A novel four-parameter log-logistic model: mathematical properties and applications to breaking stress, survival times and leukemia data

In this paper, we introduce a new continuous log-logistic extension. Several of its properties are established. A numerical analysis for skewness and kurtosis is presented. The new failure rate can be "bathtub or U shaped", "increasing", "decreasing-constant", "J shaped", "constant" and "decreasing". Many bivariate and Multivariate type distributions are derived using the Clayton Copula and the Morgenstern family. To assess of the finite sample behavior of the estimators, we performed a graphical simulation. Some useful applications are considered for supporting the new model.


1.Introduction and motivation A continues random variable (RV)
is said to has the log-logistic (LL) model if its survival/reliability function (SF) be written as ( ) = 1 − ( ) = (1 + ) − where > 0 is a shape parameter and ( ) refer to the cumulative distribution functions (CDF) of the LL model. The density function (PDF) due to (1) is given as (2) The CDF and PDF in (1) and (2) is a sub-model from the Burr model from the type-XII (BUXII) model (Burr (1942(Burr ( , 1968(Burr ( and 1973, Tadikamalla (1980) and Rodriguez (1977)). Due to Yousof et al. (2018), the CDF of the new Weibull Generalized log-logistic (WG-LL) is defined as represents the CDF of the well-known LL model with parameters and (1 + 4 ) . This work can be motivated via the following applied justifications: i.
The new model in its novel pattern will be useful in mathematical modeling of the engineering real-life datasets such as the "monotonically-increasing HRF" engineering breaking stress real-life dataset. ii.
The novel model in its novel version can be used for statistical modeling of the reliability real-life datasets such as the "monotonically-increasing HRF" reliability real-life dataset.

iii.
The new current version of the log-logistic version can be used in modeling the medical real-life datasets such as the "U-HRF" medical real-life dataset.

Properties Moments and generating function
The m th ordinary moment of is given by Then, we obtain is the second type of beta function. By fixing = 1 in , ′ , we obtain the mean of the model. The m th incomplete moment ( ( ) ) of can be expressed from (3) as Is the second type of the incomplete beta function. The moment generating function (mgf) ( ) = ( ( )) of can be derived from (3) as ! [ 4 ] (1 + 4 ) (1 + , (1 + 4 ) − ) | ( <(1+ 4 ) ) ,

Probability weighted moments (PWMs)
The (m,r) th PWM of can be expressed as

Checking flexibility numerically
In this section, the impacts of the parameters on the model mean ( 1 ′ ), variance of the model (V ( ) ), skewness of the model (S ( ) ) and kurtosis of the model (K ( )) are given below in Table 1. The impacts of for the standard LL model on the 1 ′ , V ( ) , S ( ) and K ( ) are provided in Table 2.
then we have new bivariate model as

Bivariate WG-LL via Clayton Copula
Consider the following Clayton Copula Then, setting A novel four-parameter log-logistic model: mathematical properties and applications to breaking stress, survival times and leukemia data 138 The bivariate CDF can be written as

The Multivariate extension via Copula of Clayton
The -dimensional model can be expressed as

Simulations
Assessing the behavior of the maximum likelihood estimations (MLEs) is discussed in this section. For this purpose, consider the following active algorithm: for generating 5000 group of size from the WG-LL model.   Figure 5, the biases f decrease to zero as → ∞ , the valued of the obtained MSEs of , , and decrease to zero as → ∞ . Based on this assessment, the ML method performs well and can be used in estimating the model parameter. The following Section provide some useful real data applications using the ML method for comparing the competitive models.
A novel four-parameter log-logistic model: mathematical properties and applications to breaking stress, survival times and leukemia data 139 . Figure 3: Bias and MSE for the parameter .
A novel four-parameter log-logistic model: mathematical properties and applications to breaking stress, survival times and leukemia data 140   Real-life data set I: the breaking stress data. It consists of 100 observations of breaking stress of carbon fibers (in Gba) (see Nichols and Padgett (2006)). Real-life data set II: the survival times in days of 72 pigs from guinea which was infected with the virulent tubercle bacilli (Bjerkedal (1960)). Real-life data set III: the leukemia data. It represents the times of survival, in weeks, of 33 patients suffering from the wee-known acute myelogenous leukemia.
The total time test (TTT) plots (Aarset(1987)) for the three real data sets are presented in Figure 2. It is seen that the HRFs of data sets I, II are monotonical increasing and U-HRF for data set III. We consider the following goodnessof-fit statistics: the Akaike-criterion (Ͳ 1 ), Bayesian-criterion (Ͳ 2 ), consistent-criterion (Ͳ 3 ) and Hannan-Quinn criterion (Ͳ 4 ). Generally, the smaller these statistics are, the better the fit. Tables 3, 4 and 5 give the MLEs, standard errors (SEs), confidence interval (CIs95%) with for the data set I, II and III. Tables 6, 7 and 8 give the statistics Ͳ 1 , Ͳ 2 , Ͳ 3 , and Ͳ 4 values for the data set I, II and III. Due to Table 6, Table 7 and Table 8 and Figure 3-6 the WG-LL model has the best results with small values of the Ͳ 1 , Ͳ 2 , Ͳ 3 , and Ͳ 4 .

Concluding remarks
In the present paper, we introduced a novel continuous log-logistic model. Several of its main characteristic properties such as the moments, the generating function, the weighted moments, the reversed residual life are mathematically derived. Numerical analysis for the skewness and the kurtosis is presented and useful comments are added. For the new log-logistic model, the skewness ∈(−1.081, 16.74). However, for the standard log-logistic model, the skewness ∈(0.087, 2.4853). hence, the novel model can be negative skewed and positive skewed while the standard model can only be negative skewness. For the new log-logistic model, kurtosis ∈(3.245089, 702.498). However, for the standard log-logistic model, kurtosis ∈(3.741, 29.56). The new PDF can be unimodal, symmetric, or left skewed. The new failure rate can be "bathtub or U-failure rate", "increasing failure rate", "decreasing-constant failure rate", "J-failure rate", "constant failure rate " and "decreasing failure rate".
Many bivariate and extensions are derived. To assess the estimators, we performed a graphical simulation. Three different real-life data are modeled under some statistical tests. For all these real-life datasets, we compare the novel function with many relevant extensions. The new model is better than all other competitive models in modeling breaking stress data, survival times data and leukemia data.