Abstract: Chen, Bunce and Jiang [In: Proceedings of the International Con ference on Computational Intelligence and Software Engineering, pp. 1-4] claim to have proposed a new extreme value distribution. But the formulas given for the distribution do not form a valid probability distribution. Here, we correct their formulas to form a valid probability distribution. For this valid distribution, we provide a comprehensive treatment of mathematical properties, estimate parameters by the method of maximum likelihood and provide the observed information matrix. The flexibility of the distribution is illustrated using a real data set.
Abstract: It is shown that the most popular posterior distribution for the mean of the normal distribution is obtained by deriving the distribution of the ratio X/Y when X and Y are normal and Student’s t random variables distributed independently of each other. Tabulations of the associated percentage points are given along with a computer program for generating them.
The Lindley distribution has been generalized by many authors in recent years. However, all of the known generalizations so far have restricted tail behaviors. Here, we introduce the most flexible generalization of the Lindley distribution with its tails controlled by two independent parameters. Various mathematical properties of the generalization are derived. Maximum likelihood estimators of its parameters are derived. Fisher’s information matrix and asymptotic confidence intervals for the parameters are given. Finally, a real data application shows that the proposed generalization performs better than all known ones
The normal distribution is the most popular model in applications to real data. We propose a new extension of this distribution, called the Kummer beta normal distribution, which presents greater flexibility to model scenarios involving skewed data. The new probability density function can be represented as a linear combination of exponentiated normal pdfs. We also propose analytical expressions for some mathematical quantities: Ordinary and incomplete moments, mean deviations and order statistics. The estimation of parameters is approached by the method of maximum likelihood and Bayesian analysis. Likelihood ratio statistics and formal goodnessof-fit tests are used to compare the proposed distribution with some of its sub-models and non-nested models. A real data set is used to illustrate the importance of the proposed model.
Abstract: In recent years, many modifications of the Weibull distribution have been proposed. Some of these modifications have a large number of parameters and so their real benefits over simpler modifications are questionable. Here, we use two data sets with modified unimodal (unimodal followed by increasing) hazard function for comparing the exponentiated Weibull and generalized modified Weibull distributions. We find no evidence that the generalized modified Weibull distribution can provide a better fit than the exponentiated Weibull distribution for data sets exhibiting the modified unimodal hazard function.In a related issue, we consider Carrasco et al. (2008), a widely cited paper, proposing the generalized modified Weibull distribution, and illustrating two real data applications. We point out that some of the results in both real data applications in Carrasco et al. (2008) 1 are incorrect.
The generalized exponentiated exponential Lindley distribution is a novel three parameter distribution due to Hussain et al. (2017). They studied its properties including estimation issues and illustrated applications to four datasets. Here, we show that several known distributions including those having two parameters can provide better fits. We also correct errors in the derivatives of the likelihood function.