A New Parametric Life Family of Distributions: Properties, Copula and Modeling Failure and Service Times

: A new family of probability distributions is deﬁned and applied for modeling symmetric real-life datasets. Some new bivariate type G families using Farlie–Gumbel–Morgenstern copula, modiﬁed Farlie–Gumbel–Morgenstern copula, Clayton copula and Renyi’s entropy copula are derived. Moreover, some of its statistical properties are presented and studied. Next, the maximum likelihood estimation method is used. A graphical assessment based on biases and mean squared errors is introduced. Based on this assessment, the maximum likelihood method performs well and can be used for estimating the model parameters. Finally, two symmetric real-life applications to illustrate the importance and ﬂexibility of the new family are proposed. The symmetricity of the real data is proved nonparametrically using the kernel density estimation method.


Introduction and Genesis
Statistical probability distributions are an important tool in modeling the characteristics of real-life datasets such as "symmetric" or "right" or "left" skewness, "symmetric/asymmetric bi-modality" or "multi-modality" in different applied sciences such as reliability, medicine, engineering and finance, among others. The well-known distributions such as Burr X, normal, Burr XII, Weibull, beta, gamma, Kumaraswamy and Lindley are widely used because of their simple forms and identifiability of their properties. However, in the last few decades, statisticians have focused on the more complex and flexible distributions to increase the modeling ability of these models via adding one or more shape parameters. The well-known family of distributions can be cited as follows: the Marshall-Olkin-G (MO-G) family [1], the beta-G (B-G) family [2], the Kumaraswamy-G (K-G) family [3], the Burr X-G (BX-G) family [4], the Burr XII-G (BXII-G) family [5] and a new Weibull class of distributions [6], among others.
The rest of the paper is outlined as follows: In Section 2, we derive a useful representation for the KBX-G density function. Five special models of this family are presented in Section 3 corresponding to the baseline exponential, Weibull, Rayleigh, Lomax and Burr XII distributions. We obtain in Section 4 some general mathematical properties of the proposed KBX-G family. In Section 5, simple-type copula using Farlie-Gumbel-Morgenstern (FGM) copula, modified FGM copula, Clayton copula and Renyi's entropy are presented. Maximum likelihood estimation is defined in Section 6. Section 7 provides a graphical simulation study based on Kumaraswamy Burr X Lomax model. Section 8 provides two applications. Finally, concluding remarks are listed in Section 9.

Special Models
This section presents some special KBX models based on Exponential (E), Weibull (W), Rayleigh (R), Lomax (Lx) and Burr XII (BXII) distributions. Table 1 below presents some new submodels based on the new KBX-G family. For the KBXLx, Figure 1 (right panel) and Figure 1 (left panel) give some plots of its PDF and the hazard rate function (HRF), respectively. Based on Figure 1 (right panel), the PDF of the KBX Lomax (KBXLx) can be "symmetric PDF" and "asymmetric right skewed PDF" with many useful shapes. Based on Figure 1 (left panel), the HRF of the KBXLx can be "decreasing-constant-increasing or U-shape HRF", "decreasing HRF", increasing HRF" and "J-HRF". Table 1. New Sub Models Based on the New Kumaraswamy Burr X-G (KBX-G) Family.

Moments
Let Z 2 j+k+2 be a nonnegative random variable which has the Exp-G with power parameter 2 j + k + 2. The rth moment of the KBX-G family can be obtained from (11) as can be calculated and analyzed numerically in terms of the baseline quantile function Q G (u), i.e., Q G (u) = G −1 (u) as For the KBXLx model Here, we introduce a formula for the moment generating function (MGF) as where M 2 j+k+2 (t) is the moment generating function (MGF) of Z 2 j+k+2 . Consequently, we can easily determine M X (t) from the Exp-G MGF as which can be calculated numerically from the baseline quantile function, i.e., Q G (u) = G −1 (u). Thus, for the KBXLx model we have

Conditional Moments
The sth incomplete moments of X defined by Ω s (t) for any real s > 0 can be expressed from (11) as where

BvKBX Type via FGM Copula
The joint PDF can then derived from c π (u, v)

BvKBX Type via Modified FGM Copula
Consider the modified version of FGM copula defined as (see [11]) Where ϑ(u) and ω(v) are two functions on (0, 1) with the following conditions: II Let Then,

BvKBX-FGM (Type II) model
Let ϑ(u) and ω(v) be two functional forms to satisfy all the conditions stated earlier where The corresponding BvKBX-FGM (Type II) can be derived from

BvKBX-FGM (Type III) model
Consider the following functional form for both Θ(u) and ϕ(v) which satisfy all the conditions stated earlier where Symmetry 2020, 12, 1462 8 of 20 In this case, one can also derive a closed form expression for the associated CDF of the BivKBX-FGM (Type III).

BvKBX-FGM (Type IV) model
According to [13], the CDF of the BvKBX-FGM (Type IV) model can be derived from Then,

BvKBX Type via Clayton Copula
The Clayton copula can be considered as Then, the BvKBX type distribution can be derived as

BvKBX Type via Renyi's Entropy
Consider the theorem of [14] where then, the associated BvKBX will be By fixing a and b we then have a five-dimension parameter BvKBX-type distribution.

MvKBX Extention via Clayton Copula
The m-dimensional extension from the above will be Then, the MvKBX extension can expressed as

Maximum Likelihood Estimation
Let ψ = (α, θ, λ, δ) T be the vector of parameters, then the log-likelihood function for ψ is given by The components of the score function U n ψ = U n (α), U n (θ), U n (λ), U n δ . Setting the nonlinear system of equations U n (α), U n (θ), U n (λ), U n δ equal to zero and solving the equations simultaneously yields the maximum likelihood estimation (MLE) of ψ, where these equations cannot be solved analytically; thus, we use any statistical software to solve these equations.

Simulations
To assess of the finite sample behavior of the MaxLEs under the KBXLx model, we will consider and apply the following algorithm: 1-Use to generate N = 1000 samples of size n from the KBXLx distribution; 2-Compute the MaxLEs for the N = 1000 samples. 3-Compute the standard errors (StErs) of the MaxLEs for the N = 1000 samples. 4-Compute the biases (B h ) and mean squared errors (MSEs) given for h = α, θ, λ, a, b. We repeated these steps for n = 100, . . . , 500 with α = θ = λ = a = b = 1 so computing biases, mean squared errors (MSE h ) for α, θ, λ, a, b and n = 100, . . . , 500.

Applications and Comparing Models
In this section, we provide two real-life applications to illustrate the importance, potentiality and flexibility of the KBXLx model. We compare the fit of the KBXLx with some well-known competitive models (see Table 2).
Dataset I (84 Aircraft Windshield): Times of Failure: The first real-life dataset represents the data on times of failure for 84 aircraft windshields (see [15] Many other useful real-life datasets can be found in [16][17][18][19][20][21][22][23][24][25][26][27]. For exploring the outliers, the box plot is plotted in Figure 7. Based on Figure 7, we note that no outliers were found. For checking the data normality, the Q-Q plot is sketched in Figure 8. Based on Figure 8, we note that normality nearly exists. For exploring the shape of the HRF for real data, the total time test (TTT) plot (see [28]) is provided (see Figure 9). Based on Figure 9, we note that the HRF is "increasing monotonically" for the two real-life datasets. For exploring the initial shape of real data nonparametrically, the KDE is provided in Figure 10. Figures 11 and 12 give the estimated Kaplan-Meier survival (EKMS) plot, estimated PDF (EPDF), P-P plot and estimated HRF (EHRF) for dataset I and II, respectively.

Applications and Comparing Models
In this section, we provide two real-life applications to illustrate the importance, potentiality and flexibility of the KBXLx model. We compare the fit of the KBXLx with some well-known competitive models (see Table 2).
Dataset I (84 Aircraft Windshield): Times of Failure: The first real-life dataset represents the data on times of failure for 84 aircraft windshields (see [15] Many other useful real-life datasets can be found in [16][17][18][19][20][21][22][23][24][25][26][27]. For exploring the outliers, the box plot is plotted in Figure 7. Based on Figure 7, we note that no outliers were found. For checking the data normality, the Q-Q plot is sketched in Figure 8. Based on Figure 8, we note that normality nearly exists. For exploring the shape of the HRF for real data, the total time test (TTT) plot (see [28]) is provided (see Figure 9). Based on Figure 9, we note that the HRF is "increasing monotonically" for the two real-life datasets. For exploring the initial shape of real data nonparametrically, the KDE is provided in Figure 10. Figures 11 and 12 give the estimated Kaplan-Meier survival (EKMS) plot, estimated PDF (EPDF), P-P plot and estimated HRF (EHRF) for dataset I and II, respectively.  Failure times data Service times data  Failure times data Service times data   Failure times data Service times data   Failure times data Service times data  Tables 3 and 4. Table 3 gives the MaxLEs and StErs for failure time data. Table 4 gives the −ℓ and GOF statistics for failure time data. For service time data, the analysis results are listed in Tables 5 and 6. Table 5 gives the MaxLEs and StErs for service time data, and Table 6 gives the −ℓ and GOFs statistics for the service time data. Based on Tables 4 and 6, we note that the KBXLx model gives the lowest values for the AIC, CAIC, BIC, HQIC, • and • among all the fitted models. Hence, it could be chosen as the best model under these criteria.   Tables 3 and 4. Table 3 gives the MaxLEs and StErs for failure time data. Table 4 gives the −ˆ and GOF statistics for failure time data. For service time data, the analysis results are listed in Tables 5  and 6. Table 5 gives the MaxLEs and StErs for service time data, and Table 6 Figure 11. EKMS plot, EPDF, P-P plot and EHRF for dataset I. It is worth mentioning that the presented class of stochastic distributions can have also an important utilization in the insurance industry, risks, reliability, medicine and engineering (see [36][37][38][39][40][41][42][43][44][45][46][47]).

Conclusions
A new family of probability distributions based on the Kumaraswamy and Burr X families is derived and studied. Some new bivariate type G families using Farlie-Gumbel-Morgenstern copula, modified Farlie-Gumbel-Morgenstern copula, Clayton copula and Renyi's entropy copula are derived. Some special models based on exponential, Weibull, Rayleigh, Lomax and Burr XII distributions are presented. Then, special attention is devoted to the Lomax case. Some of its statistical properties, including the quantile function, moments, incomplete moments and mean deviation, are presented. Simulation based on biases and mean squared errors is introduced. Based on this simulation, the maximum likelihood method performs well and can be used for estimating the model parameters. Finally, two real-life applications to illustrate the importance of the new family are proposed.
Supplementary Materials: The first and second datasets are given by [15]. https://onlinelibrary.wiley.com/doi/ book/10.1002/047147326X. Author Contributions: All authors contributed equally for this work. All authors have read and agreed to the published version of the manuscript.
Funding: King Saud University.