A Novel Generator of Continuous Probability Distributions for the Asymmetric Left-skewed Bimodal Real-life Data with Properties and Copulas

This paper presents a novel two-parameter G family of distributions. Relevant statistical properties such as the ordinary moments, incomplete moments and moment generating function are derived. Using common copulas, some new bivariate type G families are derived. Special attention is devoted to the standard exponential base line model. The density of the new exponential extension can be “asymmetric and right skewed shape” with no peak, “asymmetric right skewed shape” with one peak, “symmetric shape” and “asymmetric left skewed shape” with one peak. The hazard rate of the new exponential distribution can be “increasing”, “U-shape”, “decreasing” and “Jshape”. The usefulness and flexibility of the new family is illustrated by means of two applications to real data sets. The new family is compared with many common G families in modeling relief times and survival times data sets.


Introduction and motivation
Statistical literature contains various G families of distributions which were generated either by compounding common existing G families or by adding one (or more) parameters to the existing G families. These novel families were employed for modeling real data in many applied studies such as engineering, insurance, demography, medicine, econometrics, biology, environmental sciences and forecasting approaches see, for example, see Yousof  Suppose that a system has subsystems functioning independently at a given time where has ZTP distribution with parameter = 1. It is the conditional probability distribution of a Poisson-distributed random variable (RV), given that the value of the RV is not zero. The probability mass function (PMF) of is given by (2) Therefore, the unconditional CDF of the QPGW-G density function can be expressed as described in Ramos et al. The proposed family is most conveniently specified in terms of the ZTP generator applied to the generalized Weibull-G class. By inserting (1) in equation (2), the CDF of the quasi-Poisson generalized Weibull-G (QPGW-G) family is given by where = ( , , ) is the parameter vector of the QPGW-G family.
The following special cases can be considered: i.
For = 1, the QPGW-G family reduces to the quasi-Poisson generalized exponential-G family.

ii.
For = 2, the QPGW-G family reduces to quasi-Poisson generalized Rayleigh-G family.

iii.
For = 1, the QPGW-G family reduces to the quasi-Poisson Weibull-G family ). iv.
For = 2 and = 1, the QPGW-G family reduces to the reduced quasi-Poisson exponential-G family ).
The PDF of the QPGW-G family can then be expressed as A RV having PDF (4) is denoted by ∼QPGW-G ( ). Many related G families can be mentioned such as On the other hand, many common copulas are employed for deriving new bivariate type QPGW-G families such as "Farlie-Gumbel-Morgenstern (FGM) copula", "Clayton copula", and "Renyi's entropy copula". Fisher (1997) provided two major justifications as to why copulas are useful and of interest to statisticians. Firstly, as a way for studying scale-free measures of dependence.
Secondly, as a starting point for constructing new bivariate G families of distributions. Precisely, copulas are an important part of the study of dependence between two variables since they allow us to separate the effect of dependence from the effects of the marginal distributions. Further future articles could be allocated to study the new bivariate type G families.
We are motivated to present the new family since it could be useful in modeling variable real-life data as illustrated below: i. The real data which have an " increasing failure rate " (see iii. The real data sets which their Kernel density estimation are asymmetric and bimodal with right tail (see Figures  3 and 5 (bottom right plots)). iv. The real data which their PDF can be "asymmetric and right skewed shape" with no peak, "asymmetric right skewed shape" with one peak, "symmetric shape" and "asymmetric left skewed shape" with one peak (see Figure  1). v. The real-life datasets which their HRF can be "increasing", "U-shape", "decreasing" and "J-shape" (see Figure  2).
Additionally, in modeling the relief times data and the survival times of the aircraft windshield data the novel family based on the quasi Poisson generalized Weibull-exponential model is better than many other common exponential extensions such as the Marshall-Olkin exponential, Moment exponential, the Burr-Hatke exponential, Generalized Marshall-Olkin exponential, the odd Lindley exponential, Beta exponential, Kumaraswamy Marshall-Olkin exponential, Marshall-Olkin Kumaraswamy exponential, the Burr X exponential, Kumaraswamy exponential and standard exponential model under the eight criteria called Anderson-Darling Criteria, Akaike Information Criteria, Cramér-Von Mises Criteria, Hannan-Quinn Information Criteria, Bayesian Information Criteria, Consistent Akaike Information Criteria, Kolmogorov-Smirnov (KS) statistic test and its corresponding P-value.
Recently, many new articles have been studied some copulas such as Ali et al. . However, future works may be allocated to study these new models.

BQPGW-G type via Clayton copula
The Clayton copula can be considered as Let us assume that ∼QPGW-G ( 1 , 1 ) and ∼ QPGW-G ( 2 , 2 ). Then, setting Then, the BQPGW-G type distribution can be derived as

BQPGW-G type via Ali-Mikhail-Haq copula
Under the stronger Lipschitz condition, the joint CDF of the Archimedean Ali-Mikhail-Haq copula can expressed as the corresponding J-PDF of the Archimedean Ali-Mikhail-Haq copula can expressed as then for any 1 ∼ QPGW-G ( 1 , 1 ) and 2 ∼ QPGW-G ( 2 , 2 ) we have The J-PDF is straightforward then omitted.

The MvQPGW-G type
Following Nelsen (2007) and Balakrishnan and Lai (2009), a straightforward Multivariate QPGW-G ℏ-dimensional extension can be derived from

Mathematical Properties 3.1 Linear representation
In this section, we derive a useful linear representation for the QPGW-G density function. Using the power series, we expand the quantity ( ) as Then, the PDF in (4) can be expressed as Then, consider the power series Applying (6) to the quantity ( ) in (5), we get Expanding the quantity ( ) in power series, we can write Inserting the above expression of ( ) in (9), the QPGW-G density reduces to Using the generalized binomial expansion to , we can write Inserting (10) in (9), the QPGW-G density can be expressed as an infinite linear combination of exp-G density functions where * ( ) = * ( )/ = * ( ) ( ) * −1 is the exp-G PDF with power parameter * and Equation (11)  Similarly, the CDF of the QPGW-G family can also be expressed as a linear combination of exp-G CDFs given by where * ( ) is the exp-G CDF with power parameter * .

Moments
The th moment of , say , ′ , follows from equation (12) as where * denotes the exp-G RV with power parameter * . The th central moment of , say , is given by

Moment generating function and the characteristic function
The moment generating function (MGF) of can follow from equation (12) as where * ( ) is the MGF of * (for , ≥ 0). Hence, ( ) can be easily obtained from the exp-G generating function. The characteristic function (CF) of can be derived from where * ( ) is the CF of * (for , ≥ 0) and = √−1.

Incomplete moments
The th incomplete moment, say , ( ) , of can be expressed from (12) Clearly, the integral in equation (13) denotes the th incomplete moment of * .

Convex-concave analysis
Convex probability density functions play a very important role in many areas of mathematics. They are important especially in studying of the "optimization problems" where they are distinguished by several convenient properties.
In mathematical analysis, a certain PDF defined on certain n-dimensional interval is called "convex probability density function " if the line between any two points on the graph of the probability density function lies above the graph between the two points.

A special case
In this section, we will focus on the base line exponential distribution. The CDF of the standard exponential model can be expressed as For = 2, QPGW-E distribution reduces to quasi-Poisson generalized Rayleigh-exponential distribution.

v.
For = 2 and = 1 the QPGW-E reduces to the reduced quasi-Poisson exponential-exponential distribution.
The PDF of the QPGW-E distribution can then be derived using (4). Figure 1 gives some plots of the PDF of the QPGW-E distribution for some selected parameter values. Figure 2 gives some plots of the HRF of the QPGW-E distribution for some selected parameter values. Based on Figure 1, we note that the new PDF of the QPGW-E distribution can be "asymmetric and right skewed shape" with no peak, "asymmetric right skewed shape" with one peak, "symmetric shape" and "asymmetric left skewed shape" with one peak.  Based on Figure 2, it is noted that the new HRF can be "increasing", "U-shape", "decreasing" and "J-shape". In the literature there are various exponential extensions which can be used in comparison such as Beta exponential (BE) model (see Lee

Modeling failure (relief) times
The first data set is related to Gross and Clark (1975) and called the failure time data. The data represents the lifetime observations relating to relief times (in minutes) of patients receiving an analgesic. The Gross and Clark data is recently analyzed by Al-Babtain et al. (2020) and Ibrahim et al. (2020). Table 1 below gives the MLEs, standard errors (SE(s)) and corresponding confidence intervals (C.I.s) for the Gross and Clark data. Table 2 below provides the AI-C, BI-C, CAI-C, HQI-C, AD − C, CvM-C, K.S. and p-value for the Gross and Clark data. Figure 3 gives the box plot (top left), quantile-quantile plot (top right), total time in test (TTT) plot (bottom left) and non-parametric Kernel density estimation (NKDE) plot (bottom right) for the relief times data. Based on Figure 3 (top left and top right), the relief times data has one outlier observation. Based on Figure 3 (bottom left), the HRF of the relief times is "monotonically increasing HRF".    Based on Figure 3 (bottom right), NKDE of the relief times data bimodal and right skewed. Figure 4 gives the fitted density, fitted CDF, P-P plot, estimated HRF and fitted survival function for relief data. Based Figure 4, it is noted that the QPGW-E model provides adequate fits to the relief data. Based on results of Table 2, we see that the QPGW-E lifetime model is better than the exponential, Odd Lindley exponential, Marshall-Olkin exponential, Moment exponential, The Logarithmic Burr-Hatke exponential, generalized Marshall-Olkin exponential, Beta exponential, Marshall-Olkin Kumaraswamy exponential, Kumaraswamy exponential, the Burr X exponential and Kumaraswamy Marshall-Olkin exponential models with AI − C = 38.50, BI − C = 41.4, CAI − C = 39.99, HQI − C = 39.08, AD − C = 0.319, CvM − C = 0.0539, KS=0.137 and p-value=0.8469 so the new lifetime model is a good alternative to these models in modeling relief times data set.

Modeling survival times
The second data set called the survival times (in days) of 72 guinea pigs infected with virulent tubercle bacilli, observed and reported by Bjerkedal (1960). This data was recently analyzed by Ibrahim et al. (2020) and Al-Babtain et al.
(2020). Table 3 below gives the MLEs, SE(s) and corresponding confidence intervals (C.I.s) for the guinea pigs data. Table 2 below provides the AI-C, BI-C, CAI-C, HQI-C, AD − C, CvM-C, K.S. and p-value for the guinea pigs data. Figure 5 gives the box plot (top left), quantile-quantile plot (top right), the TTT plot (bottom left) and the NKDE plot (bottom right) for the survival times data. Based on Figure 5 (top left and top right), the survival times data has some outlier observations. Based on Figure 5 (bottom left), the HRF of the survival times is "monotonically increasing HRF". Based on Figure 5 (bottom right), NKDE of the survival times data bimodal and right skewed. Figure 6 gives the fitted density, fitted CDF, P-P plot, estimated HRF and fitted survival function for survival data. Based Figure 6, it is noted that the QPGW-E model provides adequate fits to the survival data.
Based on results of Table 5, it is concluded that the QPGW-E lifetime model is better than the exponential, Odd Lindley exponential, Marshall-Olkin exponential, Moment exponential, The Logarithmic Burr-Hatke exponential, generalized Marshall-Olkin exponential, Beta exponential, Marshall-Olkin Kumaraswamy exponential, Kumaraswamy exponential, the Burr X exponential and Kumaraswamy Marshall-Olkin exponential models with     Figure 6: Fitted density, fitted CDF, P-P plot, estimated HRF and fitted survival function for survival data.

Conclusions
A novel two-parameter compound G family of distributions is derived and studied. Relevant statistical properties such as the ordinary moments, incomplete moments and moment generating function are derived. Using common copulas such as "Farlie-Gumbel-Morgenstern copula", "Ali-Mikhail-Haq copula", "Clayton copula" and "Renyi copula", some new bivariate type G families are derived. A special attention is devoted to the quasi-Poisson generalized Weibull-exponential distribution as a special case.
The density of the quasi-Poisson generalized Weibull-exponential distribution can be "asymmetric and right skewed shape" with no peak, "asymmetric right skewed shape" with one peak, "symmetric shape" and "asymmetric left skewed shape" with one peak.
The hazard rate of the quasi-Poisson generalized Weibull-exponential distribution can be "increasing", "U-shape", "decreasing" and "J-shape". The usefulness and flexibility of the quasi-Poisson generalized Weibull-exponential distribution is illustrated by means of two applications to real data sets.
The new the quasi-Poisson generalized Weibull-exponential distribution is much better than many common exponential extensions in modeling relief times and survival times data sets under the eight criteria called Anderson-Darling Criteria, Akaike Information Criteria, Cramér-Von Mises Criteria, Hannan-Quinn Information Criteria, Bayesian Information Criteria, Consistent Akaike Information Criteria, Kolmogorov-Smirnov (KS) statistic test and its corresponding p-value. As a future interesting works, many new statistic tests can be used for right censored validation such as the Nikulin Rao Robson (N.R.R) goodness-of-fit statistic test and Bagdonavicius-Nikulin (Bag.N) goodness-of-fit statistic test test (see Goual