On Type II Half Logistic Weibull Distribution with Applications

In recent years, several of new improved and extended probability distributions have been discovered from the current distributions to facilitate their applications in many fields. A new three-parameter distribution, the so called the T ype II half logistic Weibull (TIIHLW), is introduced for modeling lifetime data. Some mathematical properties of the TIIHLW distribution are provided. Explicit expressions for the moments, probability weighted moments, quantile function, order statistics and Renyi entropy are investigated. Maximum likelihood estimation technique is employed to estimate the model parameters and simulation issues are presented. In addition, the superiority of the subject distribution is illustrated with an application to two real data sets.  Indeed, the TIIHLW model yields a better fit to these data than the beta Weibull, Mcdonald Weibull and exponentiated Weibull distributions. Keywords : Type II half logistic-G class; Weibull distribution, Order statistics; Maximum likelihood method. DOI : 10.7176/MTM/9-1-05


Introduction
The Weibull (W) distribution is a very popular distribution for modeling lifetime data in reliability where the hazard rate function is monotone. However, in many applied areas, the two-parameter W distribution is inadequate for modeling phenomenon with non-monotone hazard rate. Various generalizations and extensions of the W distribution have been proposed in the statistical literature to handle with bathtub shaped failure rates. Mudholkar and Srivastava (1993) and Mudholkar et al. (1996) pioneered exponentiated W (EW) distribution to analyze bathtub failure data. Xie et al. (2002) proposed a three-parameter modified W extension with a bathtub shaped hazard function. Carrasco et al. (2008) suggested the generalized modified W distribution, among others.
Recently, new generated families of continuous distributions have been attracted several statisticians to develop new models. These families are obtained by introducing one or more additional shape parameter(s) to the baseline distribution. Some of the generated families are: the beta-G (Eugene et al. (2002)), gamma-G (Zografos and Balakrishanan (2009) where  is the shape parameter. The probability density function (pdf) corresponding to (1) is given by  (2) Our motivation here is to extend the two-parameter Weibull distribution to produce a more flexible model. The new model is referred to as the Type II half logistic Weibull distribution. Based on the TIIHL-G family, we construct the TIIHLW distribution as well as we provide the main statistical distributions. The remainder of the paper is organized as follows: In Section 2, we define the TIIHLW distribution and provide its special models. In Section 3, we derive a very useful representation for the TIIHLW density and distribution functions. Further, we derive some mathematical properties of the subject distribution. The maximum likelihood method is used to estimate the model parameters in Section 4. In Section 5, simulation study is conducted to assess the performance of model parameters. In Section 6, we demonstrate the importance of the TIIHLW distribution using two real data sets. Finally, we give some concluding remarks in Section 7.

Type II Half Logistic Weibull Distribution
The cdf of the W distribution with scale parameter 0   and shape parameter 0 The pdf corresponding to (3) is given by The random variable X is said to have a TIIHLW distribution, denoted by X ~TIIHLW ( , , ), (1) as follows   21  ( ; , , ) ; , , 0 , 0. 11 The pdf corresponding to (5) is as follows

Some Statistical Properties
This section provides some statistical properties of TIIHLW distribution.

Quantile function
The quantile function of the TIIHLW distribution is obtained by inverting cdf (5) as follows Specifically, the first quartile, the median, and the third quartile are obtained by setting Q =0.25, 0.5 and 0.75, respectively, in (7). Also, the random variable X has TIIHLW distribution can be generated from (7), where Q has the uniform distribution over the interval (0,1). Furthermore, the analysis of the variability of the skewness and kurtosis on the shape parameters  and  can be investigated based on quantile measures. The Bowley skewness (see Kenney and Keeping (1962)), denoted by , B is defined by The Moors kurtosis (see Moors (1988)), denoted by M, can be defined as follows

Important Representation
The pdf and cdf expansions of TIIHLW are provided, which are useful in studying most statistical properties of TIIHLW distribution. From a generalized binomial series, it is known that, for 1, z  and  is a positive real non integer, Then, by applying the binomial theorem (8) in pdf (6), then, we have ( ) where, Inserting the expansion (11) in (10), then the pdf (8) will be converted to , where, . 2 ( 1) .

Probability Weighted Moments
Class of moments, called the probability-weighted moments (PWMs), has been proposed by Greenwood et al. (1979). This class is used to derive estimators of the parameters and quantiles of distributions expressible in inverse form. For a random variable X, the PWMs, denoted by , rs  , can be calculated according to the following relation Inserting (11) and (12) in (13), the PWMs of TIIHLW will be converted to

Moments
In this subsection we derive the th r moment for the TIIHLW distribution. If X has the pdf (11), then th r moment is obtained as follows The mean and variance of TIIHLW distribution are as follows  Figure 5 illustrates the mean and variance whose forms depend basically on the parameters  and . 

Order Statistics
Order statistics have been extensively applied in many fields of statistics, such as reliability and life testing. Let X1, X2,…, Xn be independent and identically distributed random variables with their corresponding continuous distribution function F(x). Let X(1) < X(2) <…< X(n) be the corresponding ordered random sample from a population of size n. According to David (1981), the pdf of the th r order statistic, is defined as where, (.,.) B is the beta function. The pdf of the th r order statistic for TIIHLW distribution is derived by substituting (11) and (12) in (14), replacing s with , , 0 0 0 The distribution of the smallest and largest order statistics can be obtained individually from (15) (16) By substituting (15) in (16), leads to

Rényi Entropy
The entropy of a random variable X is a measure of variation of uncertainty. It has been used in many fields such as physics, engineering and economics. According to Rényi (1961), the Rényi entropy is defined by ,

Stress-Strength Reliability
Let X1 be the strength of a system which is subjected to a stress X2, and if X1 follows TIIHLW (λ1 , δ1 , γ) and X2 follows TIIHLW (λ2 , δ2 , γ), provided X1 and X2 are statistically independent random variables, then R= P(X2 < X1), the measure of system performance (stress strength reliability measure) is given by,

Simulation Study
In this section, an extensive numerical investigation will be carried out to evaluate the performance of MLE for TIIHLW model. Performance of estimators is evaluated through their biases and mean square errors (MSEs) for ( )  (7) software. The simulation procedure is worked out as follows: Step (1): 10000 random samples of sizes 10, 20, 30, 50, 100,200 and 300 are generated from TIIHLW distribution.
Step ( Step (3): For each sample size and for each set of parameters MLE of the parameters ˆ,  and  are obtained by iterative technique.
Step (4): The biases and MSE for each sample size are calculated. Numerical results are listed in Tables 1 and 2. The values in the mentioned tables show that, in general, the mean square error for the estimates of the parameters ˆ,  and  decreases as the sample size increases.

Data Analysis
In this section, we use two real data sets to illustrate the importance and flexibility of the TIIHLW distribution. We compare the fits of the TIIHLW model with some models namely; the beta Weibull (BW) (

Example 1:
The data have been obtained from Nicholas and Padgett (2006 For the data in Example 1, Table 3 gives the MLEs of the fitted models and their standard errors (SEs) in parenthesis. The values of goodness-of-fit statistics are listed in Table 4. 60 It is noted, from Table 4, that the TIIHLW distribution provides a better fit than the other competitive fitted models. It has the smallest values for goodness-of-fit statistics among all fitted models. Plots of the histogram, fitted densities and estimated cdfs are shown in Figure 6. These figures supported the conclusion drawn from the numerical values in Table 4.

Example 2:
The second data set is obtained from Tahir  Based on first data, the MLEs of the fitted models and their SEs in parenthesis are listed in Table 3. Also, the values of goodness-of-fit statistics are presented in Table 4.  plots of the fitted densities and the histogram are given in Figure 6.  Figure 6: Estimated pdf and cdf plots for first data set By considering the second real data, MLEs of the fitted models and their SEs in parenthesis are given in Table 5. Further, the values of goodness-of-fit statistics are presented in Table 6.  It is observed, from Table 6, that the TIIHLW distribution gives a better fit than other fitted models.

Conclusion
In this paper, we propose a three-parameter model, named the TIIHLW distribution. The TIIHLW model is motivated by the wide use of the Weibull distribution in practice and also for the fact that the generalization provides more flexibility to analyze positive real-life data. We derive explicit expressions for the quantile function, ordinary and incomplete moments, order statistics and Rényi entropy. The maximum likelihood estimation of the model parameters is investigated. We provide some simulation results to assess the performance of the proposed model. The practical importance of the TIIHLW distribution is demonstrated by means of two data sets.