Continuous Univariate Distributions Research Articles

On Rereading Stein’s Lemma: Its Intrinsic Connection with Cramér-Rao Identity and Some New Identities

Now is an opportune time to revisit Stein’s (1973) beautiful lemma all over again. It is especially so since researchers have recently begun discovering a great deal of potential of closely related Stein’s unbiased risk estimate (SURE) in a number of directions involving novel applications. In recognition of the importance of the topic of Stein’s (1973; Ann Statist 9:1135–1151, 1981) research and its elegance, we include a selective review from the field. The process of rereading Stein’s lemma and reliving its awesome simplicity as well as versatility rekindled a number of personal thoughts, queries, and observations. A number of new and interesting insights are highlighted in the spirit of providing updated and futuristic versions of the celebrated lemma by largely focusing on univariate continuous distributions not belonging to an exponential family. In doing so, a number of new identities have emerged when the parent population is continuous, but they are highly non-normal. Last, but not the least, we have argued that there is no big foundational difference between the basic messages obtained via Stein’s identity and Cramer-Rao identity.

The Marshall–Olkin Transmuted-G Family of Distributions

AbstractWe introduce a new family of univariate continuous distributions called the Marshall–Olkin transmuted-G family which extends the transmuted-G family pioneered by Shaw and Buckley (2007). Special models for the new family are provided. Some of its mathematical properties including quantile measure, explicit expressions for the ordinary and incomplete moments, generating function, Rényi and Shannon entropies, order statistics and probability weighted moments are derived. The estimation of the model parameters is performed by maximum likelihood. The flexibility of the proposed family is illustrated by means of two applications to real data sets.

On empirical estimation of mode based on weakly dependent samples

Given a large sample of observations from an unknown univariate continuous distribution, it is often of interest to empirically estimate the global mode of the underlying density. Applications include samples obtained by Monte Carlo methods with independent observations, or Markov Chain Monte Carlo methods with weakly dependent samples from the underlying stationary density. In either case, often the generating density is not available in closed form and only empirical determination of the mode is possible. Assuming that the generating density has a unique global mode, a non-parametric estimate of the density is proposed based on a sequence of mixtures of Beta densities which allows for the estimation of the mode even when the mode is possibly located on the boundary of the support of the density. Furthermore, the estimated mode is shown to be strongly universally consistent under a set of mild regularity conditions. The proposed method is compared with other empirical estimates of the mode based on popular kernel density estimates. Numerical results based on extensive simulation studies show benefits of the proposed methods in terms of empirical bias, standard errors and computation time. An R package implementing the method is also made available online.

An upper bound for a cyclic sum of probabilities

In the study of nontransitive dice, seemingly paradoxically, the probability that one die rolls higher than another is not transitive. We prove a more general result which implies that a cyclic sum of ordering probabilities for n random variables can be much greater than 1, and that each probability can be much greater than 1n. In particular, we establish an upper bound for this sum and give necessary and sufficient conditions under which it is attained. We also prove that, given an absolutely continuous univariate probability distribution and ε>0, there are random variables X1,X2,…,Xn, each of which has this distribution, and for which each of the probabilities PrX1>X2,PrX2>X3,…,PrXn−1>Xn, and PrXn>X1 exceeds 1−ε.

Characterizations of Exponentiated Generalized Power Lindley and Geometric-Zero Truncated Poisson Distributions

MirMostafaee et al. (2019) proposed a continuous univariate distribution called Exponentiated Generalized Power Lindley (EGPL) distribution and studied certain properties and applications of their distribution. Akdogan et al. (2019) introduced a discrete distribution called Geometric-Zero Truncated Poisson (GZTP) distribution and provided its properties and applications. The present short note is intended to complete, in some way, the works cited above via establishing certain characterizations of the EGPL and GZTP distributions in different directions.

Fixed point characterizations of continuous univariate probability distributions and their applications

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.

On Characterizations of Four Recently Introduced Distributions: Two Continuous and Two Discrete

Oluyede et al. (2016) and Mdlongwa et al. (2017) consider the continuous univariate distributions called Dagum-Poisson (DP) and Burr XII Modified Weibull (BXIIMW), respectively, and study certain properties and applications of these distributions. Shahid and Raheel (2019) and Para and Jan (2019) proposed the univariate discrete distributions called Discrete Modified Inverse Rayleigh (DMIR) and Discrete Generalized Inverse Weibull (DGIW) and study some of their mathematical properties. The present short note is intended to complete, in some way, the works cited above via establishing certain characterizations of these distributions in different directions.

On Generating a New Family of Distributions Using the Tangent Function

In this paper, a method for generating a new family of univariate continuous distributions using the tangent function is proposed. Some general properties of this new family are discussed: hazard function, quantile function, Renyi and Shannon entropies, symmetry, and existence of the non-central n^th moment. Some new members as sub-families in the T-X family of distributions are provided. Three members of the new sub-families are defined and discussed: the five-parameter Normal-Generalized hyperbolic secant distribution (NGHS), the five-parameter Gumbel-Generalized hyperbolic secant distribution (GGHS), and the six-parameter Generalized Error-Generalized hyperbolic secant distribution (GEHS), the shapes of these distributions were found: skewed right, skewed left, or symmetric, and unimodal, bimodal, or trimodal. Finally, to demonstrate the usefulness and the capability of the distributions, two real data sets are used and the results are compared with other known distributions.

The Transmuted Odd Lindley-G Family of Distributions


 
 
 
 We propose a new generator of univariate continuous distributions with two extra parameters called the transmuted odd-Lindley generator which extends the odd Lindely-G family introduced by Gomes-Silva et al. [1]. Some mathematical properties of the new generator such as, the ordinary and incomplete moments, generating function, stress strength model, Rényi entropy, probability weighted moments and order statistics are investigated. Certain characterisations of the proposed family are estimated. We discuss the maximum likelihood estimates and the observed information matrix for the model parameters. The potentiality of the new family is illustrated by means of five applications to real data sets.
 
 
 
 

Maximum product spacings method for the estimation of parameters of linear regression

Maximum product of spacing (MPS) estimator, which is a general method for estimating parameters from observations with continuous univariate distributions, is considered as an alternative approach in linear regression modelling. We describe the basic idea of the maximum spacings estimator and apply to the linear regression problem. Moreover, we conduct a simulation and experiment study to make the comparison between MPS method and maximum likelihood estimator under various distribution assumptions. Finally, a real data set has been implemented to illustrate the performance of this estimator.

Characterizations and Infinite Divisibility of Certain Recently Introduced Distributions IV

Certain characterizations of recently proposed univariate continuous distributions are presented in different directions. This work contains a goodnumber of reintroduced distributions and may serve as a source of preventing the reinvention and/or duplication of the existing distributions in the future.

A new family of the continuous distributions: the extended Weibull-G family

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.

Characterizations of Kumaraswamy-Laplace, McDonald Inverse Weibull and New Generalized Exponential Distributions

Nassar (2016) considers an interesting univariate continuous distribution called Kumaraswamy-Laplace which has different forms on two subintervals. He studies certain properties and applications of this distribution. Shahbaz et al. (2016) consider another interesting distribution called McDonald Inverse Weibull distribution. They present some basic properties of their distribution and study the estimations of the parameters as well as discussing its application via an illustrative example. What is lacking in both papers, in our opinion, is the characterizations of these two interesting distributions. the present work is intended to complete, in some way, the works of Nassar and Shahbaz et al. via establishing certain characterizations of these distributions in four directions. We also introduce several New generalized Exponential distributions and present their characterizations as well.

Characterizations and Infinite Divisibility of Certain Recently Introduced Distributions III

Certain characterizations of recently proposed univariate continuous distributions are presented in different directions. This work may be a source of preventing reinventing and duplicating the existing distributions and calling them newly proposed distributions.

A Cramér-von Mises test-based distribution-free control chart for joint monitoring of location and scale

This paper proposes a new distribution-free control chart by integrating the powerful nonparametric two-sample Cramér-von Mises test and the exponentially weighted moving average control scheme to on-line monitoring. The proposed control chart can be used to monitor the location and the scale parameters of a univariate continuous distribution, simultaneously. The control limits based on Monte-Carlo simulation are provided in a table. The sensitivity analysis of effect of the number of reference samples on the control chart is studied in detail. Comparison results based on Monte-Carlo simulation show that the proposed chart is quite robust to non-normally distributed data, and moreover, it shows satisfactory performance in detecting various process shifts in terms of the average run length and standard deviation of run length. The application of our proposed chart is illustrated by a real data example for automobile engine piston rings.

MaxEnt Upper Bounds for the Differential Entropy of Univariate Continuous Distributions

We present a series of closed-form upper bounds of the differential entropy of univariate continuous distributions based on the maximum entropy principle. We apply those bounds to Gaussian mixture models, and study their tightness properties.

Distribution-free phase-II exponentially weighted moving average schemes for joint monitoring of location and scale based on subgroup samples

In this article, we introduce some distribution-free exponentially weighted moving average (EWMA) schemes for phase-II monitoring. Our proposed schemes are based on the well-known Lepage statistic and are referred to as the EWMA-Lepage (EL) procedures. These schemes are designed for monitoring the location and the scale parameters of an unknown but continuous univariate process at phase II. A novelty of these schemes is that a single chart can monitor both the location and scale parameters. Moreover, their distribution-free characteristic ensures that the in-control (denoted IC) properties of the proposed procedure remain invariant and known for all univariate continuous distributions. We discuss both the exact and the steady-state control limits. We obtain control limits under different situations for practical implementation of these schemes and examine the IC and out-of-control (denoted OOC) properties of the monitoring schemes through simulation studies in terms of the average, the standard deviation, the median, and some percentiles of the run length distributions. We illustrate the proposed procedure with two sets of data arriving out of manufacturing sector. We also offer some remarks and future research directions.

Distances between nested densities and a measure of the impact of the prior in Bayesian statistics

In this paper, we propose tight upper and lower bounds for the Wasserstein distance between any two univariate continuous distributions with probability densities $p_{1}$ and $p_{2}$ having nested supports. These explicit bounds are expressed in terms of the derivative of the likelihood ratio $p_{1}/p_{2}$ as well as the Stein kernel $\tau_{1}$ of $p_{1}$. The method of proof relies on a new variant of Stein’s method which manipulates Stein operators. We give several applications of these bounds. Our main application is in Bayesian statistics: we derive explicit data-driven bounds on the Wasserstein distance between the posterior distribution based on a given prior and the no-prior posterior based uniquely on the sampling distribution. This is the first finite sample result confirming the well-known fact that with well-identified parameters and large sample sizes, reasonable choices of prior distributions will have only minor effects on posterior inferences if the data are benign.

Characterizations and infinite divisibility of certain 2016-2017 univariate continuous distributions II

Characterizations and infinite divisibility of certain 2016-2017 univariate continuous distributions II

Characterizations and infinite divisibility of certain 2016 univariate continuous distributions

Characterizations and infinite divisibility of certain 2016 univariate continuous distributions