Type II Half Logistic Family of Distributions with Applications

A new family of distributions called the type II half logistic is introduced and studied.  Four new special models are presented. Some mathematical properties of the type II half logistic family are studied. Explicit expressions for the moments, probability weighted, quantile function, mean deviation, order statistics and Renyi entropy are investigated. Parameter estimates of the family are obtained based on maximum likelihood procedure. Two real data sets are employed to show the usefulness of the new family.


Introduction
The most popular traditional distributions often do not characterize and do not predict most of the interesting data sets. Generated family of continuous distributions is a new improvement for creating and extending the usual classical distributions. The newly generated families have been broadly studied in several areas as well as yield more flexibility in applications. (Eugene et al. 2002) studied the beta-family of distributions. (Zografos and Balakrishnan, 2009) suggested a generated family using gamma distribution which is defined as follows () Further, some generated families were studied by several authors, for example, the kummer beta by (Pescim et al., 2012), exponentiated generalized class by (Cordeiro et al., 2013), Weibull-G by (Bourguignon et al., 2014), exponentiated half-logistic by , the type I half-logistic by (Cordeiro et al., 2015), and the Kumaraswamy Weibull by (Hassan and Elgarhy, 2016).
In the current paper, we introduce a recently generated family of distributions using the half logistic distribution as a generator. This paper can be sorted as follows. In the next section, the type II half logistic-generated ( ) TIIHL G  family is defined. Section 3 concerns with some general mathematical properties of the family. In Section 4, some new special models of the generated family are considered. In Section 5, estimation of the parameters of the family is implemented through maximum likelihood method. An illustrative purpose on the basis of real data is investigated, in Section 6. Finally, concluding remarks are handled in Section 7.

Type II Half Logistic Family
The half logistic distribution is a member of the family of logistic distributions which is introduced by (Balakrishnan, 1985) which has the following cumulative distribution function (cdf) The associated probability density function (pdf) corresponding to (3) Gx is a baseline cdf, which depends on a parameter vector .
 The distribution function (5) provides a broadly type II half logistic generated distributions. Therefore, the pdf of the type II half logistic generated family is as follows Hereafter, a random variable X has pdf (6) will be denoted by .

Moments
Since the moments are necessary and important in any statistical analysis, especially in applications. Therefore, we derive the rth moment for the TIIHL G  family. If X has the pdf (10), then rth moment is obtained as follows Further, another formula can be deduced, based on the parent quantile function, as follow;

Moment generating function
For a random variable , X it is known that, the moment generating function is defined as Additionally; different form will be yielded by using quantile function as follows;

The mean deviation
In statistics, mean deviation about the mean and mean deviation about the median measure the amount of scattering in a population. For random variable X with pdf () fx, Depending on the parent quantile function, additional form is obtained as follows;

Order statistics
Order statistics have been extensively applied in many fields of statistics, such as reliability and life testing. Let 12 , ,..., n X X X be independent and identically distributed (i.i.d) random variables with their corresponding continuous distribution function () Fx. ... n n n n X X X    the corresponding ordered random sample from a population of size . n According to (David, 1981), the pdf of the kth order statistic, is defined as where, (.,.) B stands for beta function. The pdf of the kth order statistic forTIIHL G  family is derived by substituting (10) and (12) in (14), replacing h with G are the pdf and cdf of the TIIHL G  family, respectively.
Further, the rth moment of kth order statistics for TIIHL G  is defined family by: ::

Rényi entropy
The entropy of a random variable X is a measure of variation of uncertainty and has been used in many fields such as physics, engineering and economics. As mentioned by (Renyi 1961), the Rényi entropy is defined by By applying the binomial theory (8) in the pdf (6), then the pdf () fx  can be expressed as follows Therefore, the Rényi entropy of TIIHL generated family of distributions is given by

Some special models
In this section, we define and describe four special models of the TIIHL generated family namely, TIIHL -uniform, TIIHL -BurrXII, TIIHL -Weibull and TIIHL -quasi Lindley.

TIIHL-uniform distribution
The pdf of type II half logistic-uniform ( ) TIIHLU is derived from (6), by taking as the following ( ; ) , x Gx The corresponding cdf takes the following form The plots of pdf and hazard rate function for the TIIHLU are showed in Figures 1 and 2 respectively.

TIIHL-Weibull distribution
The cdf and pdf of TIIHL-Weibull () TIIHLW distribution are derived from (5)    , we get TIIHL  exponential distribution. The plots of pdf and hazard rate function for the TIIHLW are presented in Figures 5 and 6 respectively.

Applications to Real Data
In this section, two real data sets are employed to illustrate the importance of the suggested TIIHL G  family. Recently, considerable extensions of Weibull have been introduced, in the literature, by several authors, such as type I half logistic Weibull ( TIIHLW ) by (Corediro, et al. 2015), beta Weibull ( BW ) by (Lee et al., 2007) and Weibull Weibull (WW ) by (Bourguignon et al., 2014). We fit TIIHLW distribution to the two real data sets using MLEs and compared the suggested distribution with TIHLW , BW and WW distributions. The required numerical evaluations were implemented using MathCAD 14.

Type II Half Logistic Family of Distributions with Applications
The following table shows the MLEs of the model parameters and its standard error (S.E) (in parentheses) for data set 1.       Figure 11. Estimated cumulative densities of the Figure 12. Estimated densities of the models models for data set 2.
for data set 2.
The values in Tables 2 and 4, indicate that the type II half logistic Weibull distribution is a strong competitor to other distributions used here for fitting data sets 1 and 2. A density plot compares the fitted densities of the models with the empirical histogram of the observed data (Figures 10 and 12). The fitted density for the type II half logistic Weibull model is the closest to the empirical histogram than the other fitted models.

Conclusion
In the present paper, the new type II half logistic generated family of distributions is proposed. More specifically, the type II half logistic generated family covers several new distributions. We wish a broadly statistical application in some area for this new generation. Some characteristics of the TIIHL G  , such as, expressions for the density function, moments, mean deviation, quantile function and order statistics are discussed. The maximum likelihood method is employed for estimating the model parameters. Type II half logistic uniform, type II half logistic Weibull, type II half logistic BurrXII and type II half logistic quasi Lindley selected models are provided. Applications to real data sets validate the priority of the new family.