A New Skew Logistic Distribution and Its Properties

Brazilian Journal of Development, 2021

Cruz et al. (1999) proposed a new class of Odd Log-Logistic-G distributions in order to create a new distribution family that could extend any continuous distribution. Thus, it was thought to use the Skew t-Student distribution as a base function and create the Odd Log-Logistic Skew t-Student (OLLST) distribution. There were also some applications in regression models for data from Completely Randomized Designs (CRD), and some results of density simulation showed that the new distribution is bimodal and asymmetric.

In this paper we discuss different properties of the two generalizations of the logistic distributions, which can be used to model the data exhibiting a unimodal density having some skewness present. The first generalization is carried out using the basic idea of Azzalini [2] and we call it as the skew logistic distribution. It is observed that the density function of the skew logistic distribution is always unimodal and log-concave in nature. But the distribution function, failure rate function and different moments can not be obtained in explicit forms and therefore it becomes quite difficult to use it in practice. The second generalization we propose as a proportional reversed hazard family with the base line distribution as the logistic distribution. It is also known in the literature as the Type-I generalized logistic distribution. The density function of the proportional reversed hazard logistic distribution may be asymmetric but it is always unimodal and log-concave. The distribution function, hazard function are in compact forms and the different moments can be obtained in terms of the ψ function and its derivatives. We have proposed different estimators and performed one data analysis for illustrative purposes.

Communications in Statistics: Case Studies, Data Analysis and Applications, 2019

In environmental studies, many data are typically skewed and it is desired to have a flexible statistical model for this kind of data. In this paper, we study a class of skewed distributions by invoking arguments as described by Ferreira and Steel (2006, Journal of the American Statistical Association, 101: 823-829). In particular, we consider using the logistic kernel to derive a class of univariate distribution called the truncated-logistic skew symmetric (TLSS) distribution. We provide some structural properties of the proposed distribution and develop the statistical inference for the TLSS distribution. A simulation study is conducted to investigate the efficacy of the maximum likelihood method. For illustrative purposes, two real data sets from environmental studies are used to exhibit the applicability of such a model.

Symmetry

Skewed probability distributions are important when modeling skewed data sets because they provide a way to describe the shape of the distribution and estimate the likelihood of extreme events. Asymmetric probability distributions have the potential to handle and assess problems in actuarial risk assessment and analysis. To that end, we present a new right-skewed one-parameter distribution. In this work and for this purpose, a right-skewed probability distribution was derived and analyzed. The new distribution outperformed the exponential distribution, the Pareto distribution, the Chen distribution, and others in the field of actuarial risk analysis. Some useful key risk indicators are considered and analyzed to analyze the risks and for comparison with the competitive model. Several actuarial risk functions and indicators are evaluated and analyzed using the U.K. insurance claims data set. The process of risk assessment and analysis was carried out using a comprehensive simulation....

The skew normal distribution of Azzalini (Scand J Stat 12:171–178, 1985) has been found suitable for unimodal density but with some skewness present. Through this article, we introduce a flexible extension of the Azzalini (Scand J Stat 12:171–178, 1985) skew normal distribution based on a symmetric component normal distribution (Gui et al. in J Stat Theory Appl 12(1):55–66, 2013). The proposed model can efficiently capture the bimodality, skewness and kurtosis criteria and heavy-tail property. The paper presents various basic properties of this family of distributions and provides two stochastic representations which are useful for obtaining theoretical properties and to simulate from the distribution. Further, maximum likelihood estimation of the parameters is studied numerically by simulation and the distribution is investigated by carrying out comparative fitting of three real datasets.

In this paper we discuss six different methods to introduce a shape/skewness parameter in a probability distribution. It should be noted that all these methods may not be new, but we provide new interpretations to them and that might help the partitioner to choose the correct model. It is observed that if we apply any one of these methods to any probability distribution, it may produce an extra shape/skewness parameter to that distribution. Structural properties of these skewed distributions are discussed. For illustrative purposes, we apply these methods when the base distribution is exponential, which resulted in five different generalizations of the exponential distribution. It is also observed that if we combine two or more than two methods successively, then it may produce more than one shape/skewness parameters. Several known distributions can be obtained by these methods and various new distributions with more than one shape parameters may be generated. Some of these new distributions have several interesting properties. and Bever [3], provided a nice interpretations of Azzalini's skew normal distribution as a hidden truncation model. For other univariate and multivariate skewed distributions which have been defined along the same line of Azzalini [1], the readers are referred to Arnold and Bever [4], Gupta and Gupta [9], the recent monograph by Genton [8] and the references cited there.

Journal of the American Statistical Association, 2006

We introduce a general perspective on the introduction of skewness into symmetric distributions. Through inverse probability integral transformations we provide a constructive representation of skewed distributions, where the skewing mechanism and the original symmetric distributions are specified separately. We study the effects of the skewing mechanism on e.g. modality, tail behaviour and the amount of skewness generated. The representation is used to introduce novel classes of skewed distributions, where we induce certain prespecified characteristics through particular choices of the skewing mechanism. Finally, we use a Bayesian linear regression framework to compare the new classes with some existing distributions in the context of two empirical examples.

2019

In this paper, we introduce a new distribution for positively skewed data by combining the Birnbaum-Saunders and centered skew-normal distributions. Several of its properties are developed. Our model accommodates both positively and negatively skewed data. Also, we show that our proposal circumvents some problems related to another Birnbaum-Saunders distribution based on the usual skew-normal model, previously presented in the literature. We derive both maximum likelihood and Bayesian inference, comparing them through a suitable simulation study. The convergence of the expectation conditional maximization (for maximum likelihood inference) and MCMC algorithms (for Bayesian inference) are verified and several factors of interest are compared. In general, as the sample size increases, the results indicate that the Bayesian approach provided the most accurate estimates. Our model accommodates the asymmetry of the data more properly than the usual Birnbaum-Saunders distribution, which is illustrated through real data analysis.

Communications in Statistics - Theory and Methods, 2017

The modeling and analysis of experiments is an important aspect of statistical work in a wide variety of scientific and technological fields. We introduce and study the odd log-logistic skew-normal model, which can be interpreted as a generalization of the skew-normal distribution. The new distribution can be used effectively in the analysis of experiments data since it accommodates unimodal, bimodal, symmetric, bimodal and right-skewed and bimodal and left-skewed density function depending on the parameter values. We illustrate the importance of the new model by means of three real data sets in analysis of experiments.

Journal of Statistical Research of Iran, 2010

In this paper, we discuss a new generalization of univariate skew-Cauchy distribution with two parameters, we denoted this by GSC (λ 1 , λ 2 ), that it has more flexible than the skew-Cauchy distribution (denoted by SC (λ)), introduced by Behboodian et al. (2006). Furthermore, we establish some useful properties of this distribution and by two numerical example, show that GSC (λ 1 , λ 2 ) can fits the data better than SC (λ).