On Transmuted Four Parameters Generalized Log-Logistic Distribution

In this article we transmute the four parameters generalized log-logistic distribution using quadratic rank transmutation map to develop a transmuted four parameters generalized log-logistic distribution. The quadratic rank transmutation map helps to introduce extra parameter into the baseline distribution to enhance more flexibility in the analysis of data in various disciplines such as reliability analysis in engineering, survival analysis, medicine, biological sciences, actuarial science, finance and insurance. The mathematical properties such as moments, quantile, mean, median, variance, skewness and kurtosis of this distribution are discussed. The reliability and hazard functions of the four parameters generalized log-logistic distribution are obtained. The probability density functions of the minimum and maximum order statistics of the four parameters generalized log-logistic distribution are established and the relationships between the probability density functions of the minimum and maximum order statistics of the parent model and the probability density functions of the four parameters generalized log-logistic distribution are considered. The parameter estimation is done by the maximum likelihood method. The flexibility of the model in statistical data analysis and its applicability is demonstrated by using it to fit relevant data. The study is concluded by demonstrating that the four parameters generalized log-logistic distribution has a better goodness of fit than its parent model. We hope this model will serve as an alternative to the existing ones in fitting positive real data.


Introduction
The log-logistic distribution is very useful in the field of survival analysis. The probability density function (pdf), in its simplest form, is given by The distribution has been studied into some details by Shah and Dave [1], Tadikamalla and Johnson [2], O'Quigley and Struthers [3], Ragab and Green [4].
Balakrishnan, Malik and Puthenpura [5] studied log logistic model with the pdf given by ; , , 0, 0, 1 where is the scale parameter and is the shape parameter. Olapade (2010) studied four parameters generalized log logistic distribution with pdf given by , 0 where 0 is the extra shape parameter with location parameter 0 . He established its properties and stated some theorems relating it to some other distributions in the literature.
Aryal [6] transmuted the log logistic model of Balakrishnan, Malik and Puthenpura [5] and obtained its mathematical properties and demonstrated its flexibility in the reliability analysis.
This research work will study the transmutation of the work of Olapade [7] and this will be referred to as transmuted four parameters generalized log-logistic distribution.

Transmuted Four Parameters Generalized Log-logistic Distribution
If a random variable X follows four parameters generalized log-logistic distribution with probability density function (pdf) given in (3) and the cumulative distribution function (cdf) given by The parameter is the scale parameter, !"# are shape parameters. The corresponding transmuted four parameters generalized log-logistic distribution, using the quadratic rank transmutation map.
is given by $ + , -+ , . ./ 0./ + , , 0, (6) and the corresponding pdf is given by The parameter & is the transmutation parameter. We have the pdf of four parameters log-logistic distribution when parameter λ 0 in (7). Figure 1 illustrates the graphs of pdf and cdf of transmuted four parameters generalized log-logistic distribution for different values of parameters , , !"# &.

Moments and Quantile
The 4 56 order moment of transmuted four parameters generalized log-logistic random variable x, is given by 7(8 9 ) : 9 is the pdf of transmuted four parameters generalized log-logistic distribution. This is obtained as where ? . , . is the beta function defined by ? !, B : C.
Using the functional relationships we have the 4 56 order moments given by The first moment of the transmuted four parameters generalized log-logistic distribution which is the mean of transmuted four parameters generalized log-logistic distribution is given by and the variance is given by The O 56 quantile P of transmuted four parameters generalized log-logistic distribution is obtained from (6) as The median of transmuted four parameters generalized log-logistic distribution is obtained when O = 0.5 in (9)

Random Number Generation and Parameter Estimation
Using the method of inversion we can generate random numbers from transmuted four parameters generalized loglogistic distribution as where W~Y(0,1 . After simplifying (11) we have Equation (12) can be used to generate random numbers when the parameters , , and & are known.

Parameter Estimation
The maximum likelihood estimate of the parameters , , and & that are inherent within the transmuted four parameters generalized log-logistic distribution function is given by the following.
Let 8 , 8 ', … 8 \ be a sample of size n from the transmuted four parameters generalized log-logistic distribution. The likelihood function is given by Hence the log-likelihood function of (11) becomes By differentiating (14) with respect to parameters , , and & and equating the result to zero we have The maximum likelihood estimator l m = ( n, , o p , & m q of parameters l = ( , , , & q can be obtained by solving this non-linear system of equations. It is usually more convenient to use non-linear optimization algorithms such as quasi-Newton algorithm to numerically maximize the loglikelihood function in (13).

Reliability Analysis
The survival function, also known as the reliability function in engineering, is the characteristic of the explanatory variable that maps a set of events, usually associated with mortality or failure of some system unto time. It is the probability that a system will survive beyond a specified time.
The transmuted four parameters generalized log-logistic distribution can be a useful model to characterize failure time of a given system because of the analytical structure. The reliability function r s , which is the probability of an item not failing prior to some time s, is defined by r s 1 $ s . The reliability function of the transmuted four parameters generalized log-logistic distribution is given by

Order Statistics
We know that if 8 * 8 ' * ⋯ * 8 \ denotes the order statistics of a random sample 8 , 8 ' … 8 \ from a continuous population with cdf $~ and pdf 1~ , David 1970 gave the probability density function of 8 • as where J 1,2, … ". We have from (3) and (4) Therefore the pdf of the "su order four parameters generalized log-logistic statistic 8 \ is given by and the pdf of the 1 ƒ5 order four parameters generalized loglogistic distribution statistic 8 is given by Note that in a particular case of " 2, (17) yields and (18) yields It can be observed that "…" (8 , 8 ' and "! (8 , 8 ' in (19) and (20) are special cases of (7) for & = −1 and & = 1 respectively. Now we provide the distribution of the order statistics for the transmuted four parameters generalized log-logistic distribution variable. The pdf of the J 56 order statistics for the transmuted four parameters generalized log-logistic distribution is given by

Application
The data consists of 30 observations of March precipitation (in inches) in Minneapolis/St Paul. The data set is extracted from Usman, Haq and Talib [8]. A quasi Newton algorithm was implemented in R package and the performances of the models are shown in Table 1. Akaike Information criterion (AIC), Corrected Akaike Information criterion (AICC) and Bayesian Information criterion (BIC) were respectively used to compare the performance of the transmuted four parameters generalized log-logistic distribution (TGLL) to its parent model (GLL) in (3). It is observed that the transmuted four parameters generalized log-logistic distribution (TGLL) performed better than its parent model (GLL) in (3).

Conclusion
In this article, we have introduced a new generalization of the four parameters generalized log-logistic distribution called transmuted four parameters generalized log-logistic distribution. The distribution which is generalized by using the quadratic rank transmutation map. Some mathematical properties along with estimation issues are addressed. The hazard rate function and reliability function behaviours of transmuted four parameters generalized log-logistic distribution show that the model can be used to model positive data that follow the subject model. Further studies can be done in the area of applicability of the model in various disciplines.