Type II Half Logistic Kumaraswamy Distribution with Applications

Deanship of Scientific Research, King Abdulaziz University, Saudi Arabia Faculty of Graduate Studies for Statistical Research, Cairo University, Egypt College of Statistical and Actuarial Sciences, University of the Punjab, Lahore, Pakistan Quality Enhancement Cell, National College of Arts, Lahore, Pakistan Lahore College for Women University (LCWU), Lahore, Pakistan King Abdulaziz University, Saudi Arabia Valley High Institute for Management Finance and Information Systems, Obour, Qaliubia 11828, Egypt


Introduction
Over the last few years, inspired by the increasing demand of probability distributions in many fields, many generalized distributions have been studied. Most of them are proposed by the addition of more parameters to well-known probability distributions that exist in literature to make them flexible. For instance, Haq et al. [1] proposed and studied the generalized odd Burr III (GOBIII) family of distributions along with its important characterizations. Ahmed [2] proposed a new model derived by the transformation of the baseline model. Different shapes of failure function can be formed such as increasing and bathtub. Some mathematical properties are obtained. So many research works have been done for proposing more flexible generalized probability distributions such as Alzaatreh et al. [3], Haq et al. [4], Hashmi et al. [5], Elgarhy et al. [6], and ZeinEldin et al. [7]. Alshenawy [8] suggested a new one-parameter distribution. Several statistical properties and characteristics of the proposed distribution are derived along with estimation under Type II censoring. The research is concluded on the basis of a simulation study and real data analysis.
The Kumaraswamy (Kw) distribution was introduced and studied by Kumaraswamy [9] with unit interval, denoted by Kw(a, b), with cumulative distribution function (cdf) is and its related probability density function (pdf) is The Kw density has one of these shapes depending upon parameter values, unimodal (a, b > 1), bathtub (a, b < 1), increasing (a > 1 and b ≤ 1), decreasing (D) (a ≤ 1 and b > 1), or steady (both a and b equal to 1).
The behavior of Kw distribution is analogous to the beta distribution but simpler due to closed-form of both its pdf and cdf. Boundary behavior and the major special models are also the same in both Beta and Kw distributions. This distribution could be a good substitute in situations wherein reality the bounds are finite i.e., (0, 1).
The Kw is originally developed as a lifetime distribution. Many authors studied and developed the generalizations of Kw distribution such as exponentiated Kw distribution studied by Lemonte et al. [10], El-Sherpieny, and Ahmed [11] proposed Kw Kw distribution, transmuted Kw distribution studied by Khan et al. [12], Sharma and Chakrabarty [13] studied size biased Kw distribution, George and Thobias [14] introduced Marshall-Olkin Kw distribution, exponentiated generalized Kw distribution studied by Elgarhy et al. [6], type II Topp Leone inverted Kw by ZeinEldin et al. [15], type I half logistic inverted Kw by ZeinEldin et al. [16], truncated inverted Kw by Bantan et al. [17], and Ghosh [18] introduced bivariate and multivariate weighted Kw distributions.
Hassan et al. [19] proposed Type II half logistic-G (TIIHL-G). The TIIHL-G distribution is expressed by its cdf given by where Gðx ; ζÞ is cdf of baseline model with parameter vector ζ and Fðx ; λ, ζÞ is cdf derived by the T-X generator proposed by Alzaatreh [3]. The pdf of the TIIHL-G family is given as This article is dedicated to both its mathematical and application features. A significant portion is kept for the estimation of the parameters through various methods including maximum likelihood estimation (MLE), least-square estimation (LSE), weighted least square estimation (WLSE), percentiles estimation (PCE), and Cramer-von Mises estimation (CVE).
The core purpose of this research is to suggest a simpler and more flexible model called Type II Half Logistic Kw (TIIHLKw) distribution. This article is organized in the following manner: Section 2 is dedicated for the proposition of type II half logistic-Kw (TIIHLKw) distribution. Section 3 deals with the leading statistical properties of this model. Important binomial expansions of density and distribution functions are presented which involve binomial expansions. In Section 4, an extensive study of five different methods of estimation is carried out, with all derivations and detailed discussions. A comprehensive simulation study is conducted to compute the biases and efficiency for parameters and compare the performances of five estimation approaches stated above in the next section. Section 6 is devoted to the real data application of TIIHLKwD to show the importance of TIIHLKw distribution. Lastly, the conclusion is given in Section 7.

The TIIHLKw Distribution
In this section, we examine the usefulness, and flexibility of a new associate of type II half-logistic-G family having (0, 1), using Kw distribution as the baseline. The pdf and cdf of Kw distribution (2 shape parameters: a, b > 0) are given as follows The random variable (r.v.) X follows TIIHLKw distribution if its cdf is obtained by inserting (6) The pdf of TIIHLKw distribution is as follows The survival function of TIIHLKw distribution is The failure rate function of TIIHLKwD is The flexibility of TIIHLKw distribution can be illustrated in Figures 1, 2, and 3. The pdf plots for the TIIHLKw distribution are given in Figures 1 and 2, and plots of hrf are given in Figure 3.

Some Mathematical Properties
The properties of TIIHLKwD are derived here. After this, we consider a r.v. X follows the pdf (8) and cdf (7).
3.1. Quantile Function. The quantile function of X is denoted by QðuÞ, defined as the inverse function of Gð:Þ is G −1 ð:Þ, given in (6).

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Also, Simulated values from TIIHLKwD can be utilized in the simulation study. The r. v. u follows rectangular distribution on the interval 0 to 1, i.e., x u = QðuÞ follows TIIHLKw distribution. In particular, the first three quartiles are obtained by putting u = 1/4, 1/2, and 3/4, respectively, in (11) and (12).
On differentiation, we can have the density of quantile function 3.2. Alternate Representation. Here, we present the expansion of the pdf and cdf for TIIHLKw distribution for further mathematical manipulation. The binomial theorem, for β > 0 and |z | <1, can be expressed as Then, by applying (13) in (8), the pdf of TIIHLKwD becomes for a, λ, b > 0, 0 < x < 1. Another form of pdf (8) can be obtained by means of the following expansion for |z | <1 and β > 0. By applying (16) in pdf (8), we get where η i = 2abλð−1Þ i ði + 1Þ. Now considering (13), we have  Journal of Function Spaces Inserting this expansion in (17), we have pdf of the following form where Another formula can be formed from pdf (19), which is given in (20) using an infinite linear combination Proposition 1. Let "h" be a positive integer. The expansion of cdf can be written in following form: where Proof. The cdf ½FðxÞ h is obtained, for ' h ' an integer by using (16).
Once more binomial expansion is applied to ½FðxÞ h and it can be written as The required result is obtained by combing some expression together, completing the proof.

Probability Weighted Moments
:. The probabilityweighted moments (PWMs) are used to study some more characteristics of the probability distribution. Under the specified setting discussed above, PWMs are denoted by τ r,s , can be defined as The PWMs of TIIHLKw are obtained by substituting (21) and (22) into (25), as follows Then, where ΓðsÞ = Ð +∞ 0 t s e −t dt, s > 0: Also, βða, bÞ is beta function. 3.4. Moments. The moments have a vital role in the study of the distribution and real data applications. Now, we get the r th moment for the TIIHLKwD. Under the specified assumptions, the r th moment is obtained as where The mean (μ 1 ′ ) and variance (var) of TIIHLKw distribution can be derived as Also, the coefficients of skewness and kurtosis of TIIHLKw distribution are given by

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The summary measures, mean, variance, skewness (S k ) and kurtosis (K) values are presented in Table 1. The plots of mean and var for the TIIHLKwD are given in Figure 4 and graphs of skewness and kurtosis are presented in Figure 5 for different parameter ranges.
We see a monotonic variation in these measures caused by variation in parameters a, b, and λ.
3.5. Moment-Generating Function. The moment-generating function of X, using moments about the origin (29), is obtained as 3.6. Incomplete Moments. The r th incomplete moment of TIIHLKwD can be obtained by using (19) is where β t ð:, :Þ is incomplete beta function.
3.7. The Mean Deviation. Following are expressions used to get mean deviation about mean and median, respectively where M d = Median of X and TðqÞ = Ð q −∞ xf ðxÞdx is the initial incomplete moment. Now and, Journal of Function Spaces 3.9. Order Statistics. In statistical theory, order statistics is widely applied and practiced. Let X 1 , X 2 , ⋯, X n be r.vs. with their corresponding cdfs FðxÞ. Let X 1:n , X 2:n , ⋯:, X n:n be the related ordered r. sample of size n, then the density of r th order statistic is given as where K = 1/βðr, n − r + 1Þ and β(.,.) is the beta function. The pdf of the r th order statistic of TIIHLKwD is obtained by putting (21) and (22) in (38), changing h with v + k − 1, where More, the s th moment of r th order statistics for TIIHLKwD is given by By substituting (39) in (41), we have Then, 3.10. Rényi Entropy. Renyi (1961) proposed and used this measure. It can be obtained by So, the Rényi entropy of TIIHLKw distribution is as follows where 3.11. Stress-Strength Reliability. This subsection deals with the stress-strength parameter of TIIHLKw distribution. Let X 1 be the strength of a structure with a stress X 2 , and if X 1 follows TIIHLKwðλ 1 , a 1 , b 1 Þ and X 2 follows TIIHLKwðλ 2 , a 2 , b 2 Þ, provided X 1 and X 2 are statistically independent r.vs., Proof. The reliability is defined by Then, we can write where which completes the proof.

Inference
This section is dedicated to estimation aspects of TIIHLKw distribution, assuming that population parameters (a, b, λ) are unknown and can be estimated using different methods of estimation including ML, LS, WLS, PC, and CV.

Maximum Likelihood Estimation.
For a random sample of x 1 , x 2 , x 3 , ⋯, x n from the TIIHLKwðλ, a, bÞ distribution, the log-likelihood function for Φ = ðλ, a, bÞ is The members of UðΦÞ = ðU λ , U a , U b Þ are given below 9 Journal of Function Spaces equating (52), (53), and (54) to zero and solve them simultaneously give the ML estimates (MLEs) b Φ = ðλ∧, a∧, b∧Þ T of Φ = ðλ, a, bÞ T : The iterative algorithm is used to obtain the numerical solution of these nonlinear equations such as the Newton-Raphson method.

Ordinary and Weighted LS Estimation (LSEs and
WLSEs):. The LSEs of λ, a, and b can be obtained by minimizing the sum of squares of errors with respect to parameters. Suppose X 1 , X 2 , ⋯, X n is a random sample of size n from TIIHLKwD and suppose X ð1Þ , X ð2Þ , ⋯, X ðnÞ be the related ordered sample.

PC Estimators (PCEs).
Under the specification defined above including order statistics having relationship X ð1Þ < X ð2Þ < ⋯ < X ðnÞ . In the PC method of estimation, the estimators of λ, a, and b are derived by minimizing the following expression

The Cramer-von Mises Minimum Distance Estimation.
The CVE is another estimation method based on minimum distance. The CV estimators are obtained by minimizing, with respect to λ, a, and b.
CV minimum distance estimators provide empirical evidence that the bias of this estimator is smaller than the other minimum distance estimators.

Simulation Study
In this section, we present a Monte Carlo (MC) simulation study in order to illustrate the behavior of different estimates. Estimates of parameters and their corresponding MSEs are calculated using the same sample under different methods of estimation for four sets of parameters mentioned above for small sample size (n = 50) to sufficient large sample size, that is, n = 500.
The entries of Table 2 to Table 5 show that estimates are reliable and consistent. MSEs reduce as sample size (n) increases under each method of estimation. The summary of these four tables is presented in Table 5.
Using the entries of Table 6 for different parametric combinations, we can conclude that the MLE method outperforms than all other estimation methods (with an overall score of 18.5). Therefore, depending on the simulation study, the MLE method performs best for TIIHLKwD.

Applications
The TIIHLKw distribution aims at providing an alternative distribution to fit data on the unit interval to other distributions available in the literature. Here, we used the following probability distributions as competitor models; (ii) Transmuted Kumaraswamy (TKw) distribution   (iv) Beta distribution We use the following accuracy measures for model comparison: Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), log-likelihood, Anderson-Darling (A * ), and Cramer-von Mises (W * ). R-Language is used for numerical computations.
1 st data set. The data set was taken from Dasgupta [20], and considered n = 50 on burr (in millimeters), with hole diameter and a sheet thickness of 12 mm and 3.15 mm, respectively.
2 nd data set. This data set was taken from [21], and considered n = 48 measurements on petroleum rock samples from a petroleum reservoir.
The total test time (TTT) for both datasets are presented in Figure 6. We can observe that the shape of TTT plots is concave for both datasets, which demonstrates an increasing failure rate.
The MLEs for TIIHLKw distribution along with some adequacy measures are presented in Tables 7 and 8.
Hence, it is concluded that the new model provides the better fit. Figures 7 and 8 show the estimated densities, cdfs, estimated survival functions, and PP plots for the considered distributions of both data sets, respectively. We note that the proposed model is more appropriated to fit the data than the other competing models.

Conclusion
In this article, a new Type-II Half Logistic Kumaraswamy distribution is proposed. Some characteristics of the TIIHLKw distribution including linear combination expressions for the 13 Journal of Function Spaces density function, probability weighted moments, moments, incomplete moments, quantile function, mean deviation about mean and about median, Bonferroni and Lorenz curves, order statistics, stress-strength reliability, and Rényi entropy are derived. The ML method is used to estimate model parameters. An extensive simulation study is conducted to compare several well-known estimation methods, including the method of maximum likelihood estimation, methods of least squares and weighted least squares estimation, and method of Cramer-von Mises minimum distance estimation. The simulation study showed the reliability and efficiency of the estimates. Finally, by considering the method of maximum likelihood estimation, the new model is fitted to two practical data sets. The applications on real data sets validated the significance of the new distribution.