scholarly journals Two Useful Discrete Distributions to Model Overdispersed Count Data

2020 ◽  
Vol 43 (1) ◽  
pp. 21-48
Author(s):  
Josmar Mazucheli ◽  
Wesley Bertoli ◽  
Ricardo Puziol Oliveira
Keyword(s):  
Infinite Series ◽  
Count Data ◽  
Survival Function ◽  
Continuous Distribution ◽  
Discrete Distributions ◽  
Mathematical Expressions ◽  
Probability Mass ◽  
The Difference ◽  
Discrete Analogues ◽  
Mass Functions

The methods to obtain discrete analogues of continuous distributions have been widely considered in recent years. In general, the discretization process provides probability mass functions that can be competitive with the traditional model used in the analysis of count data, the Poisson distribution. The discretization procedure also avoids the use of continuous distribution in the analysis of strictly discrete data. In this paper, we seek to introduce two discrete analogues for the Shanker distribution using the method of the infinite series and the method based on the survival function as alternatives to model overdispersed datasets. Despite the difference between discretization methods, the resulting distributions are interchangeable. However, the distribution generated by the method of infinite series method has simpler mathematical expressions for the shape, the generating functions and the central moments. The maximum likelihood theory is considered for estimation and asymptotic inference concerns. A simulation study is carried out in order to evaluate some frequentist properties of the developed methodology. The usefulness of the proposed models is evaluated using real datasets provided by the literature.

A Novelty in Blahut-Arimoto T ype Algorithms: Optimal Control over Noisy Commu nication Channels

2020 ◽  
Author(s):  
Makan Zamanipour
Keyword(s):  
Optimal Control ◽  
Channel Coding ◽  
Joint Source Channel Coding ◽  
Mathematical Expressions ◽  
Type Algorithm ◽  
Theoretic Problem ◽  
Probability Mass ◽  
Information Constraints ◽  
First Time ◽  
Mass Functions

A probability-theoretic problem under information constraints for the concept of optimal control over a noisy-memoryless channel is considered. For our \textit{Observer-Controller} block, i.e., the lossy joint-source-channel-coding (JSCC) scheme, after providing the relative mathematical expressions, we propose a \textit{Blahut-Arimoto}-type algorithm $-$ which is, to the best of our knowledge, for the first time. The algorithm efficiently finds the probability-mass-functions (PMFs) required for .......................................

scholarly journals On a stochastic order induced by an extension of Panjer’s family of discrete distributions

Metrika ◽  
2021 ◽  
Author(s):  
Aleksandr Beknazaryan ◽  
Peter Adamic
Keyword(s):  
Recursion Formula ◽  
Stochastic Order ◽  
Reliability Theory ◽  
Discrete Distributions ◽  
Compound Distributions ◽  
Space Of Distributions ◽  
Probability Mass ◽  
Mass Functions

AbstractWe factorize probability mass functions of discrete distributions belonging to Panjer’s family and to its certain extensions to define a stochastic order on the space of distributions supported on $${\mathbb {N}}_0$$ N 0 . Main properties of this order are presented. Comparison of some well-known distributions with respect to this order allows to generate new families of distributions that satisfy various recurrrence relations. The recursion formula for the probabilities of corresponding compound distributions for one such family is derived. Applications to various domains of reliability theory are provided.

scholarly journals A Novelty in Blahut-Arimoto T ype Algorithms: Optimal Control over Noisy Commu nication Channels

2020 ◽  
Author(s):  
Makan Zamanipour
Keyword(s):  
Optimal Control ◽  
Channel Coding ◽  
Joint Source Channel Coding ◽  
Mathematical Expressions ◽  
Type Algorithm ◽  
Theoretic Problem ◽  
Probability Mass ◽  
Information Constraints ◽  
First Time ◽  
Mass Functions

A probability-theoretic problem under information constraints for the concept of optimal control over a noisy-memoryless channel is considered. For our \textit{Observer-Controller} block, i.e., the lossy joint-source-channel-coding (JSCC) scheme, after providing the relative mathematical expressions, we propose a \textit{Blahut-Arimoto}-type algorithm $-$ which is, to the best of our knowledge, for the first time. The algorithm efficiently finds the probability-mass-functions (PMFs) required for .......................................

scholarly journals Stochastic methods defeat regular RSA exponentiation algorithms with combined blinding methods

2021 ◽  
Vol 15 (1) ◽  
pp. 408-433
Author(s):  
Margaux Dugardin ◽  
Werner Schindler ◽  
Sylvain Guilley
Keyword(s):  
State Of The Art ◽  
Stochastic Methods ◽  
Rsa Algorithm ◽  
Montgomery Ladder ◽  
Channel Information ◽  
Probability Mass ◽  
The Cost ◽  
Noisy Measurements ◽  
Mass Functions ◽  
Larger Sample

Abstract Extra-reductions occurring in Montgomery multiplications disclose side-channel information which can be exploited even in stringent contexts. In this article, we derive stochastic attacks to defeat Rivest-Shamir-Adleman (RSA) with Montgomery ladder regular exponentiation coupled with base blinding. Namely, we leverage on precharacterized multivariate probability mass functions of extra-reductions between pairs of (multiplication, square) in one iteration of the RSA algorithm and that of the next one(s) to build a maximum likelihood distinguisher. The efficiency of our attack (in terms of required traces) is more than double compared to the state-of-the-art. In addition to this result, we also apply our method to the case of regular exponentiation, base blinding, and modulus blinding. Quite surprisingly, modulus blinding does not make our attack impossible, and so even for large sizes of the modulus randomizing element. At the cost of larger sample sizes our attacks tolerate noisy measurements. Fortunately, effective countermeasures exist.

A Critique and Improvement of the CL Common Language Effect Size Statistics of McGraw and Wong

2000 ◽  
Vol 25 (2) ◽  
pp. 101-132 ◽  
Author(s):  
András Vargha ◽  
Harold D. Delaney
Keyword(s):  
Effect Size ◽  
Continuous Distribution ◽  
Interval Estimation ◽  
Continuous Variable ◽  
Common Language ◽  
Exact Methods ◽  
Normal Distributions ◽  
The Difference ◽  
Discrete Values ◽  
Two Populations

McGraw and Wong (1992) described an appealing index of effect size, called CL, which measures the difference between two populations in terms of the probability that a score sampled at random from the first population will be greater than a score sampled at random from the second. McGraw and Wong introduced this "common language effect size statistic" for normal distributions and then proposed an approximate estimation for any continuous distribution. In addition, they generalized CL to the n-group case, the correlated samples case, and the discrete values case. In the current paper a different generalization of CL, called the A measure of stochastic superiority, is proposed, which may be directly applied for any discrete or continuous variable that is at least ordinally scaled. Exact methods for point and interval estimation as well as the significance tests of the A = .5 hypothesis are provided. New generalizations ofCL are provided for the multi-group and correlated samples cases.

scholarly journals On the control of the difference between two Brownian motions: a dynamic copula approach

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
Thomas Deschatre
Keyword(s):  
Cumulative Distribution Function ◽  
Survival Function ◽  
Random Variable ◽  
Cumulative Distribution ◽  
Gaussian Copula ◽  
Brownian Motions ◽  
Dynamic Copula ◽  
Copula Approach ◽  
The Difference ◽  
The Right

AbstractWe propose new copulae to model the dependence between two Brownian motions and to control the distribution of their difference. Our approach is based on the copula between the Brownian motion and its reflection. We show that the class of admissible copulae for the Brownian motions are not limited to the class of Gaussian copulae and that it also contains asymmetric copulae. These copulae allow for the survival function of the difference between two Brownian motions to have higher value in the right tail than in the Gaussian copula case. Considering two Brownian motions B1t and B2t, the main result is that the range of possible values for is the same for Markovian pairs and all pairs of Brownian motions, that is with φ being the cumulative distribution function of a standard Gaussian random variable.

scholarly journals Scholarly Book Review on Bayesian Statistics for Beginners: A Step-by-Step Approach

2020 ◽  
pp. 1-4
Author(s):  
Eahsan Shahriary ◽  
Amir Hajibabaee
Keyword(s):  
Bayesian Statistics ◽  
Open Source Software ◽  
Probability Density Functions ◽  
Bayesian Probability ◽  
Density Functions ◽  
Scholarly Book ◽  
Real World Applications ◽  
Probability Mass ◽  
Mass Functions

This book offers the students and researchers a unique introduction to Bayesian statistics. Authors provide a wonderful journey in the realm of Bayesian Probability and aspire readers to become Bayesian statisticians. The book starts with Introduction to Probability and covers Bayes’ Theorem, Probability Mass Functions, Probability Density Functions, The Beta-Binomial Conjugate, Markov chain Monte Carlo (MCMC), and Metropolis-Hastings Algorithm. The book is very well written, and topics are very to the point with real-world applications but does not provide examples for computing using common open-source software.

scholarly journals Distributions You Can Count On …But What’s the Point?

Econometrics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 9 ◽  
Author(s):  
Brendan P. M. McCabe ◽  
Christopher L. Skeels
Keyword(s):  
Count Data ◽  
Negative Binomial ◽  
Broad Class ◽  
Poisson Model ◽  
Econometric Analysis ◽  
Discrete Distributions ◽  
Optimal Test ◽  
Score Tests ◽  
Optimal Tests ◽  
Over Dispersion

The Poisson regression model remains an important tool in the econometric analysis of count data. In a pioneering contribution to the econometric analysis of such models, Lung-Fei Lee presented a specification test for a Poisson model against a broad class of discrete distributions sometimes called the Katz family. Two members of this alternative class are the binomial and negative binomial distributions, which are commonly used with count data to allow for under- and over-dispersion, respectively. In this paper we explore the structure of other distributions within the class and their suitability as alternatives to the Poisson model. Potential difficulties with the Katz likelihood leads us to investigate a class of point optimal tests of the Poisson assumption against the alternative of over-dispersion in both the regression and intercept only cases. In a simulation study, we compare score tests of ‘Poisson-ness’ with various point optimal tests, based on the Katz family, and conclude that it is possible to choose a point optimal test which is better in the intercept only case, although the nuisance parameters arising in the regression case are problematic. One possible cause is poor choice of the point at which to optimize. Consequently, we explore the use of Hellinger distance to aid this choice. Ultimately we conclude that score tests remain the most practical approach to testing for over-dispersion in this context.

Concentration inequalities for the empirical distribution of discrete distributions: beyond the method of types

2019 ◽  
Vol 9 (4) ◽  
pp. 813-850 ◽  
Author(s):  
Jay Mardia ◽  
Jiantao Jiao ◽  
Ervin Tánczos ◽  
Robert D Nowak ◽  
Tsachy Weissman
Keyword(s):  
Sample Size ◽  
Empirical Distribution ◽  
Discrete Distributions ◽  
Concentration Inequalities ◽  
Sample Sizes ◽  
Alphabet Size ◽  
Large Sample ◽  
Kl Divergence ◽  
The Difference ◽  
True Distribution

Abstract We study concentration inequalities for the Kullback–Leibler (KL) divergence between the empirical distribution and the true distribution. Applying a recursion technique, we improve over the method of types bound uniformly in all regimes of sample size $n$ and alphabet size $k$, and the improvement becomes more significant when $k$ is large. We discuss the applications of our results in obtaining tighter concentration inequalities for $L_1$ deviations of the empirical distribution from the true distribution, and the difference between concentration around the expectation or zero. We also obtain asymptotically tight bounds on the variance of the KL divergence between the empirical and true distribution, and demonstrate their quantitatively different behaviours between small and large sample sizes compared to the alphabet size.

One-Dimensional Representations of the Cycle Subalgebra of a Semi-Simple Lie Algebra

1970 ◽  
Vol 13 (4) ◽  
pp. 463-467 ◽  
Author(s):  
F. W. Lemire
Keyword(s):  
Lie Algebra ◽  
Cartan Subalgebra ◽  
Mass Function ◽  
Algebra Homomorphism ◽  
Closed Field ◽  
Noncommutative Algebra ◽  
One Dimensional ◽  
Finite Dimensional ◽  
The Difference ◽  
Mass Functions

Let L denote a semi-simple, finite dimensional Lie algebra over an algebraically closed field K of characteristic zero. If denotes a Cartan subalgebra of L and denotes the centralizer of in the universal enveloping algebra U of L, then it has been shown that each algebra homomorphism (called a "mass-function" on ) uniquely determines a linear irreducible representation of L. The technique involved in this construction is analogous to the Harish-Chandra construction [2] of dominated irreducible representations of L starting from a linear functional . The difference between the two results lies in the fact that all linear functionals on are readily obtained, whereas since is in general a noncommutative algebra the construction of mass-functions is decidedly nontrivial.

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