Gamma Generalized Logistic Distribution: Properties and Applications

In this paper we consider a new class of asymmetric logistic distribution that contains both the type I and type II generalized logistic distributions of Balakrishnan and Leung (Commun Stat Simul Comput 17(1):25–50, 1988) as its special cases. We investigate some important properties of the distribution such as expressions for its mean, variance, characteristic function, measure of skewness and kurtosis, entropy etc. along with the distribution of its order statistics. A location-scale extension of the distribution is defined and discussed the maximum likelihood estimation of its parameters. Further, two real life medical data sets are utilized for illustrating the usefulness of the model and a simulation study is conducted for examining the performance of the maximum likelihood estimators of the parameters of the distribution.


Introduction
The logistic distribution has been found several applications in various fields such as public health (Grizzle [8]), survival analysis (Plackett [13]), biology (Pearl and Reed [12]), bioassay problems (Berkson [3][4][5]) etc. For a detailed account of properties and applications of various forms of logistic distributions, refer Balakrishnan [1]. Several generalised distributions have been studied in the literature. Balakrishnan and Leung [2] studied three types of generalized logistic distributions. Wahed and Ali [18] introduced the skew logistic distribution (SLD). An extension of SLD was proposed by Nadarajah [11]. A flexible class of skew logistic distribution was studied by Kumar and Manju [10].
A continuous random variable X is said to have the standard logistic distribution (LD) if its probability density function (PDF) is of the following form, for x ∈ R = (−∞, +∞).
The cumulative distribution function (CDF), F 1 (.) of the LD is for x ∈ R. Balakrishnan and Leung [2] introduced and studied two generalized classes of logistic distributions namely the generalized logistic distribution of type I (denoted by LD I ) and type II (denoted by LD II ) respectively through the following PDFs f 2 (.) and f 3 (.) , for x ∈ R , > 0 and > 0.
The CDFs corresponds to the LD I and the LD II are respectively (1) f 1 (x) = e −x (1 + e −x ) 2 . (2) (4) f 3 (x, ) = e − x (1 + e −x ) +1 (5) F 2 (x) = 1 1 + e − x and Clearly, when = = 1 in (5), the CDF of LD I reduces to that of the LD and when = 1 in (6), the CDF of LD II reduces to that of the LD. Both these classes of distributions have applications in several areas of scientific research. Through the present paper we attempt to unify both these classes of distributions and termed it as "the gamma generalized logistic distribution (GGLD)", which is not available any where in the existing literature . The objective of the present work is to develop a more flexible class of distribution which can handle asymmetric distributions and derive some of its important properties. The paper is organized as follows. In Sect. 2, we present the definition of the GGLD and describe some important properties. A location scale extension of the GGLD is considered in Sect. 3 and in Sect. 4, two real life medical data sets are considered for illustrating the usefulness of the model compared to the LD, LD I and LD II . In Sect. 5, a generalized likelihood ratio test procedure is suggested for testing the significance of the parameters of the GGLD and a simulation study is conducted to test the efficiency of the maximum likelihood estimators (MLEs) of the distribution in Sect. 6 . We have the following representations from Gradshteyn and Ryzhik [7] , those we need in the sequel.
in which Ψ(a) = d log Γa da and C is the Euler's constant.

Definition and Properties
In this section, first we present the definition of the GGLD and discuss some of its important properties. A continuous random variable X is said to follow gamma generalized logistic distribution if its CDF is of the following form, in which x ∈ R , > 0 , > 0 and > 0.
On differentiating (9) with respect to x, we have the probability density function (PDF) of GGLD as The distribution of a random variable with CDF (9) or PDF (10) is hereafter we denoted by GGLD( , , ) . Clearly, when = 1 , the GGLD reduces to the LD I and when = = 1 , the GGLD reduces to LD II with parameter . The PDF plots of GGLD( , , ) for particular choices of its parameters , and is given in Fig. 1. From the figure it is clear that for fixed and , the distribution is positively skewed for < 1 and negatively skewed for > 1 . Furthermore, as increases the kurtosis is also increases for fixed and .

Proposition 1
The characteristic function Φ X (t) of GGLD( , , ) with PDF (10) is the following, for t ∈ R , where B(.,.) is the beta function.
Proof Let X follows GGLD( , , ) with PDF (10). Then by the definition of characteristic function, we have the following for any t ∈ R and i= √ −1. (12), to obtain Now applying binomial expansion of (1 − u ) −1 in (13) and rearranging the terms to get the following.
which gives (11), by the definition of beta integral. ◻
Proof By definition, the mean of GGLD( , , ) is by putting u = 1 + e − x −1 . Now by binomial expansion of (1 − u ) −1 , we obtain the following.
Applying (7) in the second integral term of (17) and by using integration by parts in the first integral term, one can obtain (15). By definition, the variance of GGLD( , , ) gamma generalized logistic distribution (GGLD) is in which By applying binomial expansion of (1 − u ) −1 , in (19), we obtain the following.
where and Now, by using integration by parts we have the following from (21).
In a similar way, we obtain the following from (22).
Journal of Statistical Theory and Applications (2022) 21:155-174 Applying (8) in (23), I 3 becomes, Thus, from (18) and (20) we get (16), in the light of (24), (25) and (26). Proof Galton [6] introduced the percentile oriented measure of skewness as so that 0 < g a < ∞ . Note that g a = 1 indicates symmetry, g a < 1 indicates skewness to left while g a > 1 is interpreted as skewness to right. Schmid and Trede [17] defined the percentile oriented measure of kurtosis L 0 as the product of the measure of tail T = .
Now the proof of (28) and (29) follows form (27), (30) and (31). It is quite interest to note that both skewness and kurtosis depends only on and . From Appendix it is clear that the the skewness of this distribution ranges from 0.67 to 1.82. From Fig. 1 also it is evident that this is a moderately skewed distribution. The computed values of skewness and kurtosis for different values of the parameters are given in Appendix . ◻

Proposition 5
The PDF of the kth order statistics X k∶n of GGLD( , , ) is Proof Let X 1 , X 2 , ..., X n be a random sample of size n from the GGLD( , , ) and let X k∶n be the k th order statistic for k = 1, 2, ..., n. Let F x k∶n (x) and f x k∶n (x) denotes the CDF and the PDF of X k∶n respectively. Then for x ∈ R. Now, by applying (9) and (10) in (33) to obtain (32). ◻ From Proposition (5), we have the following Corollaries.

Corollary 1
The distribution of the smallest order statistic X 1∶n based on a random sample of size n taken from a population following GGLD( , , ) is GGLD( , , n ).

Corollary 2
The PDF of the largest order statistics X n∶n is for x ∈ R, which reduces to LD I ( n, ) when = 1. (32)

Proposition 8 The hazard function is given by
The proof follows directly from the definition of survival function and hazard function and hence, omitted. The curve of hazard function is given in Fig. 2. From the curve it is seen that when , and are more than 1, the curve has a point of (35) inflection between 0.5 and 1 , and thereafter it remains stable. The point of inflection increases as any one of the parameters takes a value less than one.

Location Scale Extension
In this section we define an extended form of GGLD( , , ) by introducing the location parameter and scale parameter and discuss the maximum likelihood estimation of the parameters of extended form of GGLD( , , ).
Definition Let Z follows the GGLD( , , ) with PDF (10). Then X = + Z is said to have an "extended GGLD with parameters , , , and ", (38) When these likelihood equations donot always have solutions, the maximum of the likelihood function is reached at the border of the parameter domain. Since the MLE of the unknown parameters , , , , cannot be obtained in closed forms, there is no way to derive the exact distribution of the MLE. Therefore, we derived the second order partial derivatives of the log-likelihood function with respect to the parameters , , , , using MATHLAB software and noticed that these gave negative values for > 0, > 0, > 0, > 0, > 0 . Hence the maximum likelihood estimators of the parameters of EGGLD( , ; , , ) can be obtained by solving the above system of Eqs.

Applications
For numerical illustration we consider the following two data sets .
Data set 1. Myopia Data data set available in "https://www.umass.edu//statdata". This data set is also used by Hosmer et al. [9]. The dataset is a subset of data from the Orinda Longitudinal Study of Myopia (OLSM), a cohort study of ocular component development and risk factors for the onset of myopia in children. Data collection began in the 1989-1990 school year and continued annually through the 2000-2001 school year. All data about the parts that make up the eye (the ocular components) were collected during an examination during the school day. Data on family history and visual activities were collected yearly in a survey completed by a (40) n = ( + 1) parent or guardian. The dataset used in this text is from 618 of the subjects who had at least 5 years of follow-up and were not myopic when they entered the study. All data are from their initial exam and the dataset includes 17 variables. We have taken the continuous variable vitreous Chamber depth (VCD) in mm of 618 patients. VCD is the length from front to back of the aqueous-containing space of the eye in front of the retina. x ∈ R = (−∞, +∞) and > 0 From Table 1, it is seen that the KSS, AIC, BIC, CAIC and HQIC values are minimum for EGGLD( , ; , , ) compared to other models. Figures 3 and 4 also confirm this result. These observations reveals that the EGGLD ( , ; , , ) is relatively a better model compared to the existing models.

Testing of Hypothesis
In this section we discuss certain generalized likelihood ratio test procedures for testing the parameters of the EGGLD( , ; , , ) and attempt a brief simulation study. Here we consider the following tests. Test 1.  of Test 1, = 1, = 1 in case of Test 2 and = 1, = 1, = 1 in case of Test 3 respectively. The test statistic −2 ln Λ given in (45) is asymptotically distributed as 2 with one degree of freedom for test 1, 2 degree of freedom for test 2 and 3 degree of freedom for test 3 [14]. The computed values of lnL( ∧ Ω ;y|x), lnL( ∧ * Ω ;y|x) and test statistic in case of the two data sets are listed in Table 2. Since the critical values at the significance level 0.05 and degree of freedom one, two and three for the two tailed test are 5.024, 7.378 and 9.348 respectively the null hypothesis is rejected in all cases, which shows the appropriateness of the EGGLD to both the data sets.