New Bivariate Pareto Type II Models

Pareto type II distribution has been studied from many statisticians due to its important role in reliability modelling and lifetime testing. In this article, we introduce two bivariate Pareto Type II distributions; one is derived from copula and the other is based on mixture and copula. Parameter Estimates of the proposed distribution are obtained using the maximum likelihood method. The performance of the proposed bivariate distributions is examined using a simulation study. Finally, we analyze one data set under the proposed distributions to illustrate their flexibility for real-life applications.


Introduction
The Pareto Type-II distribution or Pearson Type-VI distribution is called Lomax distribution introduced and studied by [1]. This distribution is commonly used in reliability and many lifetime testing studies. It is also used to analyze business data. Let T be a random variable from the Pareto type II (PII) distribution with scale parameter β and shape parameter α, then the probability density function (PDF) and the cumulative density function (CDF) of PII distribution are given respectively by f(T) = αβ (1 + βt) α+1 , t > 0 (1) The survivor function (SF) is given by: The hazard rate function (HRF) and the cumulative hazard rate function (CHRF) are H(T) = α ln(1 + βt) , t > 0 Dubey [2] showed that Pareto Type II distribution can be derived as a special case of a compound gamma distribution. Bryson [3] discussed that Lomax distribution provides an excellent alternative to classical distributions such as the exponential and Weibull distributions. Ahsanullah [4] studied the record statistics of the Lomax distribution using distributional characteristics. Balakrishnan and Ahsanullah [5] acquired some repeated relations between the moments of record values for the Lomax distribution. The Lomax distribution was used as a mixing distribution for the Poisson parameter to derive the discrete Poisson-Lomax distribution [6]. Petropoulos and Kourouklis [7] considered where for j = 1, 2, f T j and F T j are given by (1) and (2), respectively, and C is the density of the bivariate Gaussian copula obtained by differentiating C, such that C = exp −1 2(1−ρ 2 ) y 2 1 − 2ρy 1 y 2 + y 2 2 2π 1 − ρ 2 (6) where y 1 = Φ −1 (v 1 ) and y 2 = Φ −1 (v 2 ). For details see, [33][34][35]. Therefore, the joint PDF of BPII distribution with PII marginal can be rewritten as where v j = F T j , j = 1, 2, given by (1), C (v 1 , v 2 ) given by (6). For more details, see [36,37]. Plots of the BPIIG distribution PDF, CDF, and contour for α 1 = 1.5 , α 2 = 2, β 1 = 0.01, β 2 = 0.03, and two values of the copula parameter ρ are presented in Figure 1.

BPIImG Distribution
The construction of BPIImG distribution depends on the mixture representation described in [25,38,39]. The idea of mixture representation is to write the density of a random variable T on (0, ∞) in the form of compound distribution as follows: where Ω is a subset of R, U is a non-negative latent random variable following a gamma distribution with shape parameter 2 and scale parameter 1, denoted by gamma (2,1). And f T|U (t|u) can be written as follows where h(t) is the HRF, and H(t) is CHRF. That is, the mixture and copula methods are combined to obtain bivariate distribution. This is conducted by constructing a bivariate gamma distribution of latent variable U = (U 1 , U 2 ) with two marginal gamma (2,1) distributions using Gaussian copula. At first stage, we obtain a bivariate gamma distribution with only unknown correlation parameter ρ such as where C (v 1 , v 2 ) is given by (6), f u j is the PDF of gamma (2,1), v j = F u j is the CDF of gamma (2,1) given by Then as a second stage, a bivariate gamma distribution in (8) is used as a mixing distribution of T 1 , T 2 , assuming that T 1 , T 2 are conditionally independent given U. The conditional PDF can be written as f(t j |u j ) = α j β j 1 + β j t j e −u j , u j > α j ln 1 + β j t j (10) And then integrate over the latent variables U to obtain the joint PDF of BPIImG distribution is as follows using the above two stages method will help in the model analysis, because we can estimate the correlation parameter ρ from the first stage (i.e., the bivariate gamma distribution). Then, estimate the other parameters from the second stage (i.e., the conditional density functions f(t j |u j )).
In addition, we can obtain approximate confidence interval (CI) of the parameters β 1 , α 1 , β 2 , α 2 , and ρ by using large sample theory and ML estimates of asymptotic distribution. That is, θ = (β 1 , α 1 , β 2 , α 2 , ρ) ∼ multivariate normal θ, I −1 (θ) , where I −1 is the inverse of the observed information matrix given by The second derivative of (13) with respect to the parameters are provided in the Appendix A. Therefore, 100(1 − γ)% approximate CI for the parameters β 1 , α 1 , β 2 , α 2 , and ρ for j = 1, 2 are given bŷ where: z γ/2 is the upper (γ/2)% of the standard normal distribution. The CI of the parameters could be adjusted for the lower bound using the method in [40].

ML Estimates of BPIIG
ML parameter estimates of the BPIIG distribution are shown in Table 1 along with the corresponding relative mean square error (RMSE). The results in Table 1 show that as the sample size increases, the RMSE of the parameters estimates become smaller. In addition, most parameters have better estimates and smaller RMSEs when the copula parameter equal to 0.80.

ML Estimates of BPIImG
Parameter estimates of BPIImG distribution using ML methods are illustrated in Table 2. In addition, the average estimates along with their RMSE over 1000 replication are reported. The results reported in Table 2 indicate that the RMSE of the parameter estimates decreases as the sample size increases. Also, we obtained better estimates of the parameters with smaller RMSE especially the estimate of ρ when the copula parameter is equal to 0.80 and the sample size is more than 150.

Models Comparison
We compared the flexibility of the BPIIG and BPIImG distributions based on RMSE, Akaike information criterion (AIC), and Bayesian information criterion (BIC) values. The results in Table 3 indicate that the BPIImG distribution has lower values of AIC and BIC. Therefore, we conclude that BPIImG distribution is more flexible and perform better than BPIIG.

Data Analysis
The American football league data obtained from the matches played on three consecutive weekends in 1986 have two variables T 1 and T 2 where; T 1 is the game time the first fields scored when the ball kicks between goalposts and T 2 is the game time the first touchdown is scored, see [41]. The histogram and the scatter plots of T 1 and T 2 are right skewed and positively correlated [29]. The sample Spearman correlation coefficient between T 1 and T 2 is 0.804 which allows using the proposed BPII distribution to model this bivariate data. Also, we conducted goodness of fit test by fitting the marginals only, see [42].
That is, the PII distribution is fitted to the marginals and the ML estimates of the parameters are: β 1 = 0.011,α 1 = 9.519,β 2 = 0.0141,α 2 = 5.3778. The plots of the fitted and the empirical CDF for the two marginals based on ML estimates are illustrated in Figure 2

Data analysis
The American football league data obtained from the matches played on three consecutive weekends in 1986 have two variables T and T where; T is the game time the first fields scored when the ball kicks between goalposts and T is the game time the first touchdown is scored, see [41]. The histogram and the scatter plots of T and T are right skewed and positively correlated [29]. The sample Spearman correlation coefficient between T and T is 0.804 which allows using the proposed BPII distribution to model this bivariate data. Also, we conducted goodness of fit test by fitting the marginals only, see [42].
That is, the PII distribution is fitted to the marginals and the ML estimates of the parameters are:  Hence, the K-S test along with the plots of the fitted and the empirical CDF in Figure 2 indicate that the BPII distribution has an appropriate fit for this bivariate data. In addition, the Gaussian copula is appropriate for this data as indicated in [29]. For more details, see [43]. Table 4 reports the ML estimates of the parameters along with the standard error (SE) of the BPIIG and BPIImG parameters. It can be seen from Table 4 that the AIC of BPIImG distribution is smaller compared to BPIIG distribution. This indicates that BPIImG distribution is more appropriate for this data. Hence, the K-S test along with the plots of the fitted and the empirical CDF in Figure 2 indicate that the BPII distribution has an appropriate fit for this bivariate data. In addition, the Gaussian copula is appropriate for this data as indicated in [29]. For more details, see [43]. Table 4 reports the ML estimates of the parameters along with the standard error (SE) of the BPIIG and BPIImG parameters. It can be seen from Table 4 that the AIC of BPIImG distribution is smaller compared to BPIIG distribution. This indicates that BPIImG distribution is more appropriate for this data. The model's comparison illustrated in [29] is re-conducted to compare BPIIG and BPIImG with Bivariate expatiated Pareto derived from the mixture and Gaussian copula (BEPmG), bivariate exponentiated generalized Weibull-Gompertz distribution (BEGWG) studied by [44], and bivariate exponentiated Gompertez distribution (BEG) using the same real data set.
The results in Table 5 show that BPIImG distribution has the lowest AIC, and BIC values compared the BEPmG, BEGWG, BEG and BPIIG distributions. Therefore, BPIImG provides a more appropriate and flexible fit for this data set.

Conclusions
In this article, we introduced BPIIG and BPIImG distributions. Parameter estimates of the proposed bivariate distributions are obtained using the ML method. A simulation study is carried out to show the performance of the proposed bivariate distributions. We concluded that the BPIImG distribution is more flexible and performs better than the BPIIG distribution. A real lifetime data is analyzed, and the results showed that the BPIImG distribution provides a more suitable fit than the BPIIG, BEPmG, BEG, and BEGWG distributions.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A
The second partial derivatives will be simplified as follows: