The Topp-Leone Weibull Distribution: Its Properties and Application

This paper presents a new generalization of the Topp-Leone distribution called the Topp-Leone Weibull Distribution (TLWD). Some of the mathematical properties of the proposed distribution are derived, and the maximum likelihood estimation method is adopted in estimating the parameters of the proposed distribution. An application of the proposed distribution alongside with some well-known distributions belonging to the Topp-Leone generated family of distributions, to a real lifetime data set reveals that the proposed distribution exhibits more flexibility in modeling lifetime data based on some comparison criteria such as maximized log-likelihood, Akaike Information Criterion (cid:1)(cid:2)(cid:3)(cid:4) = 2(cid:7) − 2 log(cid:12)(cid:13)(cid:14)(cid:15), Kolmogorov-Smirnov test statistic (cid:12)(cid:17) − (cid:18)(cid:14) and Anderson Darling test statistic (cid:12)(cid:2) ∗ (cid:14) and Crammer-Von Mises test statistic (cid:12)(cid:20) ∗ (cid:14) .


Introduction
Lifetime distributions are statistical models used for analyzing real life problems based on survival time. Many statistical distributions have been proposed to model lifetime data and the Topp-Leone distribution introduced by [14] is one of such distributions. In practice, most lifetime data sets encountered exhibits a bathtub hazard rate property and the one parameter Topp-Leone distribution happens to be the simplest distribution with such hazard rate property, but being a single parameter defined on a unit interval, its flexibility is limited in handling lifetime data sets.
Several generalizations of the distribution have been introduced to address the aforementioned drawback. These generalizations are found in the works of [1][2][3][4][5]9,10]. In this paper, we introduce a new generalization of the Topp-Leone distribution which serves as an alternative distribution among the Topp-Leone generated family of 382 distributions. We shall call the proposed distribution, "Topp-Leone Weibull Distribution (TLWD)". [14] proposed a J-shaped univariate distribution with cumulative distribution function defined by = 2 − , 0 ≤ ≤ < ∞, 0 < " < 1 (1) and the corresponding density function given by The survival function and the hazard rate function of Topp-Leone distribution are obtained using equations (1) and (2)  (4) [11] improved on the mathematical properties of the Topp-Leone distribution by deriving higher moments for the distribution and a general usefulness of the distribution. Motivations for generalizing the Topp-Leone distribution arose after the work of [11]. One of such motivation is that, suppose a random variable follow the Topp-Leone distribution then the random variable can have either a finite support 0 < < 3 or an infinite support 0 < < < ∞ . 3 Suppose we fix the parameter = 1 in equations (1) and (2), then we obtain the cumulative distribution function and the density function of the one parameter Topp-Leone distribution defined on a unit interval respectively as = 2 − , 0 < < 1 , " > 0 and Figure 1 shows the graphical illustration of the density function and the hazard rate function of the Topp-Leone distribution for varying values of the shape parameter defined in the interval 0 < " < 1 and a fixed value of the scale parameter = 1.  The plots in Figure 1(a), clearly shows that the de reversed-J shape for different values of the shape parameter defined in the interval 0 < " < 1 and a fixed value of the scale parameter indicates that the hazard rate function of TLD ex 0 < " < 1 and a non-decreasing are organized as follows: Section 2 presents some mathematical properties of the proposed distribution which include; the probability density f distribution function, Hazard rate function, Survival function, Quantile function, Moments, Moment generating function, Renyi entropy, and distribution of ordered statistics. The model parameter estimation and simulation study on the ma likelihood estimates of the proposed distribution Section 4, we applied the proposed distribution to a real data set and compared its fit with the fit of some existing Topp [13] introduced a new class of the Topp the cumulative distribution fun and the corresponding density function given by $ Density and hazard rate functions of the Topp-Leone distribution

The density and cumulative distribution functions of the proposed distribution
The plots in Figure 1(a), clearly shows that the density function of the TLD exhibits a J shape for different values of the shape parameter defined in the interval and a fixed value of the scale parameter = 1, while the plots in Figure 1(b) indicates that the hazard rate function of TLD exhibits a bathtub shape whenever decreasing shape for " 5 1. The remaining sections of this paper are organized as follows: Section 2 presents some mathematical properties of the proposed distribution which include; the probability density function, cumulative distribution function, Hazard rate function, Survival function, Quantile function, Moments, Moment generating function, Renyi entropy, and distribution of ordered statistics. The model parameter estimation and simulation study on the ma of the proposed distribution are given in Section 3. Finally, in Section 4, we applied the proposed distribution to a real data set and compared its fit with the fit of some existing Topp-Leone generated family of distributions. nsity function of the TLD exhibits a J shape for different values of the shape parameter defined in the interval while the plots in Figure 1(b) hibits a bathtub shape whenever The remaining sections of this paper are organized as follows: Section 2 presents some mathematical properties of the unction, cumulative distribution function, Hazard rate function, Survival function, Quantile function, Moments, Moment generating function, Renyi entropy, and distribution of ordered statistics. The model parameter estimation and simulation study on the maximum given in Section 3. Finally, in Section 4, we applied the proposed distribution to a real data set and compared its fit with density and cumulative distribution functions of the proposed distribution Leone generated family of distributions with

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Taking 6 as the distribution function of the Exponentiated exponential distribution developed by [8], the authors obtained the cumulative distribution function of the Topp-Leone Generalized Exponential Distribution as = :1 − ; &< = > ?2 − 81 − ; &< 9 > @ , (9) and the corresponding density function given by Using a similar approach of generating new distributions defined in equations (7) and (8), we assume the random variable D to follow the 2-parameter Weibull distribution with cumulative distribution function and probability density function respectively defined by Inserting equations (11) and (12) into (7) and (8) The series representation of equation (14) can be obtain as The graphical plots of the probability density function are shown in Figure 2.

Figure 2: Density
The plots in Figure 2 clearly indicate that the density function of the TLWD exhib a left-skewed, right-skewed, reversed J

Survival function and hazard function of the proposed distribution
Using equations (13) and (14), the survival function and the hazard rate function of the Topp Leone Weibull distribution are given by and The graphical plots of the haza the parameter value is shown in Figure 3. The graphical plots of the probability density function are shown in Figure 2.

Density function of the TLWD for fixed value of the parameters
The plots in Figure 2 clearly indicate that the density function of the TLWD exhib skewed, reversed J-shape and symmetric unimodal shapes.

function and hazard function of the proposed distribution
Using equations (13) and (14), the survival function and the hazard rate function of bull distribution are given by The graphical plots of the hazard rate function of the TLWD for different choice of the parameter value is shown in Figure 3.

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The graphical plots of the probability density function are shown in Figure 2.
of the TLWD for fixed value of the parameters.
The plots in Figure 2 clearly indicate that the density function of the TLWD exhibits shape and symmetric unimodal shapes.

function and hazard function of the proposed distribution
Using equations (13) and (14), the survival function and the hazard rate function of (16) rd rate function of the TLWD for different choice of The quantile function of a random variable is obtained by solving the system Thus, given the cumulative distribution function quantile function of the TLWD can be obtain as The median of the TLWD can be obtained by substituting

Moments of the proposed distribution
Let D be a continuous random variable with density function $ , then the d WX moment about the origin of D is defined by inserting the density function of the TLWD into equation (20), the d WX moment about the origin of the random variable D is defined by evaluating the integral part of equation (22) The first four raw moments of the TLWD are obtained from equation (23) as Furthermore, using the first four raw moments defined above, the measures of skewness x and kurtosis y are obtained as Tables 1 and 2 show the theoretical moments of the TLWD for different values of the parameters.  From Tables 1 and 2, we observed that the TLWD can be right skewed x > 0 , left skewed x < 0 and approximately symmetric x ≈ 0 . Also, at some fixed values of the parameters, the distribution can be leptokurtic y > 3 , platykurtic y < 3 as well as mesokurtic y ≈ 3 . This claim was clearly shown in Figure 2.

Moment generating function of the proposed distribution
Let D be a continuous random variable with density function $ , then the moment generating function of D is defined by inserting the density function of the TLWD into equation (24), we obtain the moment generating function of the TLWD as upon substituting equation (27) into equation (26), the moment generating function of the 390 TLWD is given by 2.6 Renyi entropy of the proposed distribution [12] defined an entropy of a random variable X as a measure of variation of uncertainty associated with the random variable X. The Renyi entropy of X with density function $ , is defined by, inserting the density function of the TLWD into equation (29), we obtain the Renyi entropy of the random variable X following the TLWD as

The distribution of the ordered statistics of the proposed distribution
Suppose that ' ':" < ' %:" < ⋯ < ' ":" is the order statistics of a random sample generated from TLWD, then the probability density function of the WX order statistics, say D = ' ":" is given by substituting the cumulative distribution function and the density function of TLWD defined in equations (13) and (14), into equation (35), we have

Maximum likelihood estimation
Let ' , % , … , " be random samples from the TLWD, then the log-likelihood function of the TLWD is defined by On differentiating the log-likelihood function with respect to the parameters, we obtain the score function as, where ¥ oe x is the score function and ) oe x is the Hessian matrix, which is the second derivative of the log-likelihood function. The Hessian matrix is defined by

Interval estimate
The asymptotic confidence intervals (CIs) for the parameters of TLWD ", F, B are 394 obtained according to the asymptotic distribution of the maximum likelihood estimates of the parameters.
Suppose oe £ = " ¬, F £ , Bis MLE of oe, then the estimators are approximately bi-variate normal with mean ", F, B and the Fisher information matrix is given by: The approximate 1 − 100 CIs for the parameters ", F and B respectively, are " ¬ ± ¯°% ±²\d " ¬ , F £ ± ¯°% ³²\d F £ and B -± ¯°% ³²\d B - where ²\d " ¬ , ²\d F £ and ²\d Bare the variance of ", F and B which are given by the diagonal elements of the variance-covariance matrix &' oe x and ¯°% is the upper ´2 percentile of the standard normal distribution.

Simulation Study
In this section, we investigate the asymptotic behaviour of the maximum likelihood estimates of the parameters of the Topp-Leone Weibull distribution (TLWD) through a simulation study. where I(·) is an indicator function and ³²\d8oe £ 9 is the standard error of the estimate oe -.
The coverage probability computes the proportion of times the confidence interval contains the true value of the parameter oe P .     Tables 3, 4 and 5 present the Monte Carlo simulation results for the average bias, mean square error and coverage probability of the 95% confidence interval of the parameter estimates of the TLWD at different choice of parameter values. Clearly from Table 3, we observe that the parameter α is positively biased, parameter λ is positively biased, while parameter θ could either be negatively or positively biased. Also from Table 4, as the sample size n increases, the values of the mean square error of the parameter estimates decreases (tends to zero). Finally, from Table 5, we observe that the coverage probabilities of the CIs are quite close to the nominal level of 95%.

Application of the Proposed Distribution
In this section, we fit the proposed distribution to a real data set alongside with some well-known lifetime distributions from the Topp-Leone Generated family of distributions with density functions given by; .
The dataset consists of the tensile strength, measured in GPa, of 69 carbon fibers tested under tension at gauge lengths of 20mm reported in [7]. Table 6 presents the dataset.

Conclusion
In this paper, we proposed a new member of the Topp distributions called the Topp properties of the proposed distribution such as; the density function, cumulative distribution function, survival function, hazard rate function, moments and related measure, moment generating function, quantile function, Renyi entropy and the distribution of ordered statistics were derived. The method of maximum likelihood estimation was used in es Finally, an application of the proposed distribution to a real lifetime data set alongside with the Topp distribution,Topp-Leone Exponential distri distribution, and Topp-Leone Nadarajah Inverse Weibull Distribution (TLIWD), Topp Leone Bur XII Distribution (TLBXIID), Topp Leone Exponential Distribution (TLED), Topp Leone Linear Exponential Distribution (TLLED) and Topp Leone Nadarajah-Haghighi Distribution (TLNHD) modeling the lifetime dataset under study. This claim was further supported by graphical illustration of the density and cumulative distribution fit of the distributions for the real as displayed in Figure 4.
Density and cumulative distribution fit of the distributions for the In this paper, we proposed a new member of the Topp-Leone generated family of called the Topp-Leone Weibull Distribution (TLWD). The mathematical properties of the proposed distribution such as; the density function, cumulative function, survival function, hazard rate function, moments and related measure, moment generating function, quantile function, Renyi entropy and the distribution of ordered statistics were derived. The method of maximum likelihood estimation was used in estimating the parameters of the proposed distribution.
Finally, an application of the proposed distribution to a real lifetime data set alongside with the Topp-Leone Bur XII distribution, Topp-Leone Inverse Weibull Leone Exponential distribution, Topp-Leone Linear Exponential Leone Nadarajah-Haghighi distribution, reveals that the 399 . Table 7 clearly ximized loglikelihood and Statistic, Anderson Darling test statistic * and demonstrates superiority over the Topp Leone Bur XII Distribution (TLBXIID), Topp Leone Exponential Distribution (TLED), Topp Leone Linear Exponential Haghighi Distribution (TLNHD) in orted by graphical illustration of the density and cumulative distribution fit of the distributions for the real for the dataset.
Leone generated family of Leone Weibull Distribution (TLWD). The mathematical properties of the proposed distribution such as; the density function, cumulative function, survival function, hazard rate function, moments and related measure, moment generating function, quantile function, Renyi entropy and the distribution of ordered statistics were derived. The method of maximum likelihood timating the parameters of the proposed distribution.
Finally, an application of the proposed distribution to a real lifetime data set Leone Inverse Weibull Leone Linear Exponential Haghighi distribution, reveals that the proposed  Topp-Leone Weibull distribution, demonstrates superiority and offers more flexibility in modeling the lifetime data set under study.