The Gompertz Gumbel II Distribution: Properties and Applications

In this paper we introduced Gompertz Gumbel II (GG II) distribution which generalizes the Gumbel II distribution. The new distribution is a flexible exponential type distribution which can be used in modeling real life data with varying degree of asymmetry. Unlike the Gumbel II distribution which exhibits a monotone decreasing failure rate, the new distribution is useful for modeling unimodal (Bathtub-shaped) failure rates which sometimes characterised the real life data. Structural properties of the new distribution namely, density function, hazard function, moments, quantile function, moment generating function, orders statistics, Stochastic Ordering, Renyi entropy were obtained. For the main formulas related to our model, we present numerical studies that illustrate the practicality of computational implementation using statistical software. We also present a Monte Carlo simulation study to evaluate the performance of the maximum likelihood estimators for the GGTT model. Three life data sets were used for applications in order to illustrate the flexibility of the new model.


Introduction
The Gumbel type-2 distribution plays an important role in Extreme value theory. The distribution can be used for modeling extreme events such as in the field of risk based engineering, flood frequency analysis, Meteorology, structural engineering, software reliability engineering, network engineering and Seismology. The distribution has not gained popularity/prominence in the area of application unlike the Weibull distribution because of its lack of fit. The Gumbel type-2 distribution can only be applied to real life data with monotonic failure rates. On the contrary, in real life situations the hazard rate of many complex phenomena that are often encountered in practice is nonmonotone and cannot be modeled by the Gumbel type-2 distribution. To address this limitation [1] proposed the Exponentiated Gumbel type-2 distribution according to Nadarajah and Kotz [2] version of Gupta, et al. [3], Okorie, et al. [4] proposed the Kumaraswamy G Exponentiated Gumbel Type-2 Distribution which was obtained by combining Exponentiated Gumbel (EG) and the kumaraswamy distribution [5,6] proposed and studied the properties of Extended Gumbel type-2 ) distribution. Motivated by some of the properties of the generalised distribution with respect to the nature of its hazard function which includes; increasing, decreasing, non-monotone and bathtub shapes as well as the tractability and flexibility of the generalised distribution with improved statistical properties. We propose and study a new distribution called the Gompertz Gumbel type-2 distribution which inherits these desirable properties with improved modeling capabilities most especially in modeling life time data.
The cumulative distribution function of the Gumbel type-2 distribution is given by With the corresponding given by Where is a shape parameter and is a scale parameter. Recently, Alizadeh, et al. [7] developed Gompertz-G family of distributions by making use of Transformed-Transformer (T-X) family of distribution, defined as The pdf corresponding to equation (3)  Where is the baseline cdf depending on the parameter is vector and are two additional shape parameters. For a specified baseline , the is defined by the (5) which represents a wide family of distributions.
The associated pdf to equation (5) is given by The new density function is most tractable when ) and have simple analytical expressions. Based on the generalization in equation (5) and (6), several flexible distribution have been proposed and studied not limited to the work of Oguntunde, et al. [8] studied the properties of Gompertz inverse exponential distribution, Khaleel, et al. [9] studied the Gompertz flexible Weibull distribution and its applications etc.

The Gompertz Gumbel Type-2 Distribution
Now, suppose that (1) and (2) are any continuous baseline cumulative distribution function (cdf) and probability density function (pdf) of Gumbel type-2 (GTT) distribution. The Gompertz Gumbel type-2 (GGTT) distribution is obtained by putting equation (1) in (5) given by And the corresponding pdf is given by Special sub-models of the GGTT distribution are recorded in Table 1. The graph of the cdf is drawn below in figure 1. taking the value of parameters and varying the values of parameters (alpha and lambda).    Figure 2. drawn below is the graph of the pdf of GGTT distribution taking , for diagram I and for diagram II. Figure 3. is the graph of the survival function for arbitrary values of the parameters and Figure 4. is the graph of the hazard function of GGTT distribution.

Quantile Function
The quantile function plays a useful role when simulating random variates from a statistical distribution. The quantile function of the distribution, say is given by: In many heavy tailed distributions, the classical measures of skewness and kurtosis may be difficult to obtain as a result of nonexistence of higher moments. In such a situation, the quantile measures could be a better option. The Bowley skewness [11] is one of the earliest measures of skewness based on quartiles of a distribution. It is defined as: Similarly, the coefficient of kurtosis can be estimated using the Moors' coefficient of kurtosis [12] is obtained based on the octiles of a distribution defined as: It should be noted that the two measures are less sensitive to outliers and they exist for distribution which moment cannot be defined. Table 1. drawn below gives the various values of Bowley Skewness ( ) and Moor kurtosis ( for arbitrary values of the parameters taking a fixed value of ( .

Mathematical Properties of Gompertz Gumbel-Type Distribution
Here, we examined the mathematical properties of distribution If | | and is a real non-integer, the following expansion exist ∑ ( *

∑
If is an integer, index i in the previous sum, stops at . Using the equations (15) and (16) we can re-write the pdf of distribution given in (8) as Finally the pdf of GGTT distribution can be written as The remaining parts of the paper is arranged as follows: section 2 presents the comprehensive study of the moments, variance, skewness, kurtosis and moment generating function of the new model; section 3 presents a study on the Renyi entropy of the new distribution; section 4 presents the Stochastic ordering of the new distribution; section 5 gives a comprehensive review of the order statistics of the new distribution which includes the and order; section 6: proposes the maximum likelihood estimation method of estimating the new model; in section 7 we carried out Monte Carlo simulation to validate the maximum likelihood estimation technique that was used to analyse the data; section 8 presents the applications of the new model and section 9 the conclusion.

Raw Moment
Here we derive an expression for raw moments of GGTT distribution as

∑ ∫
By letting and substitute in equation (20), we have Finally we have , Where ∫ the complementary incomplete gamma function. The first four raw moments for are respectively, , variance( and coefficient of variation Table 2. drawn below gives the various values of variance ( ) and coefficient of variation ( for arbitrary values of the parameters taking a fixed value of (

Moment Generating Function
The moment generating function of GGTT distribution is obtained as Applying relation in equation (16)

Incomplete Moment
The incomplete moment has important applications in different fields of study. The first incomplete moment is used in estimation of the Bonferroni and Lorenz curves which are useful in reliability, demography, insurance, seismology and medicine. The incomplete moment of the GGTT random variable is: Using the complementary incomplete gamma function, this yields: , Where ∫ the complementary incomplete gamma function.

Mean Residual Life and Mean Inactivity Time
The Mean Residual Life (MRL) or the life expectancy at age t is the expected additional life length for a unit, which is alive at age t. The MRL has several important applications in life time testing of product, life insurance, demography and economics etc. The MRL is given by: Which can also we written as { } Where, is the first incomplete moment and is the survival function. Thus, the MRL of the distribution is: The Mean Inactivity Time is the waiting time elapsed since the failure of an item on condition that the failure had occurred in (0,t). The of the random variable X is defined for t > 0 as: This can further be expressed as Substituting the first incomplete moment and the CDF of the random variable yields its MIT as:

Renyi Entropy
Entropy has been applied in the field of engineering sciences and information theory as measures of variation of uncertainty. The Renyi entropy [13] of a random variable X having the GGTT distribution is given as:

[∫ ]
Putting equation (19) in (44), we have After simple substitution, we have The Renyl entropy converges to the Shannon entropy as approaches 1. The entropy, say of the GGTT random variable is defined by: Hence,

Order Statistics
In this section, we derive closed form expressions for the pdf of the order statistic of the (EGTT) distribution. Let be a simple random sample from (GGTT) distribution with cdf and pdf given by (7) and (8), respectively. Let denote the order statistics obtained from this sample. The probability density function of is given by and are the cdf and the pdf of the GGTT distribution given in equation (7) and (8)  Using the series expansion in equation (15) and (16) and also the relation in equation (48), can be expressed as For . An expression for the order statistics and the order statistics is respectively given in equation (56) and (57).

Maximum Likelihood Estimation of the Parameters
The likelihood function of GGTT distribution is given by The log likelihood function is The nonlinear likelihood equations can be obtained for GGTT by differentiating equation (54) with respect to . The components of the score vector ] We can obtain the estimates of the unknown parameters by maximum likelihood method by setting these above nonlinear equations (55) -(58) to zero and solve them simultaneously. Therefore, statistical software can be employed in obtaining the numerical solution to the non-linear equations such as R, MATLAB, Maple etc. For the four parameters Gompertz Gumbel type-two pdf, all the second order derivatives can be obtained. Thus the inverse dispersion matrix is given by ]. This gives the approximate variance covariance matrix. By solving for the inverse of the dispersion matrix, the solution will give the asymptotic variance and covariance of the MLs for ̂ ̂ ̂ and ̂. The approximate confidence intervals for can be obtained respectively as

Simulation Study
In this sub-section, we carried out a simulation study to examine the performance of maximum likelihood estimators of Gompertz Gumbel type-2 distribution. In this context, we carried out the simulation study using the Monte Carlo simulation, as explained in the following works: Lemonte [14], Cordeiro and Lemonte [15] and De Andrade, et al. [16]. We investigated the behavior of the MLEs for the parameters of the GGTT model by generating from (12) samples sizes n = 50,100,300 and 500 with selected values for values for α,λ,δ and θ. We consider 5,000 Monte Carlo replications. The simulation process was performed in the R software using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) maximization method in the optimum script. To ensure the reproducibility of the experiment, we use the seed for the random number generator: set.seed (90) The results of the simulations are presented in Table 4. 4.1, and 4.2, including the means, Absolute Bias (AB), Standard Error (SE) and the Mean Square Error (MSE). We observed that the estimated values of the parameters are very close to the true values and also the MSE consistently decreases as the sample size increases. This is a desirable property to show the adequacy of the estimation technique.

Applications to Real Life Data
In this section, we present three examples that demonstrate the flexibility and the applicability of the GGTT distribution in modelling real world data. We fit the density functions of the GGTT distribution and compare its fits with that of Exponentiated Gumbel Type-two (EGT), Extended Gumbel type-two (EGTT) distribution and its submodels Gumbel Type-two (GT) distribution. For all the fitted models, we compute the MLEs of the model parameters (with their corresponding standard errors in parentheses) and also the values of the Akaike information criterion (AIC), Hannan-Quinn information criterion (HQIC), Consistent Akaike information criterion (CAIC), Bayesian Information Criterion (BIC), Kolmogorov-Smirnoff (KS) statistic, Anderson Darling statistic ( ) and the probability value (PV) used as methods of comparing fits of distributions to data. In general, it is considered that the smaller the values of AIC, BIC, HQIC, CAIC and ( ) and the larger the PV the better the model fit to the data.

Yarn Specimen Data Set
The third dataset consists of data on the number of cycles of failure for 25 specimens of 100 cm specimens of yarn, tested at a particular strain level by Lawless [19]. The starting points for the iterative processes in the yarn specimens' data are (109:35; 1:2920; 3:6125; 1:0; 1:0) : 15, 20, 38, 42, 61, 76, 86, 98, 121, 146, 149, 157, 175, 176,  180, 180, 198, 220, 224, 251, 264, 282, 321, 325, and 653. Table 10 gives the MLEs and Table 11 gives the selection, criteria statistics for the pig data. Figure 9.0 gives the graph of the Total Time on Test plot and the graph of the kernel density of the pig data which clearly shows that the yarn specimen data exhibits an increasing failure rate and positively skewed, also Figure 10. Gives the fitted densities of Yarn specimen data

Conclusion
In this study, a four-parameter model called GGTT distribution is proposed and its statistical properties are derived. We discussed the maximum likelihood method to estimate the model parameters and presented a Monte Carlo simulation study to evaluate the performance of the maximum likelihood estimators for the GGTT model. Finally, three applications illustrate the potential of the GGTT distribution for fitting survival data. The performance of the GGTT distribution with regards to providing good fit to the data sets is assessed by comparing it with other models including its sub-model. The results show that the GGTT model provides a more reasonable parametric fit to the data sets.

Data Availability Statement
The data used for this research are commonly and predominantly use data in our area of research. There is absolutely no conflict of interest between the authors and producers of the data because we do not intend to use these data as an avenue for any litigation but for the advancement of knowledge.

Funding Statement
The research was funded by effort or the authors.