A New Version of Cubic Rank Transmuted Gumbel Distribution

A Cubic Rank Transmuted Gumbel distribution (CTGD) in this article is a new generalization of the Gumbel distribution based on a cubic ranking transmutation map. Examined are the cubic transmuted Gumbel model’s fundamental statistical properties, such as its hazard rate function, moment-generating function, moments, characteristic function, quantile function, entropy, and order statistics. Finally, the usefulness and applicability of the CTGD using two real data sets about waiting time at a bank is described and Wheaton River flood, and the fit has been compared with Gumbel distribution (GD) and transmuted Gumbel distribution (TGD). The results show that the proposed model provides a superior fit than transmuted Gumbel distributions and Gumbel distributions.


Introduction:
The Gumbel distribution is named after Emil Julius Gumbel (1891-1966), who described the distribution in his original writings. The Gumbel distribution is a special illustration of the generalized extreme value distribution (also known as the Fisher-Tippett distribution). It is sometimes referred to as the double exponential distribution and the log-Weibull distribution [1]. Perhaps the most frequently used statistical distribution for engineering challenges is the Gumbel distribution. It is sometimes referred to as the type I extreme value distribution. Flood frequency analysis, network engineering, nuclear engineering, offshore engineering, riskbased engineering, space engineering, software reliability engineering, structural engineering, and wind engineering are a few of its latest application areas in engineering.
Over fifty applications are included in a recent book by Kotz and Nadarajah [2] that describes this distribution and includes information on accelerated life tests, earthquakes, floods, horse racing, rainfall, queues in supermarkets, sea currents, wind speeds, and track race records (to mention just a few). It is one of four EVDs that are frequently used. The other three are the Generalized Extreme Value Distribution, the Weibull Distribution, and Frechet Distribution.
The Gumbel, Frechet, and Weibull families, commonly known as type I, type II, and type III extreme value distributions, have been combined into a single family of continuous probability distributions called the generalized extreme value (GEV) distribution. The probability density function (PDF) and the cumulative distribution function (CDF) for Gumbel distribution are defined as follow: where , x ∈ R, σ , µ > 0 Some extensions of the Gumbel distribution have previously been proposed. The Beta Gumbel distribution, Nadarajah et al. [3], the Exponentiated Gumbel distribution as a generalization of the standard Gumbel distribution introduced by Nadarajah [4], and the Exponentiated Gumbel type-2 distribution, studied by Okorie et al. [5]. Transmuted Gumbel type-II distributton with applications in diverse fields of science by Ahmad et al. [6], presenting Transmuted exponentiated Gumbel distribution (TEGD) and its application to water quality data of Deka et al. [7], and transmuted Gumbel distribution (TGD) along with several mathematical properties has studied by Aryal and Tsokos [8] using quadratic rank transmutation. Quadratic rank transmuted distribution has been proposed by Shaw and Buckley [9]. A random variable X is said to have a quadratic rank transmuted distribution if its cumulative distribution function is given by: Differentiating (3) with respect to x, it gives the probability density function (pdf) of the quadratic rank transmuted distribution as: where G(x) and g(x) are the cdf and pdf respectively of the base distribution. It is very important observe that at λ = 0, we have the base original distribution. The family of quadratic transmuted distributions shown in (3) expands any baseline distribution G(x), increasing its applicability. Recently, Rahman et al. [10] introduced the cubic transmuted family of distributions. A random variable X is said to have cubic transmuted distribution with parameter λ if its cumulative distribution function (cdf) is given by: with corresponding pdf This paper is organized as follows, in Section 2 defining the cubic transmuted Gumbel distribution. Statistical properties have been discussed such as the shapes of the density and hazard rate functions, quantile function, moments and moment-generating function, Characteristic Function, and cumulant generating function in Section 3. Entropy was studied in Section 4, and order statistics in Section 5. Section 6, we address the parameters of the CRTG distribution via maximum likelihood method.
A simulation study is carried out in Section 7 to assess performance of suggested maximum likelihood estimators. An application of the CTGD to two real data sets for the purpose of illustration is conducted in section 8. Finally, in Section 9, some conclusions are declared.

A new version cubic transmuted Gumbel distribution:
The cubic rank transmuted Gumbel distribution is defined as follows: The CDF of a cubic rank transmuted Gumbel distribution is obtained by using (2) in (5).
where x ∈ R, µ > 0, and σ > 0, are a location and scale parameters respectively, λ ∈ [−1, 1] is shape parameter. It is very important note observe that at value λ = 0, the cubic rank transmuted Gumbel distribution reduce to Gumbel distribution according to the transmutation map. The probability density function (pdf) of a cubic rank transmuted Gumbel distribution is given by: where x ∈ R, µ,σ > 0, λ ∈ [-1,1] Figure   bel (CTG) distribution, the reliability function is given as: The hazard rate function can be written as the ratio of the pdf f (x) and the reliability function R(x) = 1 − F(x). That is: then we can find the hazard rate function of GTF distribution by (8) and (9): The cumulative hazard function is defined by: so the cumulative hazard function of the CTG distribution is: The reverse hazard function is: Using (12), we can write the reverse hazard function of CTG distribution as

Quantile Function:
Here we will compute the quantile function of the cubic rank transmuted Gumbel probability distribution. Theorem 3.1 Let X be random variable from the cubic rank transmuted Gumbel probability distribution with parameters σ > 0, µ > 0 and −1 ≤ λ ≤ 1. Then the quantile function of X, is given by: Proof. To calculate the quantile function of the cubic rank transmuted Gumbel probability distribution, we substitute x by x q and F(x) by q in (7) to get the equation Then, we solve the equation (15) for x q . So, let y = e (−e −( xq−µ σ ) ) Thus, (15) becomes q = (1 − λ )y + 3λ y 2 − 2λ y 3 and hence, let a = 2 λ , b = -3 λ , c = (−1 + λ ) and d = q, then the equation (16) becomes Then, where Now, let the function B(q, λ )be defined by: Hence, Take natural Logarithm to both sides to get: Then, we have the equation put q = 0.25, 0.50, and 0.75 in (14) to obtained the first quartile, median and third quartile respectively. Quantiles for selected parameter values for the CTG distribution are shown in Table 1.

Moments function:
Then the rth moment of is given by: Proof. The r th moment is given by Substituting from (8) in to (20), dx, x = µσ log y With substitution by this transformation in (21) then: Now, we calculate [µ − σ log y] r using binomial using Gamma integration: The mean and variance can be easily obtained by using r = 1, 2 in Eq. (19) such that: where γ ≈ 0.5772 is the Euler Mascheroni constant [11].

Moment Generating Function
Proof. We know that: Some moments for selected parameters values in order  Table 2, where CV, SD, CK, and CS represent the coefficient of variation, standard deviation, kurtosis, and skewness, respectively and 3D plots of kurtosis and skewness for the CTG distribution are given in Figure 4 and Figure 5. We observe that: • When the parameter λ is fix, the kurtosis and skewness of CTGD decrease as σ decreases.
• When we fix the parameters µ, the skewness and kurtosis of CTGD decrease as σ decreases.

Characteristic Function:
The cubic transmuted Gumbel distribution's characteristic function theorem is stated as follows: Theorem 3.4 Assume that the random variable X have the CTGD (µ, σ , λ ), then characteristic function, Φ x (t), is: Where i = √ −1 and t ∈ ℜ

Cumulant Generating Function:
The cumulant generating function is defined by:  Cumulant generating function of cubic rank transmuted Gumbel distribution is given by:

Rényi Entropy:
If X is a non-negative continuous random variable with pdf f (x), then the Renyi entropy of δ order (See Renyi [12]) of X is defined as, Theorem 4.1 The Rényi entropy of a random variable X ∼ CTGD (µ, σ ), with λ ̸ = 1 and λ ̸ = 0 is given by: Proof. Assume X has the pdf in (8). Then, can compute.

Shannon Entropy:
In a non-negative continuous random variable X with pdf f(x), the Shannons entropy [15] is defined as: The Expansion of the Logarithm function will be used below (Taylor series at 1), The Shannon entropy of a random variable X ∼ CTGD (µ, σ ), with λ ̸ = 1 and λ ̸ = 0 is given by: To compute the Shannon's entropy of X, substitute from (38) in (35).
substituting from (30) in (39), to get: Now, we use (33) to fined the value of the integration, so,

Order Statistics:
Let X 1 , X 2 , ..., X n be a random sample of size k from the CTG distribution with parameters µ > 0, σ > 0, and −1 ⩽ λ ⩽ 1 From (Casella and Berger [16]), the pdf of the k th order statistics is obtain by: Let X k be the k th order statistic from X ∼ CTGD (µ, σ ) with λ ̸ = 1 and λ ̸ = 0. Then pdf of the k th order statistic is given by: k+ j−1

Maximum Likelihood Estimation (MLE):
Assume X 1 , X 2 , ..., X n be random sample of size n from CTGD (µ, σ ) hen the likelihood function can be written as: Then l(µ, σ , λ ) = 1 Then, the Log likelihood function of a vector of parameters given as, Differentiate w.r.t parameters µ, σ , and λ we have.
By setting the above nonlinear equations to zero, we can use the maximum likelihood method to estimate the unknown parameters.
and solving them simultaneously. Therefore, in order to acquire the numerical solution to the nonlinear equations, can utilize statistical software. using quasi-Newton procedure,or computer packages/ softwares such as R, SAS, Ox, MAT-LAB and MATHEMATICA, We can calculate the maximum likelihood estimators (MLEs) of parameters (µ, σ , λ ).

Simulation Study:
In this section, simulation results are presented for different sample sizes of n = 100, 200, 350, 500 and 600 to check the consistency and accuracy of the maximum likelihood estimators (MLEs) for each CTG distribution parameter. The simulation was conducted N = 1000 times, and the root mean square errors (RMSEs), average bias (A Bias), and mean estimations were assessed. The mean estimations are displayed in Tables 3 and 4 together with the corresponding RSMEs and A Bias. The A Bias and RMSEs for the estimated parameter, say,θ are respectively given as:  From the results, we can verify that as the sample size n increases, the mean estimates tend to be closer to the true parameter values, whereas the RSMEs and A Bias decrease for all parameter values.

Data2: Wheaton River flood data
For the data2, this subsection includes parameter estimates (standard errors in parenthesis), goodness-of-fit statistics, plots of the fitted densities, empirical cdf, hrf graphs, probability plots, Kaplan-Meier and TTT plots.
The CTG distribution fit the data the best, as shown in Figure 9, Figure 10, and Figure 11. According to the fitted density, the CTG distribution can handle skewed data. In data2, the estimated variance-covariance matrix for the CTGD model is given by      From the results in Table 9, CTG distribution performed better than any other model. It had the lowest values for −2 log L, AIC, AICC, BIC, W * , A * , and K-S, as well as the highest p-value when compared to competing models across for Wheaton River flood data. Also These plots indicate that the CTG distribution provides a better fit than others models considered for both data.

Concluding Remarks:
This article examines the cubic rank transmuted Gumbel (CTG) distribution, a novel generalized distribution. The distribution's hazard function, quantile function, moments, distribution of the order statistics, and entropy are among the structural aspects that are examined. The model parameters are estimated using a technique called maximum likelihood estimation. To investigate the performance of the CTG distribution, a Monte Carlo simulation analysis was carried out. The importance and potential of the CTG distribution is demonstrated by examples from two real life data sets.   Funding: None.
Data Availability Statement: All of the data supporting the findings of the presented study are available from corresponding author on request.
Declarations: Conflict of interest: The authors declare that they have no conflict of interest.
Ethical approval: The manuscript has not been published or submitted to another journal, nor is it under review. * dalhirtani@gmail.com * (CT GD) CT GD