Chen Pareto Distribution: Properties and Application

In this work, a new distribution called the Chen Pareto distribution was derived using the Chen-G family of distributions. The mixture representation of the distribution was obtained. Furthermore, some statistical properties such as moments, moment generating functions, order statistics properties of the distribution were explored. The parameter estimation for the distribution was done using the maximum likelihood estimation method and the performance of estimators was assessed by conducting an extensive simulation study. The distribution was applied to a real data set in which it performed best when compared to some related distributions


Introduction
In a bid to get distributions that are more flexible and better fit to real-life data, generalization(extension) of the distributions is done by the introduction of additional parameters to the baseline distribution. Recently, researchers do such an extension by introducing the baseline distribution to a family of distribution. Several mathematical properties of the extended distributions may be easily explored using mixture forms of exponentiated-G (exp-G) of the distributions. The Pareto distribution has been extensively studied and applied to many situations. This includes health, economics, finances, survival analysis, engineering, and Actuarial sciences. The distribution gains interest due to its ability to model heavy tailed data Lee and Kim (2018). It has many applications in actuarial science, survival analysis, economics, life testing, hydrology, finance, telecommunication, reliability analysis, physics, and engineering such as Brazauskas and Serfling (2003), Farshchian Chen Pareto Distribution : Properties and Application where k and σ is the scale and shape parameter respectively Considering a baseline cumulative distribution function (c.d.f) G(x) with a corresponding probability distribution function (p.d.f) g(x), the c.d.f of the Chen-G family of distribution is and the p.d.f as It is of importance to note that W is a normalizing constant, µ is the scale parameter, and φ is the shape parameter For this research, we considered deriving a generalized form of Pareto distribution using the Chen-G family of distribution Anzagra (2020). The plan of this article is as follows. Section 2 discusses the derivation of the Chen-Pareto distribution. In section 3, we studied the mixture representation of the p.d.f and the c.d.f of the Chen-Pareto distribution. Some statistical properties of the derived distribution were considered in section 4 after which the parameter estimation of the distribution is done in section 5. Section 6 discussed the simulation study to assess the performance of the parameters of the distribution. Section 7 discussed the application to real-life data. The conclusions were done in section 8.

Derivation of the Chen-Pareto Distribution
In this section, we discussed the derivation of the c.d.f and p.d.f of the Chen-Pareto Distribution. Inserting Equation (1) in (3) and Equation (2) in (4) respectively, a random variable X is said to be distributed to Chen Pareto distribution when its c.d.f F(x) is and the corresponding p.d.f is The Chen-Pareto distribution becomes pareto distribution when µ=1 and φ=1. It becomes Chen distribution when k=1 and σ=1

Mixture Representation
In this section, we derived the mixture representation of the Chen-Pareto distribution. This analytical derivation will find use subsequently for the study of the statistical properties of the Chen-Pareto Distribution. From the p.d.f of the Chen-Pareto distribution, Equation (6) can be re-written as Using Taylor series expansion x n n! (a) k=0.5,σ=0.5,µ=2,φ=2 (b) k=0.5,σ=2,µ=0.5,φ=10 (c) k=0.5,σ=1.5,µ=4, φ=0.5 Using the Taylor expression in (9) Inserting (11) and (8) in (7), we have the mixture representation for the p.d.f as where ζ = φ(r + l + 1) and where f of the Exponentiated Pareto Distribution with φ(r + l + 1) power parameter. Therefore the corresponding c.d.f for the mixture representation of the Chen-Pareto is derived by Integrating (13) to have where Θ ζ is the c.d.f of the Exponentiated-Pareto Distribution with power parameter φ(r + l + 1)

Statistical Properties
In this section, we study some statistical properties of the Chen-Pareto distribution. This includes survival function, hazard function, moments, moment generating function, order statistics, and entropies.

Reliability Function
The survival function s(x) of a Chen Pareto distribution is given as It has the hazard function h(x) as equation (15) and the reverse hazardh(x) as equation (16). Figure 3 shows the plots of the survival function and the hazard function with assigned parameter values.

Moments
In this section, the moments of the Chen-Pareto is derived. The moments help to describe the distribution. Let X 1 , X 2 , X 3 , ..., X n be distributed to a particular distribution, the r th is defined as Therefore, the r th moment of the Chen-Pareto is given as Finally,

Moment Generating Function
In this section, we derived the moment generating function of the Chen-Pareto. It is also a function that can help in the derivation of the moments of the distribution. The moment generating function of Chen Pareto distribution is obtained as Chen Pareto Distribution : Properties and Application Expanding e tx Inserting equation (20) in (19), we have

Order Statistics
If X (1) , X (2) , X (3) , ..., X (n) is an order statistics taken from a randomly sampled values X 1 , X 2 , X 3 , ..., X n with p.d.f f(x) and c.d.f F(x), then the p.d.f of the h th samples is Substituting equations (3) and (4) in (21), we have the order statistics of the Chen-Pareto distribution as Therefore, from (22), the p.d.f of the first order, X (1) (i.e h=1) is and the p.d.f of the last order, X (n) is n−1

Shanon Entropy
Entropy measures the uncertainty of a random variable X. Shannon(1948) defined the entropy of a random variable X. The entropy B for the Chen-Pareto distribution is given as This can be estimated iteratively.

Parametric Estimation
In this section, the parametric estimation of the distribution is conducted. Let X 1 , X 2 , ..., X n be samples assumed to be distributed to the Chen Pareto distribution, the loglikelihood l is given as Differentiating l with respect to the distribution parameters, we have Setting ∂l ∂φ = 0, ∂l ∂µ = 0, ∂l ∂σ = 0, ∂l ∂b = 0 and solving the resulting equations will result in obtaining the estimatesφ ,μ,σ,k respectively. As n|∞, we obtained the asymptotic distribution of the estimates obtained from equations (27)-(30) aŝ where θ= φ, σ, µ, k ' and α= α is the variance-covariance matrix of the estimates.An asymptotic confidence interval with significance level γ for each parameter θ i is given by where αθ ,θ is the i t h diagonal element of K n (θ) −1 for i = 1, 2, 3, 4 and z γ/2 is the quantile of the standard normal distribution. The expressions of the elements of α are in Appendix 1

Simulation Studies
In this section, we carried out the simulation study to assess the performance of the maximum likelihood estimates of the Chen-Pareto distribution. The simulation study was conducted by deriving a random sample of sizes 50, 100, 200,300 from the Chen Pareto distribution for different values of the parameters. For each sample size, the maximum likelihood estimators are obtained and the procedure is repeated 10,000 times. We have computed the Average estimate, bias, and MSE of parameter estimates for these 10,000 values, and the results are presented in Tables 1, 2, 3 and 4 . The results show that the Bias is small, that is the estimated values are close to the true values. Also, the MSE reduces as the sample sizes increase for each parameter. This shows that the estimation method is adequate and consistent.    Mudholkar and Huston (1996) e.t.c. Table 5 gives the summary statistics of the dataset. The Pareto and Cubic Transmuted Pareto were fitted using this data. The M.L.E's and their respective -loglikelihood with corresponding Akaike Information Criterion, Bayesian Information Criterion are presented in Table 6. From the results on the table, using the comparison criterion, it is sufficient to say that the Chen-Pareto is the most appropriate for this data.  In statistics, generalization of distributions have attracted the interest of Statisticians in a bid to get better fits for real-life data. In this work, the Chen-Pareto distribution, which is a generalized distribution of the Pareto distribution, was studied. The Chen-Pareto distribution extends the pareto distribution by adding two shape parameters. The mixture representation of the distribution which enables us to study some statistical properties such as moments, moment generating function, order statistics, entropy, reliability functions, and quantile function was provided. The estimation of the parameters of the model has also been discussed. A simulation study to assess the adequacy of the estimation method and consistency of the parameters was conducted, and It has been shown, by means of a real data sets, that the Chen-Pareto distribution can provide better fits than the Pareto distribution.