Methods for generating new families of continuous univariate distributions

Seeking the flexibility of probability distributions in modeling real life phenomena could be adjudged as the driving force behind the development of new probability distributions. Methods for generating families of continuous univariate probability distributions have received widespread attention in recent decades. It is well established that much of practical statistical studies and developments in probability th theory have been dominated by the normal distribution for several decades. However, in the late 19 century, the increasing collection, tabulation, and publication of data by government, private institutions and agencies in demography, social sciences, biology and insurance revealed that the normal distribution was not sufficient for describing phenomena (homogenous with respect to all but random factors) in the real world situations. This reality spearheaded the need to develop other families of distributions that can be well adapted to real-life problems. In this paper, a review of families of continuous univariate probability distributions from the Pearson's family down to the composite family of distributions.

Continuous probability distributions can be generated in many ways. To deal with various types of data or information, many researchers have developed flexible distributions. Since the pioneer work of Karl Pearson in 1895 and 1901, the first order linear differential equation approach of Pearson is a major milestone among different ways for generating univariate continuous probability distributions. In this paper, we present the first order linear differential equation approach for generating continuous probability distributions. A review of Pearson system of probability distributions is provided. We provide various new classes of the generalized Pearson system of probability distributions that could be generated using the first order linear differential equation approach. Some distributional properties and characterizations are also provided. It is observed that the first order linear differential equation approach is an elegant method for generating new families of univariate continuous probability distributions. Finally, we provide prospects of further research and directions in which future progress can be made.

By extrapolating the explicit formula of the zero-bias distribution occurring in the context of Stein’s method, we construct characterization identities for a large class of absolutely continuous univariate distributions. Instead of trying to derive characterizing distributional transformations that inherit certain structures for the use in further theoretic endeavors, we focus on explicit representations given through a formula for the density- or distribution function. The results we establish with this ambition feature immediate applications in the area of goodness-of-fit testing. We draw up a blueprint for the construction of tests of fit that include procedures for many distributions for which little (if any) practicable tests are known. To illustrate this last point, we construct a test for the Burr Type XII distribution for which, to our knowledge, not a single test is known aside from the classical universal procedures.

Continuous Probability Distributions

Continuous probability distributions

In this study, we present a new family of continuous distributions via an extended form of the Weibull distribution. Some special members of the newly defined family are discussed and the new univariate continuous distributions are introduced. The mathematical properties are obtained for any members of the family such as expansions of the density, hazard rate function, quantile function, moments and order statistics. We obtain the distribution parameters by maximum likelihood method. The simulation study to evaluate the performance of the estimated parameters based on the selected member of the this new family is also given. The lifetime data example is discussed to illustrate the applicability of the distribution.

Starting from two known continuous univariate distributions, a bivariate distribution is constructed depending on a parameter which measures the degree of stochastic dependence between the two random variables. From the foregoing construction we then pass to a multivariate-type distribution, constructed using only univariate distributions and an association matrix. Some properties of the multivariate and bivariate case are studied.

Using the Gibbs sampler for conditional simulation of Gaussian-based random fields

Bibliography

We use inequalities to design short universal algorithms that can be used to generate random variates from large classes of univariate continuous or discrete distributions (including all log-concave distributions). The expected time is uniformly bounded over all these distributions. The algorithms can be implemented in a few lines of high-level language code. In opposition to other black-box algorithms hardly any setup step is required, and thus it is superior in the changing-parameter case.

We examine the problem of computing the highest density region (HDR) in a computational context where the user has access to a density function and quantile function for the distribution (e.g., in the statistical language R). We examine several common classes of continuous univariate distributions based on the shape of the density function; this includes monotone densities, quasi-concave and quasi-convex densities, and general multimodal densities. In each case we show how the user can compute the HDR from the quantile and density functions by framing the problem as a nonlinear optimisation problem. We implement these methods in R to obtain general functions to compute HDRs for classes of distributions, and for commonly used families of distributions. We compare our method to existing R packages for computing HDRs and we show that our method performs favourably in terms of both accuracy and average speed.

Most of the well-known continuous univariate distributions are characterized based on a truncated moment of a function of the 1st order statistic or of the nth order statistic.

The art of parameter(s) induction to the baseline distribution has received a great deal of attention in recent years. The induction of one or more additional shape parameter(s) to the baseline distribution makes the distribution more flexible especially for studying the tail properties. This parameter(s) induction also proved helpful in improving the goodness-of-fit of the proposed generalized family of distributions. There exist many generalized (or generated) G families of continuous univariate distributions since 1985. In this paper, the well-established and widely-accepted G families of distributions like the exponentiated family, Marshall-Olkin extended family, beta-generated family, McDonald-generalized family, Kumaraswamy-generalized family and exponentiated generalized family are discussed. We provide lists of contributed literature on these well-established G families of distributions. Some extended forms of the Marshall-Olkin extended family and Kumaraswamy-generalized family of distributions are proposed.

Moments of truncated continuous univariate distributions

Continuous Distributions (General). Normal Distributions. Lognormal Distributions. Inverse Gaussian (Wald) Distributions. Cauchy Distribution. Gamma Distributions. Chi-Square Distributions Including Chi and Rayleigh. Exponential Distributions. Pareto Distributions. Weibull Distributions. Abbreviations. Indexes.