A WEIGHTED GOMPERTZ-G FAMILY OF DISTRIBUTIONS FOR RELIABILITY AND LIFETIME DATA ANALYSIS

. This article is set to push new boundaries with leading-edge innovations in statistical distribution for generating up-to-the-minute contemporary distributions by a mixture of the second record value of the Gompertz distribution and the classical Gompertz model (weighted Gompertz model) using T-X characterization, especially used for two-sided schemes that provide an accurate model. The quantile, ordinary, and complete moments, order statistics, probability, and moments generating functions, entropies, probability weighted moments, Lin’s condition random variable, reliability in multicomponent stress strength system, reversed, and moments of residuals life and other reliability characteristics in engineering, actuarial, economics, and environmental technology were derived in their closed form. To investigate and test the flexibility, viability, tractability, and performance of the proposed Weighted Gompertz-G (WGG) generated model, the shapes of some sub-models of the WGG model were examined. The shapes of the sub-models indicated J-shapes, increasing, decreasing, and bathtub hazard rate functions. The maximum likelihood estimation of the WGG-generated model parameters was examined. An illustration with simulation and real-life data analysis indicated that the WGG-generated model provides consistently better goodness-of-fit statistics than some competitive models in the literature.

Modeling abrupt behavioural structure and scenarios has become more complicated as a result of their change-point.Though the method of generating new distribution is not new, using the weighted generator concept to generate new models is a new approach.Hence, this study is motivated to propose a model with a true reflection of the character of the data obtained.Thus, the WGG generated model tends to improve the goodness-of-fit, and the test statistics of the existing distributional models using weighted distribution characterization.
The study aim at introducing a class of generator with the aid of the weighted Gompertz model called the weighted Gompertz generator.This generated model will improve the performance, flexibility and the viability of the goodness-of-fit of the abrupt behaviourial change-point scenarios in lifetime modeling.

The Weighted Gompertz-G Distribution
Suppose a nonnegative random variable T is defined on the interval T ∈ [m, n] for −∞ < m < n < ∞ with pdf r(G(t)) such that r(G(t)) = − log[1 − G(t)] is monotonically non-decreasing; r(G(t)) is closed in the interval [m, n]; and r(G(t)) approaches m as t tends to negative infinity, and r(G(t)) approaches n as t tends to positive infinity.Thus, by [4] the cdf and the pdf of the WGG generated class of distribution can be expressed as and for t > 0, λ, β > 0, where g(t), and G(t) are the parents pdf and cdf.
The WGG generated reliability model can be expressed as The hazard rate function that corresponds to the WGG generated model is defined as The reversed hazard rate function is obtained as for t > 0, λ, β > 0.
The cumulative hazard rate function of the WGG generated function is give as:

The Quantile Function
Quantile is fundamental for the simulation and estimation of a distribution parameter(s).Hence, it is a function that associates the probability distribution function of the WGG generated model of a random variable T such that the probability of the variable being less than or equal to that value equals the probability for a uniform interval q ∈ (0, 1) is defined as where W −1 is the Lambert-W or omega function as defined in [13] and [16] such that In particular, the median is obtained when q = 0.5.
Theorem 1.The shape, characteristics, and behaviour of the WGG generated model can be examined by investigating the first and second derivatives of the log of the WGG generated pdf model.Thus, for f ′ (t) < 0.Then, then cdf F (t) will be decreasing monotonically for all values of t.The WGG generated model will be bimodal if f ′′ (t) changes its signs from negative to non-negative, viz-a-viz.
Proof.The log f (t) is give as Thus, taking the derivative with respect to the variable,we have where The second derivative was implemented to determine if the model was bimodal.Thus, the second derivative is given as

Order statistics
Order statistics are useful tools to improve the robustness of sampling plans by variables, and shorten test times of Poisson processes.
Let T (1) , T (2) , T (3) , . . ., T (k) be the order statistics for a random variable T 1 , T 2 , T 3 , . . ., T k with WGG distribution.Then, the WGG density of the u th order statistics is given as However, using the binomial expansion, and noting that S = 1 − G(t), we have the order statistics as The minimum order statistics is obtained when u = 1, and the maximum order statistics is obtained when u = k respectively.
4.1.Record value distributions of the WWG model.Let T i for i = 1, 2, 3, . . ., k be a finite sequence of independently identically distributed random variables with WGG generated cdf F (t) and a record times given as U (1) = 1 and U (k + 1) = min{j > U (k); T j > T u(k) }; k ∈ N with the random variable T u(k) (k ∈ N) as the upper record values.Then, the pdf of the i upper record value U R i = T u(k) with a special case of U R 1 = T 1 is given as 5. Sub-models Some special sub-models were considered for flexibility, viability, and tractability using the proposed WGG generated model.We present some special cases of the WGG generated family of distributions since it extends several useful distributions in the literature.For all cases listed next, we consider t, λ, β > 0. Especially submodels with increasing, decreasing shaped data with or without a flat region in modeling.These special sub-models include Burr-XII, Lomax, and Frechet distributions.

5.1.
Weighted Gompertz-G Burr-XII (WGG-B) distribution.Consider the Burr XII distribution with positive parameters θ and ρ, and cdf and pdf given as Then, inserting these expressions into Equations ( 3) and ( 4) gives the WGG-B density function with the cdf and pdf given as and Plots of the WGG-B density function for the selected parameter values are displayed in Figure 1a. Figure 1b displays the corresponding hazard rate function (hrfs) for particular values of the parameters.The shapes of the hazard rate function indicated increasing, and decreasing.

Weighted Gompertz-G Lomax (WGG-L) distribution. Consider the
Lomax distribution with positive shape parameters θ and scale parameter ρ, and cdf and pdf given as θ+1) .Then, inserting these expressions into Equations ( 3) and ( 4) gives the WGG-L density function with the cdf and pdf given as and Plots of the WGG-L density function for the selected parameter values are displayed in Figure 2a. Figure 2b displays the corresponding hrfs for some particular values of the parameters.The shapes of the hazard rate function indicated increasing, and decreasing.

Mathematical Expression
To examine the productivity of the WGG generated model, mathematical expansion of the pdf and cdf is carried out.The exponential term in (3) Also, by binomial expansion, we have Hence, the WGG generated pdf can be expressed as power function as where where Γ(•) is a gamma function.

Statistical Properties
The viability and performance of the proposed model will be investigated by examining some general statistical properties of the WGG generated model in this section.
Oftentimes, the expectation, variance, and moments of random variables can be obtained from some characteristics of the distribution function.Some of these functions are the probability generating function and the moment generating function.
Lin's condition random variable The Lin's function for a pdf f of a random variable T with a support t > 0 is defined as

Incomplete moments
The incomplete moments of the WGG generated model allow the shape of the moments of WGG generated distribution, which is of interest for many areas, including econometrics, finance, and reliability, to be visible.
The k th incomplete moment, say τ k (t) of the WGG generated moment is given as Probability generating function This is a useful mechanism for characterizing the distribution of the random variable T with the WGG generated model.It can succinctly be used to describe the sequence of the probability of the random variable T with the WGG distribution.Hence, a random variable T with a WGG distribution has the probability generating function defined as where

Moment generating function
The probability density function of the random variable T can be identified using the moment generating function instrument.This is, however, possible since the moment generating function is a non-negative integral of measurable function.Thus, for a random variable T with a WGG distribution, the moment generating function is given as Probability weighted moments One of the widely used characteristics of a distribution is called L-moments or probability weighted moments.This characteristic is used in hydrology to estimate the parameters of flood distributions.This might be because it is less sensitive to outliers, lower sampling variability, and fast convergence to asymptotic normality.The shape of the WGG generated probability distribution can also be summarized using the L-moments.Thus, L-moments are defined as: However, F v can be expressed as where Γ(•) is a gamma function.Hence, L-moments is given as where

Entropies
The heterogeneity or impurity of the target variable of Poisson process can be measured by the amount of uncertainty associated in the value of a random variable.
Thus, the Shannon entropy of WGG generated random variable T is defined as The Renyi entropy is a measure that increasingly weighs all WGG generated random events with nonzero probability.As θ approaches zero, the WGG generated Renyi entropy is given as This implies where

Moment of the residual
In reliability theory, and life testing scenarios, the additional lifetime a process or a product that a component or chain has survived up to time t is called the vitality function or residual life function or truncated moment.It can also be used to obtain the distribution function F (t). Thus, the k th moment of the residual life defined as Hence, it is expressed as where Theorem 2. Let T be a random variable with a WGG generated probability distribution function Then, Thus, differentiating with respect to y, we have Hence,

Parameter Estimation
It is intuitive to note that the parameters of the WGG generated model are descriptive measures of the entire population that determine the shape and location of the curve on the plot of the WGG generated distribution.Hence, for a better forecasting and regression analysis of the proposed WGG model to be efficient, there is a need to obtain the parameter estimates of the WGG generated model.Thus, in this section, the parameters of the WGG generated model are estimated using the maximum likelihood estimation (MLE) method.8.1.Maximum Likelihood.Let T = (T 1 , T 2 , . . ., T k ) be a random sample obtained from the WGG generated distribution with unknown parameter vector Θ = (β, λ, ψ) T .Let t = (t 1 , t 2 , . . ., t k ) be a sample value of a random sample T.Then, we can obtain the log-likelihood as The parameters of the WGG generated model are obtained by taking the first partial derivative of the log-likelihood of the WGG model with respect to each of the parameters and equate to zero.Thus, we have and However, the solutions to the nonlinear equations ( 29), (30), and (31) are obtained in closed form using numerical methods.These numerical methods are beyond the scope of this article.

Applications
The viability, tractability, and performance of the WGG generated model is examined by first performing a Monte Carlo simulation of some sub-models of the proposed model.The real-life applications of some of the sub-models of the proposed model were investigated and compared to some competitive-related models in the literature.The WGG sub-models were compared with some existing models based on their mean squared errors in the simulation cases and goodness-of-fit test statistics in life applications.9.1.Simulation study.A Monte Carlo simulation was carried out to test the flexibility and efficiency of the proposed distribution.The simulation was achieved using the quantile function in (9) to generate random data for the proposed model with 0 < q < 1 for various values of λ = 1.0, β = 1.0, θ = 0.2 and ρ = 1.0 for the Burr XII sub-model.λ = 0.9, β = 2.3, θ = 0.1 and ρ = 0.01 for the Lomax sub-model, and λ = 0.1, β = 0.1, θ = 0.3 and ρ = 0.8 for the Frechet sub-model for 1000 replicated trials.
The sample size n are taken as n = 5, 10, 20, 50, 100, 150, 200, 250,300, 350, 400, 450, and 500 The simulation studied the mean estimated (ME), biases, and mean squared errors (MSE).The result of the simulation is as shown in Table 1.In Table one, we observed that the biases converge to zero as sample sizes increase.The estimated mean also converges to the true value as the sample sizes increases.The mean square errors converge to zero.
The bias is obtained for (W = λ, β, θ, ρ) as Also, the MSE is obtained as 9.2.Life applications.In most cases of statistical modeling, the interest is to estimate the model parameters and evaluate their test statistics goodness-of-fit.Thus, in this section, the viability, tractability, and effectiveness of the proposed model is investigated with the illustration of real-life data sets.The measures of the test statistics' goodness-of-fit were examined with some existing neighbourhood models in the literature.These models in the literature include, but are not limited to, the class of Weibull, Gompertz, Kumaraswamy, and Frechet distributions.The test statistics considered include the Akaike information criterion (AIC), Anderson-Darling (A), Cramer-von Mises (W), Kolmogrov-Smirnov (KS), and p-value (p-val).The larger the p-value and the smaller the test statistics the better the model fits the data.9.2.1.Obesity Data.The first data consist of 22 obesity among children and adolescents aged 12-19 by selected characteristics: United States, selected between 2015 -2018 as reported by [9].The data are available in https://www.cdc.gov/nchs/hus/contents.htm-The descriptive statistics of the data are given in Table 2.
Table 1.The mean estimates (ME), biases and mean squared errors (MSE) for λ, β, θ and ρ with WGG generated sub-models Table 3 shows the test statistics of the goodness-of-fit measure of comparison adopted for comprehensive comparison.The descriptive statistics of the data are given in Table 4.We observed from Table 4 that the a positive kurtosis and skewness indicated that distribution is peaked and possesses thick tails, and most values are clustered around the left tail of the distribution while the right tail of the distribution is longer.Figure 6 shows the empirical histogram and cdfs of the obesity real-life data applications.9.3.Discussion.In Tables 3 and 5, we observed that the p-values of the WGG generated models are the highest with the lowest AIC test statistic in Burr XII, Lomax, and Frechet sub-models.Hence, the WGG model has provided a better alternative to making statistical distributions more flexible, and viable compared

Conclusion
Intuitively, a two-parameter weighted Gompertz-G generated distribution was examined and introduced by making use of a weighted Gompertz and the T-X characterizations.The newly developed model has found its uses in cases where two-sided abrupt changes schemes occurred in applications.The WGG model has provided a better alternative to making statistical distributions more flexible, and viable compared to the model generated by Gompertz, Weibull, Kumaraswamy, and Alpha power models.The statistical properties and estimations of the model parameters were obtained.The viability and flexibility of the WGG-generated model were demonstrated by illustration of a simulation and real data sets using their goodness-of-fit statistics.The outcomes of the WGG-generated test statistics indicated a better viable, tractable, flexible, and parsimonious generator compared to some competitive models in the literature.Hence, it can be used as a better alternative in reliability theory and extreme value theory.

Figure 1 .
Figure 1.The plots WGG-B model for selected values of parameters

Figure 2 .
Figure 2. The plots WGG-L model for selected values of parameters

Figure 3 .
Figure 3.The plots WGG-F model for selected values of parameters

Figure 5 9 . 2 . 2 .
Figure 5 shows the empirical histogram and cdfs of the obesity real-life data applications.9.2.2.Precipitations in Karachi city, Pakistan Data.The second data examined comprises 59 annual maximum precipitations in Karachi city, Pakistan, for the

Figure 4 .
Figure 4.The Empirical densities and cdfs of obesity among children and adolescents data set

Figure 5 .
Figure 5.The Empirical densities and cdfs of maximum precipitations in Karachi city, Pakistan data set Table-027.The data were measured based on height and weight.

Table 3 :
The goodness-of-fit measure of obesity among children and adolescents data set (standard errors in parentheses)

Table 3 -
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Table 4 .
The Descriptive statistics of annual maximum precipitations in Karachi city, Pakistan data set to 2 decimal points

Table 5 :
The goodness-of-fit measure of maximum precipitations in Karachi city, Pakistan data set (standard errors in parentheses)

Table 5 -
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