Pareto-Weibull Distribution with Properties and Applications: A Member of Pareto-X Family

In this paper, we have proposed a new family of distributions namely the Pareto-X family of distributions. A sub model of the proposed family called Pareto-Weibull (PW) distribution is discussed. Some important properties of the proposed PW distribution are studied. Parameter estimation of the distribution by using the maximum likelihood method is discussed. The proposed distribution has been fitted on two real data sets about environmental and biological variable. The practical applications shows that the proposed model provides better fit as compared with the other models used in the study.

An interesting method to obtain a new probability distributions has been proposed by Alzaatreh et al. (2013b). The method is based upon a "transformer" and a "transformed" distribution and is known as the T-X family of distributions. This method is illustrated below.
Let X be a random variable with density function g(x) and distribution function G(x). Also, let T be a continuous random variable with density function r(t) with support on [a, b]. The cumulative distribution function (cdf ) of the T − X family of distributions is then given as where W (G(x)) satisfies following conditions W (G(x)) ∈ [a, b], W (G(x)) is differentiable and monotonically non-decreasing, W (G(x)) → a as x → −∞ and W (G(x)) → b as x → ∞.

  
The distribution function F T −X (x) in (1) is a composite function of (R · W · G)(x) and can be written as where R(t) is the distribution function of random variable T . The corresponding density function is The density function r(t) in (1) is "transformed" into a new distribution function F T −X (x) through the function W (G(x)), which acts as a "transformer". The density function f T −X (x) in (2) is transformed from random variable T through the transformer random variable X and is called "Transformed-Transformer" or "T-X" distribution.
The main focus of this paper is to introduce a new family of distributions called the Pareto-X family of distributions by assuming that the random variable T in (1) has the Pareto distribution.
The organization of the paper follows: The Pareto-X family of distributions is proposed in Section 2. In Section 3, a special sub model of the family is presented and is named as the Pareto-Weibull (PW) distribution. In Section 4, some distributional properties of the proposed PW distribution are derived. Distribution of various order statistics are presented in Section 5. Maximum likelihood estimation of the model parameters is discussed in Section 6. Section 7 contains some real data applications of the proposed PW distribution. Finally, some concluding remarks are given.

The Pareto-X Family of Distributions
Let T be a Pareto random variable with density function r(t) defined on the support on [a, ∞) , a ≥ 0 and without loss of generality, we assume that a = x m . Also let W (G (x)) = x m − log (1 − G (x)), where x m ∈ R + , then the distribution function of the proposed Pareto-X family of distributions is obtained from (1) as The density function corresponding to (3) is where h (x) is the hazard function of X.
The Pareto distribution with parameter (x m , α) has the cumulative distribution function R(t) = 1 − xm t α , t ≥ x m and density function r(t) = αx α m t α+1 , t ≥ x m . Using these in (3), the distribution function of the Pareto-X family of distributions is where x m ∈ R + and α ∈ R + are the scale and shape parameters respectively. The density function of the Pareto-X family of distributions is easily obtained from (4) as Various members of the proposed Pareto-X family of distributions can be studied by using different base distributions G(x) in (5). Some special cases of the proposed Pareto-X family of distributions are given in Table 1.
In the following we have obtained a specific member of the Pareto-X family of distributions by using the Weibull baseline distribution.

The Pareto-Weibull Distribution
The Pareto-Weibull (P W ) distribution is obtained by using Weibull baseline distribution in (5). For this, suppose that the random variable X follows the Weibull distribution with cdf where λ ∈ R + and k ∈ R + are the scale and shape parameters respectively. Now, using (6) in (5) the distribution function of the PW distribution is where x m , λ ∈ R + are the scale parameters and α, k ∈ R + are the shape parameters of the distribution. The density function of the PW distribution is readily obtained from (7) and is given in the definition below.
Definition: A continuous random variable X is said to have a Pareto-Weibull distribution if its probability density function is where x m , λ ∈ R + are the scale parameters and α, k ∈ R + are the shape parameters of the distribution.

Special Cases:
(i) The distribution function of the PW distribution given in (7) reduces to the distribution function of the Paretoexponential distribution for k = 1.
(ii) A Pareto-exponential distribution developed by Waseem and Bashir (2019) is considered as a special case of (7) for k = 1 and x m = 1. Some of the possible shapes for the density and distribution functions of the proposed PW distribution are given in Figure 1. It has been observed from the Figure that the proposed distribution has the capability to capture different behaviours in datasets.

Distributional Properties
In the following we will discuss some important properties of the proposed PW distribution.
Pareto-Weibull Distribution with Properties and Applications: A Member of Pareto-X Family.

Moments
The moments of random variable are useful in studying various properties of its distribution. In the following we will give the moments of the proposed PW distribution.
Definition: Let X be a random variable having the PW distribution then its rth raw moment is given as Mean can be obtained by setting r = 1 in (9) and is The variance of the distribution is obtained as The higher moments of the distribution can be obtained by using r > 2 in (9).

Moment Generating Function
The moment generating function is useful in obtaining moments of a random variable. The MGF for PW distribution is given in the following theorem.
Theorem 4.1. Let random variable X follows the PW distribution then the moment generating function, M X (t) is where t ∈ R.
Proof. The moment generating function is defined as where f (x) is given in (8). Using the series representation of e tx given in Jeffrey and Zwillinger (2007), we have Using E(X r ) from (9) in (11), we have (10).

Characteristic Function
The characteristic function (CF) plays a vital role both in probability theory and applied statistics. The characteristic function completely describe a probability distribution and this always exist. The CF for PW distribution is given in the following theorem.
Theorem 4.2. Let X follows the PW distribution then the characteristic function, ϕ X (t) is where i = √ −1 is the imaginary unit and t ∈ R.
Proof. The proof is simple.

Reliability Analysis
The reliability function, used by Ebeling (2004), or survival function, used by Carpenter (1997), is simply the complement of the cumulative distribution function. The reliability function is useful in survival analysis and engineering. The reliability function for the PW distribution is The hazard function is the ratio of the density function to the reliability function and for the PW distribution, it is given as The reliability and hazard rate functions of the PW distribution are given in Figure 2 above. We can see that the hazard rate function shows both increasing and decreasing behavior.

Quantile Function and Median
The quantile function, denoted by x q , is useful in obtaining quantiles of a distribution. This function is also useful in generating the random sample from any distribution. The quantile function is obtained by solving F (x) = q for x, see for example Rahman et al. (2020). The quantile function for the PW distribution is readily obtained by solving (7) for x and is The meadian is obtained by using q = 0.5 in above equation and is The lower and upper quartiles can also be obtained by using q = 0.25, 0.75 in (12) respectively.

Generating Random Sample
Random sample is often required in simulation studies. The random sample from PW distribution can be generated by using the following expression, see Rahman et al. (2020), where u ∼ U (0, 1). One can generate random sample from PW distribution by using (13) for various values of the model parameters.

Order Statistics
Order statistics are widely used in many fields like economics, geology etc. The distribution of order statistics is given below. Let X 1:n ≤ X 2:n ≤ · · · ≤ X n:n denote the order statistic of a random sample X 1 , X 2 , · · · , X n from a continuous distribution F X (x). The density function of X r:n is given as where f X (x) is density function corresponding to F X (x). The density function of the rth order statistic for the PW distribution is obtained by using its density and distribution function in (14) and is where r = 1, 2, · · · , n. Using r = 1 in (15), the density function of the smallest order statistic X 1:n for PW distribution is Again, using r = n in (15), the density function of the largest order statistic X n:n , is

Estimation and Inference
The parameter estimation is an essential step in fitting distribution to some real data. The maximum likelihood method of estimation is, perhaps, the most popular method to estimate the parameters of a distribution. In this section, we have discussed the maximum likelihood estimation for parameters of the PW distribution. For this, suppose x 1 , x 2 , · · · , x n is a random sample of size n from the PW distribution. The likelihood function is The log-likelihood function is The maximum likelihood estimator of x m is the first-order statistic x (1) . The derivatives of (16) with respect to λ, α and k are Now setting, ∂l ∂λ = 0, ∂l ∂α = 0 and ∂l ∂k = 0 and solving the resulting nonlinear system of equations the maximum likelihood estimateΘ = λ ,α,k ′ of Θ = (λ, α, k) ′ can be obtained. Also, as n → ∞, the asymptotic distribution of the M LEs λ ,α,k are given by, see for example Aryal and Tsokos (2011), The asymptotic variance-covariance matrix V , of the estimatesλ,α andk is obtained by inverting Hessian matrix; see Appendix. An approximate 100(1 − α)% two sided confidence intervals for λ, α and k are given by:λ where Z α is the αth percentile of the standard normal distribution.

Real-life Applications
In this section we have given two real data applications of the PW distribution.

Floyd River Data
We considered this dataset for the Floyd River, located in James, Iowa, USA, which provides the consecutive annual flood discharge rates for the year 1935 − 1973. The dataset has been previously used by Akinsete et al. (2008). For the source and details of the data, see Mudholkar and Hutson (1996 Table 3 contains the estimated values of the model parameters alongside the standard errors. The estimated distribution function of the PW distribution is plotted alongside the empirical distribution, for the Floyd river data, in the left panel of Figure 3 below. This figure also contains the fitted distribution function of the competing models. Various model selection criteria like Log-likelihood, Akaike's information criterion (AIC), corrected Akaike's information criterion (AICc), Bayesian information criterion (BIC) are shown in Table 4. The results of this table shows that the PW distribution is the best fit for this data.

Bladder Cancer Data
The dataset consist of a set of remission times collected from cancer patients in a bladder cancer study, see Lee and Wang (2003   The summary statistics of the data are presented in Table 2. The estimated values of the model parameters alongside the standard errors are given in Table 5. The estimated distribution function of the PW distribution alongside the empirical distribution function are given in the right panel of Figure 3. Table 6 presents the computed values for different selection criterion. The results of the selection criterion indicate that the PW distribution is most suitable fit for this data.

Concluding Remarks
In this paper, a new Pareto-X family of distributions has been introduced. A four-parameters sub-model of the proposed family called the Pareto-Weibull distribution is studied in detail. The distributional properties of the proposed PW distribution including moments, moment generating function, characteristics function, quantile function, random number generation, reliability functions and the distribution of order statistics are discussed. The maximum likelihood estimation of the parameters is done. Finally, two applications of the proposed PW distribution are given by using real data sets. We have found that the proposed PW distribution is a suitable model for modeling of the data sets.