Half-Logistic Xgamma Distribution: Properties and Estimation under Censored Samples

+is paper proposed a new probability distribution, namely, the half-logistic xgamma (HLXG) distribution. Various statistical properties, such as, moments, incomplete moments, mean residual life, and stochastic ordering of the proposed distribution, are discussed. Parameter estimation of the half-logistic xgamma distribution is approached by the maximum likelihoodmethod based on complete and censored samples. Asymptotic confidence intervals of model parameters are provided. A simulation study is conducted to illustrate the theoretical results. Moreover, the model parameters of the HLXG distribution are estimated by using the maximum likelihood, least square, maximum product spacing, percentile, and Cramer–von Mises (CVM) methods. Superiority of the new model over some existing distributions is illustrated through three real data sets.

Hassan et al. [20] proposed a class of probability distribution generated by a half-logistic random variable with the following cdf: where λ is the shape parameter and G(x; ζ) is a baseline cdf, which relies on a parameter vector ζ. e pdf corresponding to (2) is given by Recently, Sen et al. [21] proposed the xgamma (XG) distribution, following the idea of Lindley distribution. e XG model is a special finite mixture from exponential with a scale parameter (θ) and gamma with a scale parameter (θ) and shape parameter (3). e pdf and cdf of the XG distribution are, respectively, given by and Sen et al. [21] investigated the structural properties of the XG distribution, and they have found that in many cases that the XG distribution has more flexibility than the exponential distribution. e extensions of XG distribution were studied in [22,23] and [24], among others.
Our objective here is two folds: First, based on the TIIHL-G class, we propose a new distribution related to xgamma distribution. We call the new model as the half-logistic xgamma (HLXG) distribution, and we study its several statistical properties. Second, the maximum likelihood (ML) method is employed to estimate the model parameters of the HLXG based on complete and censored samples. Further, (1 − v)% asymptotic confidence intervals (CIs) of the model parameters are constructed. Simulation and application issues are considered. e rest of the paper is organized as follows. Sections 2 and 3 provide the pdf, cdf, hazard rate function (hrf), and structure properties of the HLXG model. In Section 4, point and approximate CIs of model parameters are derived under complete, type I censoring (TIC) and type II censoring (TIIC). Also, in the same section, the behavior of the estimates is studied via a simulation study. Real data sets are analyzed to demonstrate the flexibility of the HLXG distribution over some known distributions in Section 5. e article ends with some concluding remarks.

The Half-Logistic Xgamma Distribution
In this section, we provide a more flexible model by adding one extra shape parameter to the XG model for improving its goodness-of-fit to real data. e motivations of the HLXG distribution are (i) to obtain a more flexible pdf with right skewed, unimodal, and reversed J-shape; (ii) to be capable of modeling decreasing, increasing, and semibathtub hazard rate shapes; and (iii) to provide more flexibility to model the various types of data.

Definition 1.
A random variable X is said to have a HLXG distribution, if its cdf is given by where A(θ, x) � (1 + θ + θx + (θ 2 x 2 )/2/(1 + θ))e − θx . e corresponding pdf is given as follows: A random variable X with pdf (7) will be denoted by HLXG∼(λ, θ). For λ � 1 in (7), we obtain the MO extended xgamma distribution with δ � 0.5 as a special new model. Further, the sf and hrf of the HLXG are given, respectively, by e pdf and hrf plots for the HLXG are displayed in Figure 1 for some given values of parameters. Figure 1 reveals that the pdf of X is quite flexible and can take asymmetric forms, among others. Also, the hrf can be increasing, decreasing, and semibathtub shapes. In general, they reinforce the importance of the HLXG model to fit real lifetime data.

Main Properties
In this section, we give some important properties of the HLXG distribution such as ordinary and incomplete moments, stochastic ordering, mean residual life, and mean waiting time, among others.

Expansion.
is section provides a useful expansion of the HLXG pdf due its complicated form to obtain its structure properties. Since the generalized binomial series is then, by applying (8) in (7), we have Employing the binomial expansion in the last term of (9), Again, we use the binomial expansion more than one time; then, we have Discrete Dynamics in Nature and Society

Moments and Related Statistics.
In this section, we obtain moments and some related measures of the HLXG distribution with pdf defined in (10). e s th moment about the origin of the random variable X has the HLXG distribution obtained as follows: After simplification, the s th moment of the HLXG distribution is given by where Γ(·) is the gamma function. Furthermore, the s th central moment of a given random variable X is defined by Additionally, the s th lower incomplete moment, say π s (t), of the HLXG distribution is given by which leads to where c(., t) is the lower incomplete gamma function. e first incomplete moment, say π 1 , for s � 1 in (15 is obtained. e Lorenz and Bonferroni curves are the essential applications of the first incomplete moment. e Lorenz curve, say LO (t), and the Bonferroni curve, say BO (t), of the HLXG are obtained, respectively, as follows:

Mean Residual Life and Mean Waiting Time.
e mean residual life (MRL) function at age t measures the expected remaining lifetime of an individual of age t. e MRL of X is given by Hence, the MRL of X can be obtained as follows: e mean waiting time (MWT) of an item failed in an interval [0, t] is defined as Hence, the MWT of the HLXG distribution is determined as follows: 3.4. Stochastic Ordering. Shaked and Shanthikumar [25] stated that for independent random variables X and Y with cdfs, F X and F Y , respectively, X is said to be smaller than Y in the following ordering, if the following is available: We have the following chain of implications among the various partial orderings discussed above: Theorem 1. Let X∼HLXG (λ 1 , θ 1 ) and Y∼HLXG (λ 2 , θ 2 ). If λ 1 ≥ λ 2 and θ 1 ≥ θ 2 , then X ≤ lr Y, X ≤ hr Y, X ≤ mrl Y, and X ≤ sr Y.
Proof. It is sufficient to show that f X (x)/f Y (x) is a decreasing function of x; the likelihood ratio is therefore, is decreasing in x, and hence X ≤ lr Y. Similarly, we can conclude for X ≤ hr Y, X ≤ mrl Y, and X ≤ sr Y. e HLXG quantile function, say Q(u) � F − 1 (u), is straightforward to be computed by inverting (6) as follows: where x q � Q(u). e uniroot function of the R software can be used to solve nonlinear equation (25), numerically, and then the HLXG random variable X can be generated, where u has the uniform distribution on the interval (0, 1).

Parameter Estimation under Censored Samples
In survival analysis and the industrial life testing model, it is necessary to minimize the cost and/or the duration of a lifetesting experiment, so one may choose to terminate the test early, which results in the so-called censored sampling scheme. TIC and TIIC are the most popular types of censoring. e life testing experiment is ended at a predetermined time, say T, for the TIC, while for the TIIC, the test is ended when a predetermined number of failures, say r, is reached. In the following sections, the maximum likelihood (ML) estimators of the model parameters of the HLXG model based on complete, TIC, and TIIC are provided. Approximate CIs are constructed. Further, estimators of survival function and hazard rate function for different mission times are given.

Maximum Likelihood Estimators via TIIC.
Let X (1) < X (2) <· · ·< X (n) be a TIIC of size r from a life test on n items whose lifetimes have the HLXG model with parameters λ and θ. e log-likelihood function, denoted by ln ℓ 1 , of r failures and (n − r) censored values is given by Discrete Dynamics in Nature and Society (i) , and A(θ, x (r) ) � (1 + θ + θx (r) + θ 2 x 2 (r) /2/1 + θ)e − θx (r) . e first partial derivatives of the log-likelihood function with respect to λ and θ are obtained as follows: where zA θ, Putting z ln ℓ 1 /zλ and z ln ℓ 1 /zθ equal to zero and solving these equations numerically provide the ML estimator of λ and θ, respectively. Note that for r � n, we obtain the ML of the model parameters in case of a complete sample.

Maximum Likelihood Estimators via TIC.
Suppose that n items, whose lifetimes have the HLXG distribution, are placed on a life test, and the test is terminated at a specified time T before all n items have failed. e number of failures r and all failure times are random variables. e log-likelihood function, based on TIC, is given by where A(θ, T) e first partial derivatives of the log-likelihood function with respect to λ and θ are obtained as follows: where Putting z ln ℓ 2 /zλ and z ln ℓ 2 /zθ equal to zero and solving these equations numerically provide the ML estimators of λ and θ.

Approximate Confidence Intervals.
In this section, approximate confidence intervals (CIs) of the model parameters for the HLXG distribution are obtained. e asymptotic variances and the covariance matrix of the ML estimates are given by the elements of the inverse of the Fisher information matrix I ij (θ) � E − z 2 ln ℓ 1 /zλ zθ . Unfortunately, the exact mathematical expressions for the above expectation are very difficult to obtain. erefore, the Fisher information matrix is given by I ij (θ) � − z 2 ln ℓ/zλ zθ , which is obtained by dropping the expectation on operation E and replacing λ and θ with λ and θ, respectively (see [26]). e asymptotic variance covariance matrix F for the maximum likelihood estimates can be written as follows: Hence, the approximate 100 (1 − v)% two-sided CIs for λ and θ are, respectively, given by where Z ]/2 is the upper ]/2 th percentile of the standard normal distribution. By a similar way, we can find the approximate 100 (1 − v)% two-sided CIs for λ and θ under TIIC and a complete sample.

Simulation Study.
is section provides a simulation study to assess the behavior of the estimators in case of complete, TIC, and TIIC. Mean square errors (MSEs), biases, lower bounds (LBs) of the CIs, upper bounds (UBs) of the CIs, and average lengths (ALs) of 90% and 95% are calculated via Mathematica 9.
e following algorithm is designed as follows:  Tables 2-4 based  on complete and TIIC, while Tables 5-7 Table 8,  while Table 9 contains numerical results based on TIC.
From Tables 2-9, we observed the following: e Biases, MSEs, and average length of λ and θ decrease as n increases As the censoring level time values, T, increase, the biases, MSEs, and ALs of λ and θ decrease e biases, MSEs, and ALs of λ and θ decrease as the number of failures r increases e lengths of the CIs become narrower as n increases e ALs of the CIs increase as the confidence levels increase from 90% to 95%. Also, as the number of failures r increases, the AL of CIs decreases As the mission time increases, the estimates of sf decrease while the hrf estimates increase for all sampling schemes As the sample size increases, the sf and the hrf estimates are increasing e MSEs, biases, and ALs are increasing as θ increases from 0.5 to 1.2 at λ � 0.5

Data Analysis
In this section, we fit our proposed HLXG model to the following three real data sets. Furthermore, parameter estimation of the three real data sets by using the ML, least square (LS), maximum product spacing (MPS), percentile (PE), and Cramer-von Mises (CVM) methods is provided. e first data consists of 100 observations of breaking stress of carbon fibres (in Gba) given by Nichols and Padgett [27]. e data are given as follows: e third data set is censored data discussed by Sickle-Santanello et al. [29] and given by Klein and Moeschberger [30]. e data consist of death times (in weeks) of patients with cancer of the tongue with an aneuploid DNA profile. Data under study were based on the effects of ploidy on the prognosis of patients with cancers of the mouth. Patients were selected in whom a paraffin-embedded sample of the cancerous tissue had been taken, and the times to reinfection for patients with sexually transmitted diseases were calculated. Follow-up survival data were obtained on each patient. e tissue samples were examined using a flow cytometer to determine if the tumor had an aneuploid (abnormal) or diploid (normal) DNA profile using a technique discussed by Sickle-Santanello et al. [29]. e observations are 1, 3, 3, 4, 10, 13, 13, 16, 16, 24, 26, 27, 28, 30, 30,        The profile log-likelihood  The profile log-likelihood  We estimate the model parameters by using the ML method. We compare the goodness-of-fit of the models using − 2 ln L where L denotes the log-likelihood function evaluated at the ML estimates, Akaike information criterion (AIC), corrected Akaike information criterion (CAIC), Hannan-Quinn information criterion (HQIC), and Bayesian information criterion (BIC). We also reported the Kolmogorov-Smirnov (K-S) and Anderson-Darling (A * ) statistics and their corresponding P values. e pdfs of the IL, GIL, EIL, MOIL, and IW models are, respectively, given by x, α, β, λ > 0.  Tables 10 and 11, respectively, for the second data set in Tables 12 and 13, respectively, and for the third data set, in Tables 14 and 15, respectively. e loglikelihood for the first, second, and third data sets are given in Figures 2-4, respectively.
It is observed from Tables 10-15 that the HLXG distribution has the smallest values of − 2 ln L, AIC, CAIC, BIC, HQIC, K-S, and A * but the largest P value among all other competitive models. Hence, it is evidenced that our proposed model performed the best for all three data sets. e relative histogram and the fitted densities and the plots of the fitted empirical cdf and the empirical sf and p − p plots of the three data sets are displayed in Figures 5-7, respectively. ese figures supported the above conclusions to some extent.
Tables 16-18 provide the ML, LS, MPS, PC, and CV estimates of the unknown parameters of the HLXG distribution based on four methods of estimation. Also, values of − 2 ln L, AIC, BIC HQIC, and CAIC are listed. From these tables, we conclude that all estimation methods performed well.

Some Concluding Remarks
A new probability distribution related to xgamma, called the half-logistic xgamma distribution, is proposed in this paper. Some important statistical properties, including moments, incomplete moments, mean residual life, mean waiting time, quantile function, and stochastic ordering of the suggested model, are derived. Parameter estimation of model parameters is discussed via the maximum likelihood method from complete and censored samples. Further, survival and hazard rate function estimates are obtained for different sample sizes. Also, asymptotic confidence intervals of parameters are constructed. Good performance and accuracy of the maximum likelihood estimators of the parameters are examined through a simulation study. Based on maximum likelihood, least square, maximum product spacing, percentile, and Cramer-von Mises (CV) methods, we estimate the model parameters of the HLXG distribution using three real data sets. Furthermore, three real data sets are analyzed to demonstrate the superiority of the proposed half-logistic xgamma distribution compared to some existing distributions.

Data Availability
In order to obtain the numerical data set used to carry out the analysis reported in the manuscript, please contact Mahmoud Elsehetry at ma_sehetry@hotmail.com.

Conflicts of Interest
e authors declare no conflicts of interest.