On Transmuted Type II Generalized Logistic Distribution with Application

Introducing extra parameters into the baseline distribution has been a huge breakthrough in research as this enhances more flexibility of the existing models. One of the recent methods is the use of transmutation map which has attracted the interest of many researchers in the last decade. This article investigates the flexibility of transmuted type II generalized logistic distribution. The well-known type II generalized logistic distribution is transmuted using quadratic rank transmutation map to develop a transmuted type II generalized logistic distribution. The map enables the introduction of additional parameter into its parent model to make it more flexible in the analysis of data in various disciplines such as biological sciences, actuarial science, finance and insurance. Some statistical properties of the model are considered and these properties include the moment, quantiles and functions of minimum and maximum order statistics. The estimation issue of the subject model is addressed using method of maximum likelihood estimation. The model is applied to real life data to demonstrate its performance and the comparison of the result of the subject model with its parent model was done using Akaike Information criterion (AIC), Corrected Akaike Information criterion (AICC) and Bayesian Information criterion (BIC) respectively. It is believed that the results from this research work will be of immense contributions in this field and other related disciplines in modelling real data.


Introduction
Several probability models have been studied over the years with applications in different fields of endeavor which include biological science, actuarial science, clinical science and other related disciplines. Logistic distributions are probability models that are popularly used in modeling as normal distribution.
Its cumulative distribution function (cdf) in its simplest form is given by with the location parameter ≤ and scale parameter > 0.
The model in (1) has gained generalizations by different authors which include Balakrishnan and Leung [1] which introduced shape parameter into the model and this has birthed type I generalized logistic distribution, type II generalized logistic distribution and type III generalized logistic distribution.
The transmutation of many baseline distributions has been studied by many authors in view of making them more flexible in analyzing data arising in various fields. Shaw and Beckley [2] developed a quadratic rank transmutation map. The map was applied to normal, exponential and uniform distribution respectively through which extra parameter was introduced. This map was applied to Gumbel distribution by Aryal [3], to weibul distribution by Aryal [4] and to the loglogistic distribution by Aryal [5]. Others that applied the same map to other probability models include Merovci et al [6] to some well-known distributions, Merovci et al [7] to Lindley-geometric distribution, Merovci et al [8] to generalized Rayleigh distribution, Merovci et al [9] to Pareto distribution, Merovci et al [10] to Lindley distribution, Adeyinka et al [11] to a generalized log-logistic distribution, Adeyinka et al [12] to a half logistic distribution, Adeyinka et al [13], Adeyinka [14] to logistic distribution, Adeyinka [15] to type I generalized logistic distribution. AL-Kadim et al [16] developed a cubic rank transmutation map and applied it to Weilbul distribution. Granzotto et al [17] suggested another cubic rank transmutation map and applied it to Weibul and log-logistic distributions. Rahman et al [18] obtained the general form of the work of Shaw et al [2].
This research article will focus on the transmuted form of type II generalized logistic distribution and its performance will be investigated with real data.

Transmuted Type II Generalized Logistic Distribution
Suppose a random variable X has the type II generalized logistic distribution with probability density function (pdf) and its cumulative distribution function (cdf) given by respectively where " > 0 is the shape parameter. The corresponding transmuted type II generalized logistic distribution, using the quadratic rank transmutation map suggested by Shaw et al [2], is given by which is the cdf of transmuted type II generalized logistic distribution. The corresponding pdf is obtained by differentiating (5) with respect to and it is given by with transmutation parameter % . It is observed that when parameter λ = 0 in (5)
Suppose , is the pdf of transmuted type II generalized logistic distribution, on substitution, (7) The integral part of (10) is equivalent to a beta function which can be written as Similarly, the characteristic function of the transmuted type II generalized logistic distribution can be written as Other properties such as mean, variance, skewness and kurtosis of the model can be obtained from (11).
The S 5T -quantile of the random variable / is given by If $ 1 is the cdf of transmuted type II generalized logistic distribution (13) becomes which on solving for V in (14) gives The median of the model is obtained by making S = 0.5 in (15) which becomes

Random Number Generation
By method of inversion random numbers can be generated from a given distribution with cdf given by where F is uniformly distributed on 0, 1 . Suppose $ 1 is the cdf of transmuted type II generalized logistic distribution, on substitution (17) becomes On solving for in (18) For known values of parameters " IM9 % random numbers can be generated from (19).

Order Statistics
Given a random sample / , / & … / g from a continuous population with cdf $ 1 and pdf , 1 , the arrangement / ≤ / & ≤ ⋯ ≤ / g denotes order statistics. The probability density function of / i is given by where l = 1, 2, … M. This is given in David [19] The pdf of the l 5T order statistics of the type II generalized logistic random variable / i is given by which is obtained from (2) with the pdf of the largest observation / g given by

Estimation of Parameters
Consider a sample / , / & … / g with sample size M which is drawn from transmuted type II generalized logistic distribution whose pdf is given in (5). Its likelihood function is given by o = " g Each estimate of the inherent parameters " and % can be obtained by taking the derivative of (30) with respect to each parameter and equate the result to zero to obtain The system of equations obtained in (31) and (32) are nonlinear in parameter. Therefore, the maximum likelihood estimator z { = ", | % { } of parameters z = ", % } cannot be obtained analytically and as a result numerical method, quasi-Newton algorithm, is used to maximize the loglikelihood function in (30).

Application
This data is originally considered by Badar and Priest [21] and also used by Gupta et al [20]. It is the strength measured in GPA, for single carbon fibers and impregnated 1000carbon fiber tows. The fibers were tested under tension at gauge lengths of 1, 10, 20 and 50 mm. Impregnated tows of 1000 fibers were tested at gauge lengths of 20, 50, 150 and 300 mm. To illustrate this single fibers data set of 10 mm in gauge lengths with sample size 63. The data are: Using quasi Newton algorithm in R package, the results of the analysis are shown in Table 1. The test of the performance of each model is done with Akaike Information criterion (AIC), Corrected Akaike Information criterion (AICC) and Bayesian Information criterion (BIC) respectively. The results show that the transmuted type II generalized logistic distribution (TGLD) has a better goodness of fit than its parent model (GLD) which is given in (2). The following are defined as and with k given as the number of parameters in the model and n as the sample size and LL as the maximized value of log likelihood function.

Conclusion
A new probability model has emerged from the type II generalized logistic model and this is called transmuted type II generalized logistic distribution. It has been demonstrated that the model is more flexible than its parent model. The mathematical properties which include moment generating function, quantiles, pdfs of the smallest and largest observation and characteristic function of the model are established and the estimation of parameters inherent in the subject model is properly considered. A real life data is used to illustrate the applicability of the model and compared the performance to its parent model. This model is recommended in the analysis of data in various disciplines where the real data follow this model.