A TYPE II HALF LOGISTIC EXPONENTIATED-G FAMILY OF DISTRIBUTIONS WITH APPLICATIONS TO SURVIVAL ANALYSIS

Statisticians have created and proposed new families of distribution by extending or generalizing existing distributions. These families of distributions are made more flexible in fitting different types of data by adding one or more parameters to the baseline distributions. In this article, we present a new family of distributions called Type II half-logistic exponentiated-G family of distributions. We discuss some of the statistical properties of the proposed family such as explicit expressions for the quantile function, probability weighted moments, moments, generating function, survival and order statistics. The new family’s sub-models were discussed. We discuss the estimation of the model parameters by maximum likelihood. Two real data sets were employed to show the usefulness and flexibility of the new family.

The aim of this paper is to develop and investigate the Type II Half-Logistic Exponentiated-G family of distributions (TIIHLEt-G).

MATERIALS AND METHOD Type II Half-Logistic Exponentiated-G Family
The Exponentiated G (Et-G) family of distributions as defined in Ibrahim et.al., (2020a)  where α > 0 is the shape parameter and h(x; β) and H(x; β) are the pdf and cdf of the baseline distribution with parameter vector β. Hassan et.al., (2017) proposed the Type II half logistic family (TIIHL) of distributions with cdf defined as: where λ > 0, x > 0 and G(x; ξ) and g(x; ξ) are the cdf and pdf of the baseline distribution with parameter vector ξ.

Proposition
The cdf of a new family of distribution that extends the TIIHL family called Type II Half-Logistic Exponentiated-G Family of distributions is given as _ 2 ( ; ) ( ; , , ) Let the Exponentiated G family be the baseline family with cdf and pdf given in equations (1) and (2) respectively, then the proposed Type II Half Logistic Exponentiated-G family of probability distributions has cdf given as: z  and  is a positive real non integer. The density function of the TIIHLEt-G family is then obtained by using the binomial theorem (9) to (6).

Statistical Properties
In this section, we derived statistical properties of the new family of distribution. Probability weighted moments Greenwood et.al., (1979) introduced a class of moments known as probability weighted moments (PWMs). This class is used to derive inverse form estimators for the parameters and quantiles of a distribution. The PWMs, represented by , rs  , can be derived for a random variable X using the following relationship.
The PWMs of TIIHLEt-G is derive by substituting (11) and (14) into (15), and replacing h with s, as follows Another formula will be obtained by applying the quantile function, as follows:

Moments
Since the moments are necessary and important in any statistical analysis, especially in applications. Therefore, we derive the r th moment for the new family.
By using the important representation of the pdf in equation (11), we have ( 1) 1 0 0

Moment generating function (mgf)
The Moment Generating Function of x is given as: Another formula will be obtained by applying the quantile function, as follows:

Mean deviation
The mean deviation about the mean Additional form is derived as follows, depending on the parent quantile function.

Reliability function
The reliability function which is also known as survivor function that gives the probability that a patient will survive longer than specified period of time. It is defined as Obtaining survival probabilities for different values of time x provides crucial summary information from survival data.

Hazard function
The hazard function is the probability of an event of interest occurring within a relatively short time frame and is defined as: The hazard function also known as conditional failure rate, gives the instantaneous potential per unit time for the event of interest to occur, given that the individual has survived up to time x.

Quantile Function
The quantile function is a vital tool to create random variables from any continuous probability distribution. As a result, it has a significant position in probability theory. For x, thequantile function is F(x) = u, where u is distributed as U(0,1).The TIIHLEt-G family is easily simulated by inverting equation (5)  H  is the quantile function of the baseline cdf H(x; β). The first quartile, the median and the third quartile are obtained by putting u = 0.25, 0.5 and 0.75, respectively in equation (4.6).

Order Statistics
Many areas of statistics including reliability and life testing, have made substantial use of order statistics. Let X1, X2, ..., Xn be independent and identically distributed random variables with their corresponding continuous distribution function F (x). Let X1, X2,.., Xn be n independently distributed and continuous random variables from the TIHLEt-G family of distribution. Let Fr:n(x) and fr:n(x), r = 1, 2, 3, ..., n denote the cdf and pdf of the r th order statistics Xr:n respectively. David (1970) gave the probability density function of Xr:n as:  (5) and equation (6)

Sub-Models
In this section, we describe two sub-models of the TIIHLEt-G family namely, TIIHLEt-Exponential and TIIHLEt-Loglogistic respectively.

Type II Half-Logistic Exponentiated Exponential (TIIHLEtE) Distribution
The cdf and pdf of Exponential distribution which is our baseline distribution with parameter  It can be seen from figure 1 that the TIIHLEtE distribution has a positively skewed shape.   .3, September, 2021, pp 177 -190 185 The TIIHLEtLL distribution has CDF and PDF given as; 21 ( ; , , ) , 0, , , 0 (45) 11

A Type II Half
x TIHLEtLL x Moreso, the following are the reliability function, hazard rate function and the quantile function respectively: 11 ( ; , , )  It can be seen from figure 3 that the TIIHLEtLL distribution has a positively skewed shape. The TIIHLEtLL distribution hazard plots in figure 4 illustrate that the hazard function has a bathtub-shaped failure rate.

Parameter Estimation
In this paper, we explore the maximum likelihood technique to estimate the unknown parameters of the TIIHLEt-G family for complete data. Maximum likelihood estimates (MLEs) have appealing qualities that may be used to generate confidence ranges and provide simple approximations that function well in finite samples. In distribution theory, the resulting approximation for MLEs is easily handled, either analytically or numerically. Let x1, x2, x3, ..., xn be a random sample of size n from the TIIHLEt-G family. Then, the likelihood function based on observed sample for the vector of parameter (λ, α, β) T is given by 00 00 ( ) log(2) log( ) log( ) log ( , ) ( 1) log ( , ) ( 1) log ( , ) 2 log 1 ( , )  .3, September, 2021, pp 177 -190 187 In this section, we fit the TIIHLEtE distribution to two real data sets and give a comparative study with the fits of the Type II half logistic exponential (TIHLE) by Elgarhyet.al.,(2018), Topp-Leone exponential distribution(TLEx) by Al-Shomrani et.al.,(2016), Kumaraswamy exponential distribution(KEx) by Adepoju and Chukwu (2015), Exponentiated exponential Distribution(ExEx) by Gupta and Kundu (1999), and Logistic-X exponential distribution (LoEx) by Oguntunde et.al., (2018), as comparator distributions for illustrative purposes.
The TIHLE distribution proposed by Elgarhy et.al.,(2018) has probability density function given as:  Adepoju and Chukwu (2015) has pdf defined as: The ExEx distribution pioneered Gupta and Kundu (1999) has pdf given as:

  
And the LoEx distribution developed by Oguntunde et.al., (2018) has pdf given as: The two datasets that were used as examples in the application demonstrate the new family of distributions' flexibility, applicability and 'best fit' compared to the above comparator distributions in modeling the data sets experimentally. The R programming language is used to carry out all of the computations.

Data set 1
The first data set as listed below represents the daily confirmed cases of COVID-19 positive cases record in Pakistan from March 24 to April 28, 2020, previously used by Al-Marzouki, et.al., (2020): 108, 102, 133, 170, 121, 99, 236, 178, 250, 161, 258, 172, 407, 577, 210, 243, 281, 186, 254,336,342,269,520,414,463,514,427,7 96,555,742,642,785,783,605,751,806.   Table 1 presents the results of the Maximum Likelihood Estimation of the parameters of the proposed distribution and the five comparator distributions. Based on the goodness of fit measure, the proposed distribution reported the minimum AIC value, though followed closely by the TIIHLE. The visual inspection of the fit presented in Figure 5, also confirms the superiority of the proposed distribution amongst its comparators. Thus the proposed distribution 'best fit' COVID 19 data set amongst the range of distributions considered.

Conclusion
A new family of continuous distributions called the Type II Half Logistic Exponentiated-G (TIIHLEt-G) family was proposed and studied. Some of the statistical properties of the proposed family, such as explicit quantile function expressions, probability weighted moments, moments, generating function, survivor functions, and order statistics were investigated. Some of the new family's sub-models were discussed. The method of maximum likelihood was used to estimate the model parameters. In comparison to well-known models, two real data sets were used to highlight the importance and flexibility of one of the sub model. The findings reveal that the new model appears to be superior to the existing models considered and, therefore, provides new distribution to model data in many applications.