The Weibull Logistic-exponential Distribution: Its Properties and Applications

In this paper , a new generalized distribution known as Weibull Logistic-Exponential Distribution (WLED) is proposed using the T - R { Y } framework. Several mathematical properties of this new distribution are studied. The maximum likelihood estimation method was used in estimating the parameters of the proposed distribution. Finally, an application of the proposed distribution to a real lifetime data set is presented and its fit was compared with the fit obtained by some comparable lifetime distributions.


Introduction
The logistic model is a symmetric distribution that has found application in vast areas of growth model in human populations. [11] and [9] used the logistic function in modelling data related to population. Several generalizations of the logistic distribution have been introduced in literature to improve the symmetric and tail (heavy & light) properties of the distribution. Examples of such generalization of logistic distribution are found in the works of [3], [7], [8], [14], [13], and [4]. This paper is motivated by the flexibility of the generalized distribution in terms of exhibiting and accommodating diverse kind of data sets. The proposed distribution have found an application in finance for modelling risk management and explain the return of an investment due to the strictly platykurtic nature of kurtosis ( 3).

CDF and PDF of the Proposed Weibull-Logistic {Exponential} Distribution (WLED)
Let T, R and Y be random variables from a known probability distribution with the cumulative distribution functions defined by ( ) ( ), Y Q p [2] gave a unified definition of the random variables reported in [1] by defining the cumulative distribution function of a random variable X as and the corresponding density function defined as Let R be a random variable following a logistic distribution with pdf and cdf define by Then the cdf of the proposed distribution is gotten by using the quantile function of a random variable Y defined by the relation.
By following the technique in Equation (1) we obtain the cdf of the proposed distribution.
The corresponding pdf of the WLED can be obtained by finding the derivative of its cdf in (2) with respect to , this is given by: The graphical presentation of the density function of WLED for some fixed values of the parameters is shown in Figure 1.

Survival function
Let X be a continuous random variable with density function $ and cumulative distribution function . The survival (reliability) function of the proposed WLED are defined by: where is the cdf of the WLED define in Equation (2).

Hazard rate function
Let X be a continuous random variable with density function $ and cumulative distribution function . The hazard rate (failure rate) function of the proposed WLED are defined by: The graph of the hazard rate function of the WLED for different values of , and is given in Figure 2.  Figure 2, it is clearly shown that the hazard rate function of the WLED at different parameters values is strictly increasing. This implies that the hazard rate function of WLED shows an increasing property.

Quantile function of the WLED
The quantile function of the WLED can be expressed in a closed form and it can be obtained by equating the cdf in (2) to : and solving for as shown below: Hence the quantile function of WLED is given by: The median of the WLE distribution can be obtained by setting = = 0.5 in equation (5) which gives

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From the quantiles, it is observed that all the values falls within the support (entire real line) which shows the validity of the proposed distribution.

Moments of the Weilbull logistic exponential distribution
Let X ~ WLED, then the rth moment about the origin of X is given by Proof. Let X be a random variable following the Weilbull logistic exponential distribution with parameters , and , then where $ is as defined in (3) above. Hence, Simplifying further yields

Numerical computation on the use of WLED
Here a study is carried out to obtain the mean, standard deviation, median, skewness and kurtosis of the Weilbull logistic exponential distribution for varying parameters values.  Table 3. Skewness and Kurtosis of the WLED. From Table 2 we observed that when the shape parameter r and the rate parameter λ are held constant, the mean, standard deviation, and median increases as the scale parameter k increase. This assertion clearly supports and maintains the ideal behinds the graphical illustration of the density function of the WLE distribution in Figure 1.

Moment generating function (MGF) of the WLED
Let V be a continuous random variable with density function $ , then the mgf of V following the WLED is defined by; Hence,

Shannon entropy of the WLED
[12] defined the entropy of a random variable, V as a measure of variation of uncertainty denoted by g 5 = TU− log $ W. The Shannon entropy of the WLED is given as: where h ≅ 0.5772 is the Euler's constant.
Proof. See Appendix I.

Rényi entropy of WLED
The entropy of a random variable, X, denoted by m n o is defined as a measure of the uncertainty about the outcome of a random experiment. Let X be a random variable with Then the Rényi entropy of WLED is defined by: where $ is the pdf of the WLED defined in (3) above.
Simplified further yields Proof. See Appendix II.
[5] provided some essential properties of Rényi entropy: (i) The Rényi entropy can be negative; (ii) For any o -< o • , " p ‚ ≤ " p and equality holds if and only if X is a uniform random variable.
Numerical computations of the Rényi entropy of the WLED for varying values of parameter are shown in Table 4. , the Rényi entropy " p " say ," p and " p ‚ /, satisfies the condition o -< o • " p ‚ ≤ " p as suggested by [5].

Maximum Likelihood Estimates
Let 1 2 3 , , , ..., n x x x x be random samples from the WLED with density function defined in equation (3), then the likelihood function is given by Taking the derivatives with respect to the various parameters, the following are obtained: 1 1 log 1 log 1 log 1 log log The maximum likelihood estimator φ of ϕ can be derived by using Newton Raphson's iterative method given by the relation: Proof. See Appendix III.

Application and Discussion of Results
In this section, some generalized probability distribution such as Exponentiated-Weilbull Distribution, Weibull-Exponential Distribution, and Logistic-Exponential Distribution are applied in fitting real life data set and the result is being compared with the proposed Weilbull-Logistic Exponential Distribution. The comparison criterion used in the study were, Alkaike Information Criteria (AIC), and Kolmogorov-Smirnov test (K-S) and the Log-likelihood. The data set used represents the breaking stress of carbon 208 fibers of 50 mm length in (GPa). The data was obtained from [6]. The data set is unimodal with (skewness = -0.128 and kurtosis = 0.1261208). The data is shown in Table. Source: [6]. The Weibull Logistic-exponential Distribution: Its Properties and Applications Earthline J. Table 6 reveals the summary statistics for the 50mm in length data set. The parameter estimates, Log Criterion (AIC), Kolmogorov the distributions were estimated for the data set Weibull-logistics exponential distribution gives the best fit for the dataset and thus exhibits superiority over the examined lifetime distributions considered in mode lifetime data set. This decision was further supported by examining the probability probability plots and the density plot distributions for the real lifetime data sets.  Table 6 reveals the summary statistics for the breaking stress of carbon fibers of data set. The parameter estimates, Log-likelihood, Akaike Information Criterion (AIC), Kolmogorov-Sminov Statistic (K-S) with their corresponding p the distributions were estimated for the data set. The Table indicates that the proposed logistics exponential distribution gives the best fit for the dataset and thus exhibits superiority over the examined lifetime distributions considered in mode lifetime data set. This decision was further supported by examining the probability probability plots and the density plot distributions for the real lifetime data sets.   Table indicates that the proposed logistics exponential distribution gives the best fit for the dataset and thus exhibits superiority over the examined lifetime distributions considered in modeling this lifetime data set. This decision was further supported by examining the probabilityprobability plots and the density plot distributions for the real lifetime data sets.
probability plot for the data set.

Concluding Remark
In this paper, a new distribution known as Weibull logistic-exponential distribution is proposed. Several Mathematical properties of this new distribution are study which include the cumulative distribution function, survival function, the density function, hazard rate function, moment generating function, quantile function, Shannon and Rényi entropy are obtained, the maximum likelihood estimation method was used in estimating the parameters of the proposed distribution. The application of the proposed distribution to a real lifetime data set reveals its superiority over some well-known generalized distribution. Let log(1 ) .