The Weibull-exponential {Rayleigh} Distribution: Theory and Applications

This study introduces a new distribution in the family of generalized exponential distributions generated using the transformed-transformer method. Some properties of the distribution are presented. The new distribution has three parameters and they are estimated numerically using the BGFS iterative method implemented in R software. Two real sets of data are adopted to demonstrate the flexibility and potential applications of the new distribution.


Introduction
Many sets of data are being generated in various fields of human endeavour on daily basis. Some of these data sets are quite unique in nature that there are yet no known existing statistical distributions that adequately describe such data. This has given rise to new probability distributions. In doing this, many methods have been adopted by various researchers. These methods, based on the period of development, have been majorly grouped into two: methods before 1980 which include differential equation, translation and quantile methods and those adopted since 1980 which are referred to as contemporary methods (these include those obtained by adding parameters to a known distribution and/or compounding of distributions (Lee et al. [17]). Although the methods used in generating new distributions prior to 1980 are still being explored by researchers in 66 generating new distributions, most of the new distributions are generated using the contemporary methods.
Among the recent methods, this study will adopt the transformed-transformer method developed by Alzaatreh et al. [5]. Here, they used a random variable X, with pdf, ( ) is monotonically increasing, with the corresponding density function given as Some of the studies using the transformed-transformer method include those by Alzaatreh et al. [6], Al-Aqtash et al. [1], Merovci and Elbatal [18], Oguntunde et al. [21], Fatima and Ahmad [13], Osatohanmwen et al. [22].
Hence, the associated pdf to ( ) The Again, assuming the probability density function of the three random variables exist and are represented respectively as then the cdf of the new class of distributions is given by with the corresponding pdf as The concept of T-R{Y} framework was also adopted in subsequent studies by Alzahal et al. [10], Almheidat et al. [4], Tahir et al. [25], Alzaatreh et al. [8], Alzaatreh et al. [9], Aldeni et al. [2], Jamal et al. [16], Hamed et al. [15], Ekum et al. [11].
Other sections in this article are organized as follows. In Section 2, the Weibull-Exponential {Rayleigh} is introduced. Some features of the distribution and the estimation of its parameters are studied and presented in Sections 3 and 4 respectively. The use of data to demonstrate the flexibility and application of the new distribution is achieved in Section 5. Finally, the article is summarized and concluded in Section 6.

The Weibull-exponential {Rayleigh}
Suppose T follows the Weibull (λ, α) distribution with cdf, pdf and quantile function respectively given as Similarly, let R follow the exponential distribution with cdf, pdf and quantile function respectively given as Finally, let Y follow the standard Rayleigh distribution with cdf, pdf and quantile function respectively given as ( ) where ( ) Hence, the cdf of the Weibull-exponential {Rayleigh} is given by Substituting (11), (17) and (18) into (6) gives the corresponding pdf of the Weibull-Exponential {Rayleigh} distribution as is a special case of WER distribution when 1 2 σ = and 1 λ = 4β is a special case of WER distribution when 4 σ = and 1 λ = (iii) Exponential ( ) 2β results from WER distribution when 2 σ = and 1 λ = (iv) When 1, σ = the WER distribution reduces to Exponential-Exponential {Rayleigh} with parameters λ and 2β Equations (16) and (19) respectively give the cdf and pdf of the 3-parameter Weibullexponential {Rayleigh} distribution with all the three parameters ( ) , , β σ λ as shape parameters.

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Some shapes of the distribution density function are given in Figure 1. The superimposed density plots in Figure 2 for different values of the parameters of the WERD indicate that the WERD can be right-skewed or approximately normal.

Properties of the Weibull-Exponential {Rayleigh} Distribution
Here, we discuss some of the properties of the Weibull-exponential {Rayleigh} distribution The Weibull-exponential {Rayleigh} Distribution: Theory and Applications a. Hazard Function: From (16) and (19), the hazard function of the Weibullexponential {Rayleigh} distribution is obtained as Proof.
(i) Given that T is a Weibull random variable with pdf which is the result given in (19). Thus X assumes the Weibull-Exponential {Rayleigh} distribution with parameters β, σ and λ.

c. Quantile Function:
The quantile function of the T-R{Y} distribution defined in (5) is where ( ) 1 .
To obtain the median of the Weibull-exponential {Rayleigh} distribution, 0.5 p = is substituted in (22).
The rth crude moment of a random variable X, say r m′ is given by The rth crude moment of Weibull-exponential {Raylegh} is given by Proof. From Theorem 1,  [14]). Therefore, the first four crude moments of Weibull-Exponential {Rayleigh} distribution are The standard deviation (SD) is given by Therefore, the standard deviation of Weibull-exponential {Rayleigh} is

b. Coefficient of Skewness (CS) and Coefficient of Kurtosis (CK):
Based on moments, the coefficient of skewness is given by

CS
Similarly, the coefficient of kurtosis is given by c. Shannon Entropy: This measures the level of variation of uncertainty of a random variable, Shannon [23]. The Shannon entropy of the T-R{Y} class of distributions is given by where T η is the Shannon entropy of T.

Theorem 4. The Shannon entropy of the Weibull-exponential {Rayleigh} distribution is
Proof.

d. Mode: The mode of Weibull-exponential {Rayleigh} is given by
The first derivative of (19) is factorizing and equating to zero

Estimation of Weibull-exponential {Rayleigh} Parameters using Maximum Likelihood Method
Suppose 1 2 , , ..., n x x x is a random sample of size n, the log-likelihood function of the Weibull-exponential {Rayleigh} distribution is By partially differentiating (32) with respect to each of the parameters, we obtain ( ) Since equating (33)-(35) to zero and simultaneously solving for each of the parameters cannot result to a close form of solution, the numerical method, Broyden-Fletcher-Goldfarb-Shanno (BFGS) iterative method embedded in R is adopted.

Applications
This section presents an application of the Weibull-Exponential {Rayleigh} distribution using two real datasets. The fit of the Weibull-Exponential {Rayleigh} distribution (WERD) to the data sets is compared with those of Weibull-Exponential (WED) (Oguntunde et al. [21], Weibull-Rayleigh distribution (WRD) (Merovci and Elbatal [18]), Exponential-Rayleigh distribution (ERD (Fatima and Ahmad [13]), Exponential distribution (ED) and Rayleigh distribution (RD) with respective PDFs The measures adopted for the goodness-of-fit testing are the parameter estimates, log-likelihood function, Akaike Information Criteria (AIC) and Kolmogorov-Smirnov Statistic (K-s) and p-values obtained with the aid of R software. A distribution with smaller AIC and K-S but higher log-likelihood and p-values relative to other distributions gives a better fit to the dataset. The results reported in Table 1 reveal that the Weibull-Exponential {Rayleigh} distribution performed better than the other distributions compared with as it has the highest value of log-likelihood (LL) and p-value. Also, the AIC and K-s values of the distribution are the smallest when compared with those of other distributions in fitting the data.    Values in Table 2 indicate that, for the given set of data, the Weibull-Exponential {Rayleigh} distribution (WERD) gives the best fit based on the values of the LL, AIC, Ks and p-value.

Summary and Conclusion
In this article, a new submodel belonging to the class of generalized exponential distributions is defined and studied using the T-R{Y} approach introduced by Aljarrah et al. [3]. Some features of the submodel are derived. The estimates of the parameters of the submodel are obtained with use of maximum likelihood method. Two real data sets are adopted in demonstrating the flexibility and potential applications of the new distribution.