Skellam reference manual (original) (raw)
This manuscript illustrates the implementation and testing of nine statistical distributions, namely Beta, Binomial, Chi-Square, F, Gamma, Geometric, Poisson, Student's t and Uniform distribution, where each distribution consists of three common functions-Probability Density Function (PDF), Cumulative Density Function (CDF) and the inverse of CDF (inverseCDF).
The Kummer Beta Normal: A New Useful-Skew Model
Journal of data science: JDS
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 goodness-of-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.
Objective Bayesian Analysis of Skew-tDistributions
Scandinavian Journal of Statistics, 2012
We study the Jeffreys prior and its properties for the shape parameter of univariate skew-t distributions with linear and nonlinear Student's t skewing functions. In both cases, we show that the resulting priors for the shape parameter are symmetric around zero and proper. Moreover, we propose a Student's t approximation of the Jeffreys prior that makes an objective Bayesian analysis easy to perform. We carry out a Monte Carlo simulation study that demonstrates an overall better behaviour of the maximum a posteriori estimator compared with the maximum likelihood estimator. We also compare the frequentist coverage of the credible intervals based on the Jeffreys prior and its approximation and show that they are similar. We further discuss location-scale models under scale mixtures of skew-normal distributions and show some conditions for the existence of the posterior distribution and its moments. Finally, we present three numerical examples to illustrate the implications of our results on inference for skew-t distributions.
A class of skewed distributions with applications in environmental data
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.
Statistical distributions and their application
2013
This thesis was written during my two-year term, between July 1954 and July 1956, as a Research Scholar of the Australian National University. This thesis is divided into two parts, the first consisting of chapters 1, 2, 3 on curve-fitting and the evaluation of certain integrals, and the second of chapters 4 and 5 on the testing of various unconnected hypotheses. The work in the first chapter
Handbook of the Normal Distribution (Statistics, a Series of Textbooks and Monographs
This book contains a collection of results relating to the normal distribution. It is a compendium of properties, and problems of analysis and proof are not covered. The aim of the authors has been to list results which will be useful to theoretical and applied researchers in statistics as well as to students. Distributional properties are emphasized, both for the normal law itself and for statistics based on samples from normal populations. The book covers the early historical development of the normal law (Chapter 1); basic distributional properties, including references to tables and to algorithms suitable for computers (Chapters 2 and 3); properties of sampling distributions, including order statistics (Chapters 5 and 8), Wiener and Gaussian processes (Chapter 9); and the bivariate normal distribution (Chapter 10). Chapters 4 and 6 cover characterizations of the normal law and central limit theorems, respectively; these chapters may be more useful to theoretical statisticians. A collection of results showing how other distributions may be approximated by the normal law completes the coverage of the book (Chapter 7). Several important subjects are not covered. There are no tables of distributions in this book, because excellent tables are available elsewhere; these are listed, however, with the accuracy and coverage in the sources. The multivariate normal distribution other than the bivariate case is not discussed; the general linear v vi Preface model and regression models based on normality have been amply documented elsewhere; and the applications of normality in the methodology of statistical inference and decision theory would provide material for another volume on their own.
Sampling Methods for Wallenius' and Fisher's Noncentral Hypergeometric Distributions
Communications in Statistics-simulation and Computation, 2008
Several methods for generating variates with univariate and multivariate Walleniu' and Fisher's noncentral hypergeometric distributions are developed. Methods for the univariate distributions include: simulation of urn experiments, inversion by binary search, inversion by chop-down search from the mode, ratio-of-uniforms rejection method, and rejection by sampling in the τ domain. Methods for the multivariate distributions include: simulation of urn experiments, conditional method, Gibbs sampling, and Metropolis-Hastings sampling. These methods are useful for Monte Carlo simulation of models of biased sampling and models of evolution and for calculating moments and quantiles of the distributions.
Objective Bayesian Analysis of Skew-tDistributions
Scandinavian Journal of Statistics, 2012
We study the Jeffreys prior and its properties for the shape parameter of univariate skew-t distributions with linear and nonlinear Student's t skewing functions. In both cases, we show that the resulting priors for the shape parameter are symmetric around zero and proper. Moreover, we propose a Student's t approximation of the Jeffreys prior that makes an objective Bayesian analysis easy to perform. We carry out a Monte Carlo simulation study that demonstrates an overall better behaviour of the maximum a posteriori estimator compared with the maximum likelihood estimator. We also compare the frequentist coverage of the credible intervals based on the Jeffreys prior and its approximation and show that they are similar. We further discuss location-scale models under scale mixtures of skew-normal distributions and show some conditions for the existence of the posterior distribution and its moments. Finally, we present three numerical examples to illustrate the implications of our results on inference for skew-t distributions.
The Kumaraswamy skew-normal distribution
Statistics & Probability Letters, 2015
We propose a new generalization of the skew-normal distribution referred to as the Kumaraswamy skew-normal. The new distribution is computationally more tractable than the Beta skew-normal distribution with which it shares some properties.