International Journal of Probability and Statistics

p-ISSN: 2168-4871    e-ISSN: 2168-4863

2021;  10(2): 46-62

doi:10.5923/j.ijps.20211002.02

Received: Jul. 10, 2021; Accepted: Jul. 24, 2021; Published: Aug. 15, 2021

 

A New Two-Parameter Compound G Family: Copulas, Properties and Applications

Abdelrahman M. Khedr , Zohdy M. Nofal , Yehia M. El Gebaly

Department of Statistics, Mathematics and Insurance, Benha University, Egypt

Correspondence to: Abdelrahman M. Khedr , Department of Statistics, Mathematics and Insurance, Benha University, Egypt.

Email:

Copyright © 2021 The Author(s). Published by Scientific & Academic Publishing.

This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/

Abstract

In the work, we propose and study a new two-parameter compound G family of continuous distributions. Relevant statistical properties are mathematically derived. Many new G families of bivariate distributions are presented using Renyi's copula, Clayton copula, Ali-Mikhail-Haq copula, Farlie-Gumbel-Morgenstern copula, and modified Farlie-Gumbel-Morgenstern copula. Based on a special case we presented a new Lomax extension and studied its relevant statistical properties. The maximum likelihood method is used and employed for estimating the model parameters. Two applications to real-life data sets are presented for illustrating the superiority of the new family.

Keywords: Poisson Family, Order Statistics, Farlie-Gumbel-Morgenstern, Topp Leone Family, Maximum Likelihood Estimation, Clayton copula, Generating Function, Moments, Ali-Mikhail-Haq copula

Cite this paper: Abdelrahman M. Khedr , Zohdy M. Nofal , Yehia M. El Gebaly , A New Two-Parameter Compound G Family: Copulas, Properties and Applications, International Journal of Probability and Statistics , Vol. 10 No. 2, 2021, pp. 46-62. doi: 10.5923/j.ijps.20211002.02.

Article Outline

1. Introduction
2. Copula
    2.1. BQPTLG-G type via Clayton Copula
    2.2. BQPTLG-G Type via REC
    2.3. BQPTLG-G Type via FGMC
    2.4. BQPTLG-G Type via Modified FGMC
    2.5. BQPTLG-G type via AMHC
3. Mathematical Properties
    3.1. Linear Representation
    3.2. Ordinary Moments
    3.3. Moment of the Residual Life
    3.4. Moment of the Reversed Residual Life
    3.5. Probability Weighted Moments
    3.6. Order Statistics
4. Studying a Special Model
5. Estimation
6. Modeling Real-Life Data of Aircraft Windshields
7. Conclusions

1. Introduction

The statistical literature contains many new G families of continuous distributions which have been generated either by merging (compounding) common G families of continuous distributions or by adding one or more parameters to the G family. These novel G families have been employed for modeling real-life datasets in many applied studies such as insurance, engineering, econometrics, biology, medicine, statistical forecasting, and environmental sciences see Gupta et al. (1998) for the exponentiated-G family, Marshall and Olkin (1997) for the Marshall-Olkin-G family, Eugene et al. (2002) for beta generalized-G family, Yousof et al. (2015) for the transmuted exponentiated generalized-G family, Nofal et al. (2017) for the generalized transmuted-G family, Rezaei et al. (2017) for the Topp Leone generated family, Merovci et al. (2017) for the exponentiated transmuted-G family, Brito et al. (2017) for the Topp-Leone odd log-logistic-G family, Yousof et al. (2017a) for Burr type X G family, Aryal and Yousof (2017) for exponentiated generalized-G Poisson family, Hamedani et al. (2017) for type I general exponential class of distributions, Cordeiro et al. (2018) for Burr type XII G family, Korkmaz et al. (2018a) for the exponential Lindley odd log-logistic-G family, Korkmaz et al. (2018b) for the Marshall-Olkin generalized-G Poisson family, Yousof et al. (2018) for Burr-Hatke family of distributions, Hamedani et al. (2018) for the extended G family, Hamedani et al. (2019) for the type II general exponential G family, Nascimento et al. (2019) for the odd Nadarajah-Haghighi family of distributions, Yousof et al. (2020) for the Weibull G Poisson family, Karamikabir et al. (2020) for the Weibull Topp-Leone generated family, Merovci et al. (2020) for the Poisson Topp Leone G family, Korkmaz et al. (2020) for the Hjorth's IDB generator of distributions, Alizadeh et al. (2020a) for flexible Weibull generated family of distributions, Alizadeh et al. (2020b) for the transmuted odd log-logistic-G family, Altun et al. (2021) for the Gudermannian generated family of distributions and El-Morshedy et al. (2021) for the Poisson generated exponential G family among others.
Due to Rezaei et al. (2017), the cumulative distribution function (CDF) of the Topp Leone generated G (TLG-G) family of distributions can be expressed as
(1)
The corresponding probability density function (PDF) to (1) is
(2)
where refers to the parameters vector of the new family and refers to the parameters vector of any base-line model. For , the TLG-G family reduces to the one-parameter Topp Leone G (TL-G) family. Suppose that be an independent and identically random variables (iid RVs) with common CDF follows the TLG-G family and be RV with probability mass function
defining
then
(3)
Using equations (2) and the last equation, we can write
(4)
Equation (4) can be called as quasi Poisson Topp Leone generated-G (QPTLG-G) family of distributions. The corresponding PDF of the QPTLG-G family can be expressed as
(5)
The QPTLG-G family may be useful in modeling:
I. The "monotonically increasing hazard rate" real-life datasets as illustrated in Section 6 (Figures 1 and 2 (top left plots)).
II. The real-life datasets which do not have extreme values as shown in Section 6 (Figures 1 and 2 (bottom right plots) and (bottom left plots)).
III. The real-life datasets which their nonparametric Kernel density estimations are left skewed bimodal and right skewed bimodal as given in Section 6 (Figures 1 and 2 (top right plots)).
The QPTLG-G family proved adequate superiority against many other well-known G families as illustrated below:
I. In modeling the times failure of the aircraft windshield items, the QPTLG-G family is better than the odd log-logistic-G family, the generalized mixture-G family, the transmuted Topp-Leone-G family, the Gamma-G family, the Burr-Hatke-G family, the McDonald-G family, the exponentiated-G family, the Kumaraswamy-G family, and the proportional reversed hazard rate-G family under the consistent- information criteria, Akaike information criteria, Hannan-Quinn information criteria and Bayesian information criteria.
II. In modeling the times of service of the aircraft windshield items, the QPTLG-G family is better than the odd log-logistic-G family, the generalized mixture-G family, the transmuted Topp-Leone-G family, the Gamma-G family, the Burr-Hatke-G family, the McDonald-G family, the exponentiated-G family, the Kumaraswamy-G family, and the proportional reversed hazard rate-G family under the consistent- information criteria, Akaike information criteria, Hannan-Quinn information criteria and Bayesian information criteria.

2. Copula

For modeling of the bivariate real data sets, we shall derive some new bivariate QPTLG-G (Bv-QPTLG-G) type distributions using “Farlie-Gumbel-Morgenstern copula” (FGMC for short) copula (see Morgenstern (1956), Farlie (1960). Gumbel (1960 and 1961), Johnson and Kotz (1975 and 1977)), modified FGMC (see Balakrishnan and Lai (2009)) which contains for internal types, ” Clayton copula ” (see Nelsen (2007)), “Renyi's entropy copula (REC) (Pougaza and Djafari (2010))” and “Ali-Mikhail-Haq copula (AMHC)” (see Ali et al. (1978)). The multivariate QPTLG-G (Mv-QPTLG-G) type can be easily derived based on the Clayton copula. However, future works may be allocated to study these new models.

2.1. BQPTLG-G type via Clayton Copula

Let and . Then depending on the continuous marginals and the Clayton copula can be expressed as
where
Let and
Then, the BQPTLG-G type distribution can be obtained from . A straightforward multivariate extension via Clayton copula can be derived.

2.2. BQPTLG-G Type via REC

The REC can be derived using the continuous marginal functions and as follows

2.3. BQPTLG-G Type via FGMC

Considering the FGMC, the joint CDF can be written as
where the continuous marginal function , and where , which under the grounded minimum condition and and under the grounded maximum condition. Clearly, the grounded minimum (maximum) conditions are valid for any copula. Setting and . Then, we have
Then, the joint PDF can be expressed as
where
or
where the two function and are PDFs corresponding to the joint CDFs and

2.4. BQPTLG-G Type via Modified FGMC

The modified formula of the modified FGMC can expressed as
with and where and are two continuous functions where . Then, let
and
Then, for we have
where
and
The following four types can be derived and considered:
Type I:
The new bivariate version via modified FGMC type I can written as
Type II:
Consider and which satisfy the above conditions where
and
Then, the corresponding bivariate version (modified FGMC Type II) can be derived from
Type III:
Let and . Then, the associated CDF of the BQPTLG-G-FGM (modified FGMC type III) as
Type IV:
Using the quantile concept, the CDF of the BQPTLG-G-FGM (modified FGMC type IV) model can be obtained using
where and .

2.5. BQPTLG-G type via AMHC

Under the “stronger Lipschitz condition” and following Ali et al. (1978), the joint CDF of the Archimedean Ali-Mikhail-Haq copula can written as
the corresponding joint PDF of the Archimedean Ali-Mikhail-Haq copula can be express as
then for any and we have
and

3. Mathematical Properties

3.1. Linear Representation

In this section, useful representation for the QPTLG-G PDF (5) is presented. First, expanding the quantity where
Then,
Compiling the expansion of in to (5), we have
(6)
Consider the power series
(7)
which holds for and real non-integer. Using (7), the QPTLG-G class in (6) can be written as
Which can be summarized as
(8)
where
and
and Equation (8) reveals that the density of the QPTLG-G family can then be expressed as a linear representation of exp-G PDFs. Also, the CDF of the QPTLG-G family can also be expressed as a mixture of exp-G CDFs. By integrating (8), we have
(9)
where is the CDF of the exp-G family with power parameter

3.2. Ordinary Moments

The ordinary moment of where follows QPTLG-G family with parameters is given by . Then we obtain
(10)
where denotes the density of the exp-G model with power parameter The expected value can be derived from when in (10). The integrations in and can be computed numerically for most parent distributions. The central moment of , variance skewness kurtosis and dispersion index measures can be derived using well-known relationships. The incomplete moment, say of can be expressed from (9) as . Then
(11)
The mean deviation about the mean and mean deviation about the median of are given by
and
respectively, where is the median, is obtained from (4) and is the first incomplete moment given by (11) with as
where is the first incomplete moment of the exp-G distribution. The moment generating function of can be derived as
where is the moment generating function of

3.3. Moment of the Residual Life

The moment of the residual life, say
Then, the moment of the residual life of can be given as
Therefore, using (8) we have
The life expectation can then be defined by
which represents the expected additional life length for a unit which is alive at age The MRL of can be obtained by setting in the last equation.

3.4. Moment of the Reversed Residual Life

The moment of the reversed residual life, say
The moment of the reversed residual life of can be given as
Then, the moment of the reversed residual life of becomes
The mean inactivity time (MIT) is given by
and it refers to the waiting time elapsed since the failure of an item on condition that this failure had occurred in .

3.5. Probability Weighted Moments

The PWM of following the QPTLG-G family, say is formally defined by
Using equations (4) and (5), we can write
where
and is defined above. Then, the th PWM of can be expressed as

3.6. Order Statistics

Let be a random sample from the QPTLG-G family of distributions and let be the corresponding order statistics. The PDF of order statistic, say can be written as
(12)
where is the beta function. Substituting (4) and (5) in equation (12) and using a power series expansion, we get
where
and
Then, the PDF of can be written as
Then, the density function of the QPTLG-G order statistics is a mixture of exp-G PDFs. Based on the last result, we note that the main properties of follow from those properties of and . For example, the moments of can be expressed as
(13)
Analogously to the ordinary moments we can derive the L-moments. However, the L-moments can also be estimated via a linear combination of the order statistics. Whenever the mean of the distribution exists, the L-moments are also existing. Based on the moments in equation (13), explicit expressions for the L-moments as infinite weighted linear combinations of the means of suitable QPTLG-G order statistics can be derived. The L-moments can be expressed as a linear function of the expected order statistics and can be defined by
where

4. Studying a Special Model

In this section, we shall focus on a new special QPTLG-G model based on the Lomax distribution called the QPTLG-G Lomax (PTLGL) distribution. Below, we present two theorems related to the exp-L distribution. The two theorems are employed in deriving the mathematical properties in Table 1.
Table 1. Theoretical results of the PTLGL model
     
Theorem I:
Let be a random variable having the exp-L distribution with power parameter Then the CDF of the exp-L model can be expressed as
Then, the ordinary moment of is given by
where
is the complete beta function.
Theorem 2:
Let be a random variable having the exp-L distribution with power parameter Then, the incomplete moment of is given by
where
is the incomplete beta function.

5. Estimation

Let be a random sample from the QPTLG-G distribution with parameters . For determining the maximum likelihood estimators (MLEs) of we have the log-likelihood function
The components of the score vector are
and
Setting the nonlinear system of equations and and solving them simultaneously yields the MLE. To solve these equations, it is usually more convenient to use nonlinear optimization methods such as the quasi-Newton algorithm to numerically maximize

6. Modeling Real-Life Data of Aircraft Windshields

Two real-life data applications to illustrate the importance and flexibility of the family is presented under the L case. The fits of the PTLGL are compared with other Lomax extensions shown in Table 2. However, many other Lomax extensions can be used in comparison such as the five-parameter Lomax distribution (Mead (2016)), new generalized of the Lomax distribution (Oguntunde et al. (2017)), the generalized odd Lomax Generated Family (Marzouk et al. (2019)) and the Zubair-inverse lomax distribution (Falgore (2020)). Table 3 below gives the goodness-of-fit (G-O-F) statistic tests which are used for comparing competitive models.
Table 2. The competitive models
     
Table 3. The goodness-of-fit (G-O-F) statistic tests
     
The 1st real-life data set (aircraft windshield consists of 84 aircraft windshield item) represents the data of failure times of 84 aircraft windshield. The 2nd real-life Data set (aircraft windshield consists of 63 aircraft windshield item) represents the data of service times of 63 aircraft windshield. The two real data were reported by Murthy et al. (2004). The “nonparametric Kernel density estimation (KDE)” tool is employed for exploring the initial PDF shape. The “normality” is also checked by the plot of the “Quantile-Quantile” (Q-Q). The initial HRF shapes explored via the “total time in test (TTT)” plot. The extremes are explored by the “box plot”. Based on Figures 1 and 2 (top left plots), it is shown that the HRFs are "monotonically increasing HRF" for the two data sets. Based on Figures 1 and 2 (top right plots), it is noted that the PDFs are asymmetric functions for the two data sets. Based on Figures 1 and 2 (bottom left plots), it is noted that the “normality” is exists. Based on Figures 1 and 2 (bottom right plots), we proved that no extremes are spotted.
Figure 1. TTT, NKDE, Q-Q and box plot for the 1st data
Figure 2. TTT, NKDE, Q-Q and box plot for the 2nd data
Tables 4 and 6 gives the MLEs and the corresponding standard errors (SEs) for the two real-life datasets. Tables 5 and 7 list the four G-O-F statistic tests for the two real-life datasets. Figures 3 and 4 give the Probability- Probability (P-P) plots, Kaplan-Meier Survival (KMS) plot, estimated PDF (E-PDF), estimated CDF (E-CDF) and estimated HRF (E-HRF) plot for the two data sets respectively. Based on Tables 5 and 7, it is noted that the PTLGL model gives the lowest values for all G-O-F statistics with AICr = 269.8712, CAICr = 270.6404, BICr = 282.0253 and HQICr = 274.7570 for the 1st data, and AICr = 208.582, CAICr = 209.6347, BICr = 218.2977 and HQICr = 212.7966 for the 2nd data among all fitted competitive models. So, it could be selected as the best model under these G-O-F criteria.
Table 4. MLEs and SEs for 1st data
     
Table 5. G-O-F statistics for 1st data
     
Table 6. MLEs and SEs for 2nd data
     
Table 7. G-O-F statistics for 2nd data
     
Figure 3. EPDF, EHRF, P-P, KMS plots for the 1st data set
Figure 4. EPDF, EHRF, P-P, KMS plots for the 2nd data set

7. Conclusions

A new compound G family of distributions called the Poisson Topp Leone generated-G (QPTLG-G) family is defined and studied. The QPTLG-G family is constructed by compounding the Poisson and the Topp Leone generated G families. Special case based on the Lomax model called the Poisson Topp Leone generated Lomax (PTLGL) model is studied and analyzed. Relevant properties of the PTLGL model including moment of the residual life, ordinary moments, Moment of the reversed residual life, incomplete moments, probability weighted moments, order statistics and mean deviation are derived and numerically analyzed. Several new bivariate QPTLG-G families using the “Clayton copula”, “Farlie-Gumbel-Morgenstern copula”, “modified Farlie-Gumbel-Morgenstern copula”, “Ali-Mikhail-Haq copula” and “Renyi’s entropy copula” are investigated. Two different applications to real-life datasets are presented to illustrate the applicability and importance of the QPTLG-G family. For the two real datasets: The “initial density shapes” are explored by the nonparametric Kernel density function estimated, the “normality condition” is checked by the “Quantile-Quantile plot”, the shape of the hazard rates are discovered by the “total time in test” graphical tool, the extremes are explored by the “box plots”. Based on the two applications, the PTLGL distribution gives the lowest values for all statistic tests where AICr = 269.8712, CAICr = 270.6404, BICr = 282.0253 and HQICr = 274.7570 for the failure times data; AICr = 208.582, CAICr = 209.6347, BICr = 218.2977 and HQICr = 212.7966 for the service times data.

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