Generalized Kumaraswamy Generalized Power Gompertz Distribution: Statistical Properties, Application, and Validation Using a Modified Chi-Squared Goodness of Fit Test

A new six-parameter continuous distribution called the Generalized Kumaraswamy Generalized Power Gompertz (GKGPG) distribution is proposed in this study, a graphical illustration of the probability density function and cumulative distribution function is presented. The statistical features of the Generalized Kumaraswamy Generalized Power Gompertz distribution are systematically derived and adequately studied. The estimation of the model parameters in the absence of censoring and under-right censoring is performed using the method of maximum likelihood. The test statistic for rightcensored data, criteria test for GKGPG distribution, estimated matrix Ŵ , Ĉ , and Ĝ , criteria test 2 n Y , alongside the quadratic form of the test statistic is derived. Mean simulated values of maximum likelihood estimates γ̂ and their corresponding square mean errors are presented and confirmed to agree closely with the true parameter values. Simulated levels of significance for ( ) 2 n Y γ test for the GKGPG model against their theoretical values were recorded. We conclude that the null hypothesis for which simulated samples are fitted by GKGPG distribution is widely validated for the different levels of significance considered. From the summary of the results of the strength of a specific type of braided cord dataset on the GKGPG model, it is observed that the proposed GKGPG model fits the data set for a significance level ε = 0.05. How to cite this paper: Maxwell, O., Onyedikachi, I.P., Aidi, K., Akpa, C.I. and SeddikAmeur, N. (2022) Generalized Kumaraswamy Generalized Power Gompertz Distribution: Statistical Properties, Application, and Validation Using a Modified Chi-Squared Goodness of Fit Test. Applied Mathematics, 13, 243-262. https://doi.org/10.4236/am.2022.133019 Received: January 22, 2022 Accepted: March 15, 2022 Published: March 18, 2022 Copyright © 2022 by author(s) and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access


Introduction
The Gompertz distribution is a continuous probability distribution often applied in lifetime data analysis to describe the distribution of the science such as biology [1], gerontology [2], adult lifespans by demographers [3], actuaries [4], marketing [5], network theory [6] and computer science [7]. The Gompertz distribution has a convex hazard function. It is a flexible distribution, skewed to the right and the left, and a generalization of the exponential distribution.
Statistics show that a powerful transformation is a series of functions used to create a monotonous data transformation using power functions. Applied to the random variable, the technique is useful in stabilizing variance, making the data more normal distribution-like, improving the validity of association measures like the Pearson correlation between variables, and providing a more flexible model by adding a new parameter named power parameter. The works of Ieren et al. [35], Ghitany et al. [36], and Rady et al. [37] prove this fact. Ieren et al. [35] proposed the power Gompertz  performance than the Gompertz model, Ghitany et al. [36] introduced the power Lindley distribution. This model provides more flexibility than Lindley distribution when applied to lifetime data, Rady et al. [37] proposed the Power Lomax distribution, when applied to bladder cancer data, the proposed Power Lomax distribution exhibited a much more flexible model than the Lomax distribution.
To produce a more flexible distribution for a highly skewed dataset, our focus in this paper is to present an extension of the power Gompertz distribution using the generalized Kumaraswamy generalized family of distribution [8], the result-

Formation of the Generalized Kumaraswamy Generalized Power Gompertz Distribution (GKGPG)
The Power Gompertz (PG) distribution [35] with positive parameter , α β and θ has pdf and cdf given by: and: ( ) The cdf of the Generalized Kumaraswamy Generalized (GK-G) family is de- The corresponding pdf of the GK-G family is given by: The hazard rate function (hrf) of the GK-G family is given by: Hence the pdf and cdf of the newly proposed Generalized Kumaraswamy Generalized Power Gompertz (GKGPG) distribution is given by: And: where 0, 0 1,

Graphical Description of the Generalized Kumaraswamy Generalized Power Gompertz Distribution (GKGPG)
Here, we graphically illustrate the probability density function, and cumulative distribution function of the generalized kumaraswamy generalized power Gompertz distribution at different parameter values.
Remarks: Figure 1 represents the behavior of the density plot the effect of the different parameter values. The probability density function of the generalized kumaraswamy generalized power Gompertzdistribution is unimodal; it is also decreasing, and right skewed, depending on the indicated parameter values.
Remarks: Figure 2 represents the cdf plot, clearly, the cdf approaches one (1) as X tends to infinity and equals zero when X tends to zero.

Asymptotic Behavior
This section examines the limiting behavior of the GKGPG distribution as X → ∞ and as 0 X → .
For the cdf,

Quantile Function
The quantile function (qf) of X, say ( ) ( ) Equation (3) numerically, and it is given by: where ( ) By substituting Equations (12) in (13), we obtain the quantile function of the GKGPG distribution as: This above derived function is used to obtain certain moments, such as Skewness and Kurtosis, as well as the median of the distribution and generation of random variables from the distribution concerned.

Skewness and Kurtosis
The analysis of the Skewness and Kurtosis variability on the shape parameters can be examined on the basis of quantile action. The weaknesses of the conventional measure of Kurtosis are well known. Kenney and Keeping [40] gives the Bowely Skewness based on quantiles as: Moors et al. [41] gave the Moors quantile based Kurtosis as: is obtainable using the equation of the quantile function as given in Equation (14).

Reliability Analysis of the GKGPG Distribution
The Survival function of the generalized kumaraswamy generalized power Gompertz distribution is given as (Figure 3). where The Hazard failure of the generalized kumaraswamy generalized power Gompertz distribution is given as (Figure 4).
where  (3) and (4) respectively. Then the probability density function

( )
: i n f x of the i th order statistics of the GKGPG distribution is given by: By substituting Equations (6) and (7) into the i th order statistics of the GKGPG distribution, we have that: Hence the minimum order statistics ( ) 1 X for the GKGPG distribution is given by: Similarly, the maximum order statistics ( ) n X for the GKGPG distribution is given by:

Maximum Likelihood Estimation
Here, the parameters of the GKGPG distribution are estimated using the method of maximum likelihood. Let 1 2 , , , n X X X  be random samples distributed according to the GKGPG distribution, the likelihood function is obtained by the relationship:

Estimation under Right-Censored Data
The hypothesizing test will be discussed under complete and censored data, however, the MPS is only defined for complete data, since the MLE is usually consid- In this case, the log-likelihood is obtained as follow: The Monte Carlo technique or other iterative methods can be used to determine the values of ˆ, , ,, a b c α β and θ .

Test Statistic for Right Censored Data
Let 1 , , n X X  be n i.i.d. random variables grouped into r classes i I . To assess the adequacy of a parametric model F₀: When data are right censored and the parameter vector β is unknown, Bagdonavicius and Nikulin [38] proposed a statistic test Y 2 based on the vector: This one represents the differences between observed and expected numbers of failures ( j U and j e ) to fall into these grouping intervals γ is the maximum likelihood estimator of γ on initial non-grouped data. Under the null hypothesis H₀, the limit distribution of the statistic Y 2 is a chisquare with ( ) r rank = Σ degrees of freedom. The description and applications of modified chi-square tests are discussed in Voinov et al. [42].
The interval limits j p for grouping data into j classes j I are considered as data functions and defined by: Such as the expected failure times j e to fall into these intervals are

Criteria Test for GKGPG Distribution
For testing the null hypothesis H₀ that data belong to the GKGPG model, we construct a modified chi-squared type goodness-of-fit test based on the statistic Y 2 . Suppose that τ is a finite time, and observed data are grouped into r s > sub-intervals

Estimated Matrix Ŵ and Ĉ
The components of the estimated matrix Ŵ are derived from the estimated matrix Ĉ which is given by: And:

Maximum Likelihood Estimation
We generated 10000 N = right censored samples with different sizes  Table 1.
The maximum likelihood estimated parameter values, presented in Table 1, agree closely with the true parameter values.
O. Maxwell et al.
We choose 7 r = . The results are reported in Table 2.
The null hypothesis H₀ for which simulated samples are fitted by GKGPG distribution is widely validated for the different levels of significance. Therefore, the test proposed in this work, can be used to fit data from this new distribution.

Application
In this section, we apply the results obtained through this study to real data set from reliability (Crowder et al. [44]), previously used by [45] [46] [47]. In an experiment to gain information on the strength of a certain type of braided cord after weathering, the strengths of 48 pieces of cord that had been weathered for a specified length of time were investigated. The observed right-censored streng-  Data are grouped into 7 r = intervals j I . We give the necessary calculus in Table 3.
Then we obtain the value of the statistic test 2 n Y :

Conclusion
This research has successfully introduced and studied a six-parameter continuous distribution called the generalized Kumaraswamy generalized power Gompertz distribution. The plots of the probability density and cumulative distribution function have been analyzed. We have also derived some properties of the new distribution such as asymptotic behavior, quantile function for median, Skewness, and Kurtosis, and reliability analysis. The distribution of order statistics estimation of parameters based on censored and uncensored random samples using Maxi-

Formation of the Generalized Kumaraswamy Generalized
Defined in this paper has three shape parameters which control its Skewness, Kurtosis and tails. It can therefore be applied in more real-life situations. Maximum likelihood estimates are discussed, and modified chi-square goodness-of-fit tests for right censoring are constructed. The statistical test provided in this article can be used to fit unknown parameters and censorship into this model and its sub-models. The results and efficacy of the proposed test are shown in an important simulation study.

Conflicts of Interest
The authors declare no conflict of interest.