The Modified Power Series Inverted Exponentiated Generalized Class of Distributions : Statistical Measures , Model Fit , and Characterization

Within the master thesis [1], the author considered the following random variable $$T=X^{-1}-1,$$ where $X$ follows the Kumaraswamy distribution, and obtains a so-called inverted Kumaraswamy distribution, and studies some properties and applications of this class of distributions in the context of the power series family [2]. Within the paper [3], they introduced the exponentiated generalized class of distributions and obtained some properties with applications. Based on these developments we introduce a class of modified power series inverted exponentiated generalized distributions and obtain some of their properties with applications. Some characterization theorems are also presented. Avenues for further research concludes the paper.


Introduction
The exponentiated class of distributions first appeared in [4], with the following representation for its CDF where α, β > 0.
It is well known that the inverse distribution is the distribution of the reciprocal of a random variable [8]. If we consider the random variable Y = 1 X , where X has CDF F (x), then the reciprocal distribution, F Y (y) has CDF 1 − F X 1 y .
In some recent works, authors have considered the following modification of the transformation Y = 1 X , that is where X has CDF F (x). In this case, the reciprocal distribution, F Y * (y * ) has CDF F Y * (y * ) = 1 − F X 1 y * + 1 .
The power series class of distributions was proposed and studied in [2]. This class of distributions includes binomial, geometric, logarithmic and Poisson distributions as special cases. However, these distributions may not be useful when a random variable takes the value of zero with high probability, that is, zero-inflated. In such situations, it is more appropriate to consider the distribution which is truncated at zero.
Power series distributions are usually motivated by the stochastic representations Z = min(X 1 , · · · , X N ) if the components are in series, or if the components are in parallel, and many lifetime data admit such representations, and the related distributions are very useful in modeling such data. We suppose X 1 , · · · , X N are independent and identically distributed random variables from a parent distribution with PDF f (x) and CDF F (x), and consider N to be a discrete random variable from a power series distribution (truncated at zero) and whose PDF is given by P (N = n) = a n λ n C(λ) , n = 1, 2, 3, · · · where C(λ) = ∞ n=1 a n λ n , a n depends on n, and λ > 0. C(λ) is finite, and its first, second, and third derivatives with respect to λ are defined and given by C (λ), C (λ), and C (λ), respectively. The table below represents some useful quantities including a n , C(λ), C −1 (λ), C (λ), C (λ), and C (λ), respectively, for the Poisson, geometric, logarithmic and binomial (with m being the number of replicas) distributions which belong to the power series family of distributions.  If X (1) = min{X 1 , · · · , X N ), then the conditional CDF of X (1) |N = n is given by The CDF of the power series class of distributions associated with some parent distribution with PDf f (x) and CDF F (x) is the marginal CDF of X (1) , which is given by From [3], the Exponentiated (Standard Uniform) Generalized distribution has cdf where α, β > 0 are two additional shape parameters. Suppose X follows the Exponentiated Standard Uniform Generalized Distribution, and consider the transformation discussed earlier By the transformation technique, the cdf of Y * is given by for α, β > 0. The cdf above, represents the cdf of the Inverted Exponentiated (Standard Uniform) Generalized distribution.
Based on the stochastic representation X (1) = min{X 1 , · · · , X N ) we study some properties and applications of a certain modified power series inverted Exponentiated Generalized distribution (by relaxing the domain space of y * -see new family defined ).

Organization of Manuscript
This manuscript is organized as follows. In Section 3, we present the PDF and CDF of the new family. In Section 4, we introduce some sub-models of the new family and demonstrate some of their reliablity properties visually. In Section 5, some statistical measures are presented, and they include expansion formulas for the CDF and PDF, quantile function, ordinary moments, Renyi entropy, moment generating function, stochastic ordering, incomplete moments, inequality measures, mean residual life, order statistics, and moments of order statistics. The representation of the distribution as a transformed random variable is also discussed. In Section 6, estimation of model parameters in the modified Poisson-X family of distributions is discussed. In Section 7 we assess the performance of the maximum likelihood method in estimating model parameters. For this, a sub-model of the broad family is considered, namely, the modified Poission inverted exponentiated generalized distribution (MPIEG), and a monte carlo simulation study is conducted. The result indicates that using the method of maximum likelihood in estimating model parameters is adequate. In Section 8, we show sub-models of the new family are useful in fitting real-life data coming from the chemical/material sciences. In Section 9, we present some characterizations of the Modified Poisson-X distribution. Section 10 concludes the paper, and some directions for further research are proposed.

The New Family Defined
The PDF of the new family is given by where C and C are some useful quantities from Table 1, and ∂x . The CDF of the new family is given by where C is some useful quantity from Table 1, and F (x) is as given above. The parameter space for λ is (−∞, 0) ∪ (0, ∞).

Some Sub-Models
We introduce four sub-models and visualize some of their basic reliability properties. The reliability properties include survival function (SF), hazard rate function(HF), and the cumulative hazard rate function (CHF).

Modified Binomial Inverted Exponentiated Generalized Distribution
Here we assume the number of replicas is given by m = 2. Thus, the PDF of the Modified Binomial Inverted Exponentiated Generalized distribution is given by and the CDF is given by The survival function, hazard rate function, and the cumulative hazard rate function, respectively, are given by

Modified Logarithmic Inverted Exponentiated Generalized Distribution
The PDF of the Modified Logarithmic Inverted Exponentiated Generalized distribution is given by and the CDF is given by The survival function, hazard rate function, and the cumulative hazard rate function, respectively, are given by

Some Statistical Measures
Here we obtain some properties for the Modified Poisson-X distribution, that is, we consider Explicitly, the CDF can be written as where λ ∈ (−∞, 0) ∪ (0, ∞) and F (x) is some baseline CDF.

CDF Power Series
Theorem 5.1. The CDF of the Modified Poisson-X distribution admit the following representation

PDF Power Series
By differentiation, the PDF of the Modified Poisson-X family is given by Theorem 5.2. The PDF of the Modified Poisson-X distribution admit the following representation where λ ∈ (−∞, 0) ∪ (0, ∞), and the random variable X has PDF f (x) and CDF F (x).

Transformation of Random Variables
Theorem 5.3. Let U be a random variable uniform on [0, 1). Then the following random variable follows the Modified Poisson-X class of distributions is the quantile function of the distribution with CDF F, and λ ∈ (−∞, 0) ∪ (0, ∞).

Quantile Function
Theorem 5.4. The quantile function of the Modified Poisson-X class of distributions is given by , and 0 < p < 1.

Ordinary Moments
If X is standard uniform, then the PDF of the Modified Poisson-Standard Uniform distribution is given by Thus we have the following is the Euler Gamma function, and Γ(·, ·) is the incomplete gamma function.

Renyi Entropy
Let X be a random variable with PDF f (x). By definition, the Renyi entropy [9] is defined as where δ > 0, and δ = 1. The main result of this section is the following whose proof is omitted.

Moment Generating Function
Given a random variable X, one defines the moment generating function as where E[·] is an expectation. Now using the series expansion for e zX , one can write where µ r is the rth non-central moment of the random variable X. Thus from Theorem 5.5, the following is an immediate consequence where λ ∈ (−∞, 0) ∪ (0, ∞), Γ(·) is the Euler Gamma function, and Γ(·, ·) is the incomplete gamma function.

Stochastic Ordering
Stochastic ordering is a common concept to show the ordering mechanism in lifetime distributions [10]. A random variable X is said to be stochastically smaller than the random variable Y in the likelihood ratio order (X ≤ lr Y ) if the ratio of the associated PDFs given by The main result is the following whose proof we omit.
Theorem 5.9. Let X follow the Modified Poisson Standard Uniform distribution with parameter λ 1 , and let Y follow Poisson Standard Uniform distribution with parameter λ 2 . Let f X (x) and f Y (y) denote the PDF's of X and Y, respectively. If λ 2 < λ 1 , then X ≤ lr Y .

Incomplete Moments
The rth incomplete moment is given by M r (y) = y −∞ x r f (x)dx. For the Modified Poisson-X family of distributions, we have the following Theorem 5.10. The rth incomplete moment for the Modified Poisson-X family of distributions, can be expressed as where λ ∈ (−∞, 0) ∪ (0, ∞), and the random variable X has PDF f (x) and CDF F (x).

Inequality Measures
The Lorenz curve, L F (y) for incomplete moments is defined as L F (y) = where λ ∈ (−∞, 0) ∪ (0, ∞), the random variable X has PDF f (x) and CDF F (x), and µ is the mean of the Poisson-X family.
The Bonferroni curve, B F (y) is defined as B F (y) = L F (y) F (y) . For the Modified Poisson-X family of distributions, we have the following Theorem 5.12. The Bonferroni curve of the Modified Poisson-X family can be expressed as where λ ∈ (−∞, 0) ∪ (0, ∞), the random variable X has PDF f (x) and CDF F (x), and µ is the mean of the Poisson-X family.

Mean Residual Life
The mean residual life is defined asm(y) = 1 where λ ∈ (−∞, 0) ∪ (0, ∞), the random variable X has PDF f (x) and CDF F (x), and µ is the mean of the Poisson-X family.

Order Statistics
The pdf for the pth order statistics X p:n of an independent identically distributed random sample, X 1 , X 2 , · · · , X n , of size n, f Xp:n (x), is given by for p = 1, 2, · · · , n.
Lemma 5.14. The pdf for the pth order statistics X p:n of an independent identically distributed random sample, X 1 , X 2 , · · · , X n , of size n, f Xp:n (x), can be expressed as Lemma 5.15. Let F (x) be given by the CDF of the Modified Poisson-X family of distributions. Then F (x) p−1+j can be expressed as and F (x) is some baseline CDF.
Corollary 5.16. The PDF in Theorem 5.2 can be expressed as and F (x) is some baseline CDF, and f (x) is some baseline PDF.
By combining Lemma 5.14, Lemma 5.15, and Corollary 5.16, we get the following Theorem 5.17. The pdf, f Xp:n (x), for the pth order statistics X p:n of an independent identically distributed random sample, X 1 , X 2 , · · · , X n , of size n, from the Modified Poisson-X family is given by

Moments of Order Statistics
The rth non-central moments of the pth order statistics is given by For the Poisson-X family of distributions we have the following using Theorem 5.17.
Theorem 5.18. The rth noncentral moments of the pth order statistics for the Poisson-X family is given by

Parameter Estimation
The method of maximum likelihood is used in this paper to estimate model parameters. Here we discuss this method for the Modified Poisson-X family of distributions. Suppose x 1 , x 2 , · · · , x n is a random sample of size n from the Modified Poisson-X family of distributions. It can be shown that the total log-likelihood function is given by where ψ is a vector of parameters associated with the baseline distribution. Partial differentiation of the total log-likelihood function with respect to model parameters gives the following as the score functions Equating the score functions to zero and numerically solving the system of equations using techniques such as the quasi Newton-Raphson method, gives the maximum likelihood estimates for the model parameters. Let = (λ; ψ), for the purposes of constructing confidence intervals for the parameters in the Poisson-X family of distributions, the observed information matrix, call it J( ), can be used due to the complex nature of the expected information matrix. The observed information matrix is given by The elements of the observed information matrix are given below When the usual regularity conditions are satisfied and that the parameters are within the interior of the parameter space, but not on the boundary, the distribution of √ n( − ) converges to the multivariate normal distribution N p+1 (0, I −1 ( )), where I( ) is the expected information matrix, and it is assumed that ψ = (ψ 1 , · · · , ψ p ). The asymptotic behavior remains valid when I( ) is replaced by the observed information matrix evaluated at J( ). The asymptotic multivariate normal distribution N p+1 (0, J −1 ( )) is a very useful tool for constructing an approximate 100(1 − ξ)% two-sided confidence intervals for the model parameters, where ξ is the significance level.

Monte Carlo Simulation Study
In this section we show that the method of maximum likelihood is adequate in estimating the parameters in the power series family of inverted exponentiated generalized distributions. For this, a Monte Carlo simulation study is carried out to assess the performance of the estimation method in the Modified Poisson inverted exponentiated generalized distribution (MPIEG). Samples of sizes 200, 400, 500, and 700, are drawn from the MPIEG distribution, and the samples have been drawn for the following set of parameters The maximum likelihood estimators for the parameters α, β, and λ are obtained. The procedure has been repeated 400 times, and the mean and standard deviation for the estimates are computed, and the results are summarized in Tables 4-6 for each of sets I, II, and III, respectively, considered above. From the table above, we find that the simulated estimates are close to the true values of the parameters and hence the estimation method is adequate. We have also observed that the estimated standard deviation consistently decrease with increasing sample size as seen in the graphs below.  From the table above, we find that the simulated estimates are close to the true values of the parameters and hence the estimation method is adequate. We have also observed that the estimated standard deviation consistently decrease with increasing sample size as seen in the graphs below.  From the table above, we find that the simulated estimates are close to the true values of the parameters and hence the estimation method is adequate. We have also observed that the estimated standard deviation consistently decrease with increasing sample size as seen in the graphs below. Table 9: Decreasing standard deviation for increasing sample size.
Overall the simulation study conducted, indicated that using the method of maximum likelihood in estimating model parameters is adequate.

Application to Real-Life Data
Here we demonstrate usefulness of the new family to the breaking stress of carbon fibers data, Table 2 [10]. The PDF's and CDF's of the distributions considered are already given in Section 4. We assume the modified parameter and domain space of Section 4. Using the R software, we report below in Table 10, the estimates for the parameters in each of the four distributions alongside their standard errors. The fitted CDF's to the carbon fibers data are shown below  Table 2 [10].
The measures of goodness of fit we consider include Akaike information criterion (AIC), Bayesian information criterion (BIC), -2Log-Likelihood, and second-order Akaike Information Criterion (AICc) and they are reported in Table 11. Whilst it appears from the fits above, that all the distributions are competitive in fitting the breaking stress of carbon fibers data, Table 11 reveals that the MGIEG distribution is most compatible with this data set, and hence can be considered the best in this instance.

Some Characterization Theorems
The characterization of statistical distributions plays a major role in stochastic modeling. In this section we present some characterizations of the Modified Poisson-X family of distributions. Our first characterization theorem is based on a simple relationship between two truncated moments, and for related works in this direction, the reader is referred to [11]- [16].
At first, we recall the following which will be useful in Section 9.1 Theorem 9.1. [12] Let (Ω, Σ, P) be a given probability space, and let I = [a, b] be an interval for some a < b (a = −∞, b = ∞ might as well be allowed). Let X : Ω → I be a continuous random variable with probability distribution function F, and let q 1 and q 2 be two real functions on I such that is defined with some real function η. Assume that q 1 , q 2 ∈ C 1 (I), and η ∈ C 2 (I), and F is twice continuously differentiable and strictly monotone increasing on the set I. Finally, assume that the equation ηq 2 = q 1 has no real solutions in the interior of I. Then F is uniquely determined by the functions q 1 , q 2 , η. In particular, where the function s is a solution of the differential equation and C is a constant chosen to make I dF = 1.
Remark 9.2. The characterization based on the ratio of two truncated moments is stable in the sense of weak convergence, and for more details see [17].
In Section 9.2 we employ a single function ψ of X and state characterization results in terms of ψ(X). At first we recall the following two results that will be useful in the sequel.  → (a, b) be a continuous random varable with CDF F . Let ψ(x) be a differentiable function on (a, b) with lim x→a + ψ(x) = δ > 1 and ∈ (a, b).

Characterization Based on Two Truncated Moments
The main result here is the following which we state without proof Proposition 9.5. Let X : Ω → R be a continuous random variable, and let q 2 (x) = 1 and where λ ∈ (−∞, 0) ∪ (0, ∞) and F (x) is some baseline CDF. The PDF of X is where f (x) is the PDF of the baseline distribution iff the function η of Theorem 8.1 is given by η(x) = 1 2 − e λ(1−F (x)) − 1 2(e λ − 1) .
If q 2 is given by the previous Proposition, then we have the following Corollary 9.6. Let X : Ω → R be a continuous random variable. The PDF of X is iff there exists functions q 1 and η defined in Theorem 8.1 satisfying the following differential equation Remark 9.7. The general solution of the above differential equation is given by where D is a constant. One set of appropirate functions is given by the previous Proposition with D = 0.

Characterization Based on Conditional Expectation
The main results here are as follows.

Further Recommendations
The Chen distribution [19] has CDF given by where γ, β, t > 0. We suggest obtaining some properties and applications of a so-called power seres inverted Chen class of distributions (using the CDF of Section 1 which is based on the stochastic representation X 1 = min{X 1 , · · · , X n }) which is given by where C is some useful quantity from Table 1, and with G being the CDF of the Chen distribution.