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

Wahid Shehata,Murtadha Mansour Abdullah,Mohamed K A Refaie

doi:10.18187/pjsor.v18i1.3268

Abstract

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.

Highlights

Introduction and motivationA continues random variable (RV) Y is said to has the log-logistic (LL) model if its survival/reliability function (SF) be written asSb(y) = 1 − Sb(y) = (1 + yb)−1, (1)where b > 0 is a shape parameter and Sb(y) refer to the cumulative distribution functions (CDF) of the LL model.The density function (PDF) due to (1) is given as sb(y) = byb−1(1 + yb)−2. (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, 1968 and 1973), Tadikamalla (1980) and Rodriguez (1977))
Due to Fig. 1(left chart) it is seen that the novel PDF of the WG-LL model can be monotonical and unimodal PDF, symmetric PDF or negative skewed PDF
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

Summary

1.Introduction and motivation

A continues random variable (RV) Y is said to has the log-logistic (LL) model if its survival/reliability function (SF). Where b > 0 is a shape parameter and Sb(y) refer to the cumulative distribution functions (CDF) of the LL model. Figure 1: graphical charts of the novel PDF and it corresponding HRF. 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. 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. 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

Copula

Simulations

Real data modeling and analysis