathematical and statistical properties such as moment generating function, order statistics, probability weighted moments and Rényi entropy. The estimations of the model parameters are obtained by maximum likelihood method. One application of the proposed family empirically prove its flexibility to model the real data sets, in particular we verified that a special case of the GMOPG family can provide better fits than its sub models and other models generated from well-known families.
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