ODD GOMPERTZ-G FAMILY OF DISTRIBUTION, ITS PROPERTIES AND APPLICATIONS

In this research paper, we introduced a novel generator derived from the continuous Gompertz distribution, known as the odd Gompertz-G distribution family. We conducted an in-depth analysis of the statistical characteristics of this family, including moments, moment-generating functions, quantile functions, survival functions, hazard functions, entropies, and order statistics. Within this family, we also derived a specific distribution called the odd Gompertz-Exponential distribution. To evaluate the reliability of the distribution's parameters, we employed Monte Carlo simulations. Furthermore, we assessed the applicability of this newly proposed distribution family by examining its performance on real-world data and the results demonstrate that the new model (OG-E) outperformed its comparators under consideration.


MATERIAL AND METHOD The new family
(  −1) ; 0 <  < ∞, ,  > 0 (2) The Cumulative distribution function (, Φ) with survival function (; Φ) = 1 − (; Φ) of the parent distribution depends on a parameter vector Φ and assuming a random variable T relates to a system having a baseline G distribution.So the odd T that a system will not work at particular time interval is given by ( ) ( ) G t G t .The random variable T of the odds using Gompertz model is given by Replacing x in the Gompertz cdf by the odd ratio ( ) ( ) G t G t , the cdf of the novel family, OG-G is as follows 3) The corresponding pdf to (3) is given by ( ) ( ; ) 1 ( ; ) ( ; ) 1 1 ( ; ) 2 ( ; ) ( ; , , ) 1 ( ; ) (4) where(, Φ) is the pdf of any parent distribution andΦis the parameter vector, therefore a random variable X with density function and distribution function in equations ( 3) and ( 4) is denoted by ~ − (, , Φ).

Validity Check of the OG-G family of distributions
It is significant to ascertain whether the pdf of OG-G family of distributions as given in equation ( 4) establishes a valid probability density and this can be realized by ensuring that its integral over the domain of X equals to unity.i.e 0 ( ; , , ) 1 and as  → 0;  → 0and as  → ∞;  → ∞ Now equation ( 5) can be written as Hence, the pdf of the OG-G family of distributions be valid as required.

Useful expansion
In this section, we will explore a valuable expansion of the distribution functions for the OG-G family.

Proposition:
The expression that provides a linear representation of the OG-G family of distributions is as follows: From the power series expansion By binomial expansion, Using generalize binomial theorem, ( ) denotes the cdf of the Exponentiated-G distribution with power parameter ( + ) > 0 By differentiating (13), the pdf of x can be given in the mixture form as; , ( ) 0 0 ( ) ( ) denotes the Exponentiated-G density function with power parameter () kl + .

Moments of the OG-G Family
The rth moment of a random variable X that follows the Odd Gompertz-G (OG-G) family is given as:

Moment-Generating Function of the OG-G Family
The moment-generating function of a random variable X that follows the Odd Gompertz-G (OG-G) family is given as:

Quantile Function of OG-G Family
The quantile function of the OG-G family is obtained by inverting the CDF in equation (3).Say Where, 1 G − is the quantile function of any continuous parent distribution and u is considered as a uniform random variable on the interval (0, 1).

Entropies of the OG-G Family
Entropy is a measure of variation or uncertainty of a random variable X (Renyi 1961).The entropy of the OG-G family is defined statistically as follow: Where 0 Z  and 1 Z 

Order Statistics of the OG-G Family
Let  1 ,  2 , . . .,   be a random sample from the OG-G distribution and  1: ≤  2: ≤. . . : denote the corresponding order statistics, then the ith order statistic is given as:

Survival and Hazard Rate Function of the OG-G Family
The survival function and hazard function are respectively given as: ( ; )

Graph of the Special Sub-Model of the OG-G Family
The plot of the probability density function, hazard function, survival function and cumulative distribution function of the Odd Gompertz-Exponential (OG-E) distribution is given as;

Monte Carlo Simulation
The well-known class of computational algorithms known as "Monte Carlo simulation" is applied to a replicated random sample in order to produce numerical results so as to address the problem of risk in modeling lifetime data.

Simulation Study
To appraise the consistency of the OG-ED model, simulation training was conceded out using Monte Carlo Simulation method.This study aimed to calculate mean, bias, and root mean square error of the estimated model parameters obtained through maximum likelihood estimation.Simulated data was generated using the quantile function described in equation ( 18), and this process was repeated 1,000 times for various sample sizes: n = 50, 100, 250, 500, and 1,000.The parameters were held constant at a specific value for each of these simulation runs.

Applications
Here, we exhibit the potentiality of the Odd Gompertz-Exponential Distribution (OG-ED) using a real data set from a previous studies, see Arslan et al. (2019).The maximum likelihood estimates, as well as goodness-of-fit measures, were computed via R software and compared with Weibull Exponential (WE), Gompertz Exponential (GE), Kumaraswammy Exponential (KE), Exponentiated Weibull-Exponential (EW-E) and Exponential (E) distribution.We employ the Akaike Information Criterion (AIC), which has the following mathematical expression in order to identify which of the competing models is the best: 1 .6, 3.5, 4.8, 5.4, 6.0, 6.5, 7.0, 7.3, 7.7, 8.0, 8.4, 2.0, 3.9, 5.0, 5.6, 6.1, 6.5, 7.1, 7.3, 7.8, 8.1, 8.4, 2.6, 4.5, 5.1, 5.8, 6.3, 6.7, 7.3, 7.7, 7.9, 8.3, 8.5, 3.0, 4.6, 5.3, 6.0, 8.7, 8.8, 9.0.Table 2 displays the outcomes of the maximum likelihood estimation regarding the parameters of the new distribution and five other reference distributions.When assessing the goodness of fit, it was observed that the proposed distribution exhibited the lowest AIC value, with GE coming in a close second.A visual examination of the fit, as depicted in Figure 5, further validates that the proposed distribution outperformed its comparator distributions.Consequently, among the various distributions under consideration, the proposed distribution is deemed the most suitable for modeling the failure time of turbocharger of one engine dataset.The results demonstrate that the new model (OG-E) outperformed the existing ones under consideration, suggesting its utility as a new distribution for modeling data in a wide range of applications.

CONCLUSION
Consider the probability density function (pdf) and cumulative distribution function (cdf) of the Gompertz distribution as defined by lanert (2012) with  as scale parameter and  as shape parameter are respectively known as: These equations are intractable and can only be solved using a numerical iterative method.

Figure 1 :Figure 4 :
Figure 1: pdf of the Odd Gompertz-Exponential Distribution 2 + 2.Where L stands for log-likelihood function, k is the number of model parameters.The data set used for the analysis is obtained from the work ofArslan et al. (2019)  and it represents the time to failure (10^3/h) of turbocharger of one engine as seen below:

Figure 5 :
Figure 5: Histogram Plots of the Distribution of time to failure of turbocharger of one engine Data.
We define the odd Gompertz-G as a new family of continuous distribution.Some statistical characteristics of the new family, like the explicit quantile function, moments, momentgenerating functions, survival function, hazard function, entropies, and distribution of order statistics, are investigated.Additionally, specific sub-model within this novel family was deliberated.The technique of maximum likelihood is employed to estimate the parameters of the model.Simulation training was conducted in order to assess the effectiveness of the offered distribution.To showcase the significance and adaptability of the sub-model, a real-life dataset is used in analyzing and comparing the well-known competing models.

Table 1 : Average Values of the MLEs, Biases and RMSEs of the OG-ED
Table1above indicates that biases and RMSEs tend to approach zero as the sample size rises.This trend suggests that the estimates become more accurate and reliable, converging towards the initial (true) values, it demonstrates that the estimates are both efficient and consistent as the sample size grows.