Uniform conditional stochastic order

1980 ◽  
Vol 17 (1) ◽  
pp. 112-123 ◽  
Author(s):  
Ward Whitt
Keyword(s):  
Probability Measure ◽  
Real Line ◽  
Standard Form ◽  
Stochastic Order ◽  
Probability Measures ◽  
Monotone Likelihood Ratio ◽  
Lifetime Distributions ◽  
Probability Mass ◽  
Mass Functions ◽  
Standard Probability

One probability measure is less than or equal to another in the sense of UCSO (uniform conditional stochastic order) if a standard form of stochastic order holds for each pair of conditional probability measures obtained by conditioning on appropriate subsets. UCSO can be applied to the comparison of lifetime distributions or the comparison of decisions under uncertainty when there may be reductions in the set of possible outcomes. When densities or probability mass functions exist on the real line, then the main version of UCSO is shown to be equivalent to the MLR (monotone likelihood ratio) property. UCSO is shown to be preserved by some standard probability operations and not by others.

Uniform conditional stochastic order

1980 ◽  
Vol 17 (01) ◽  
pp. 112-123 ◽  
Author(s):  
Ward Whitt
Keyword(s):  
Probability Measure ◽  
Real Line ◽  
Standard Form ◽  
Stochastic Order ◽  
Probability Measures ◽  
Monotone Likelihood Ratio ◽  
Lifetime Distributions ◽  
Probability Mass ◽  
Mass Functions ◽  
Standard Probability

One probability measure is less than or equal to another in the sense of UCSO (uniform conditional stochastic order) if a standard form of stochastic order holds for each pair of conditional probability measures obtained by conditioning on appropriate subsets. UCSO can be applied to the comparison of lifetime distributions or the comparison of decisions under uncertainty when there may be reductions in the set of possible outcomes. When densities or probability mass functions exist on the real line, then the main version of UCSO is shown to be equivalent to the MLR (monotone likelihood ratio) property. UCSO is shown to be preserved by some standard probability operations and not by others.

Probability measures with trivial Stam groups

1982 ◽  
Vol 91 (3) ◽  
pp. 477-484
Author(s):  
Gavin Brown ◽  
William Mohan
Keyword(s):  
Probability Measure ◽  
Real Number ◽  
Real Line ◽  
Probability Measures ◽  
The Real ◽  
Interesting Observation ◽  
Image Position ◽  
Positive Integers

Let μ be a probability measure on the real line ℝ, x a real number and δ(x) the probability atom concentrated at x. Stam made the interesting observation that eitheror else(ii) δ(x)* μn, are mutually singular for all positive integers n.

Multivariate monotone likelihood ratio and uniform conditional stochastic order

1982 ◽  
Vol 19 (3) ◽  
pp. 695-701 ◽  
Author(s):  
Ward Whitt
Keyword(s):  
Conditional Probability ◽  
Likelihood Ratio ◽  
Probability Distributions ◽  
Stochastic Order ◽  
Total Positivity ◽  
Probability Measures ◽  
Monotone Likelihood Ratio ◽  
Fkg Inequality ◽  
The Fkg Inequality

Karlin and Rinott (1980) introduced and investigated concepts of multivariate total positivity (TP2) and multivariate monotone likelihood ratio (MLR) for probability measures on Rn These TP and MLR concepts are intimately related to supermodularity as discussed in Topkis (1968), (1978) and the FKG inequality of Fortuin, Kasteleyn and Ginibre (1971). This note points out connections between these concepts and uniform conditional stochastic order (ucso) as defined in Whitt (1980). ucso holds for two probability distributions if there is ordinary stochastic order for the corresponding conditional probability distributions obtained by conditioning on subsets from a specified class. The appropriate subsets to condition on for ucso appear to be the sublattices of Rn. Then MLR implies ucso, with the two orderings being equivalent when at least one of the probability measures is TP2.

Multivariate monotone likelihood ratio and uniform conditional stochastic order

1982 ◽  
Vol 19 (03) ◽  
pp. 695-701 ◽  
Author(s):  
Ward Whitt
Keyword(s):  
Conditional Probability ◽  
Likelihood Ratio ◽  
Probability Distributions ◽  
Stochastic Order ◽  
Total Positivity ◽  
Probability Measures ◽  
Monotone Likelihood Ratio ◽  
Fkg Inequality ◽  
The Fkg Inequality

Karlin and Rinott (1980) introduced and investigated concepts of multivariate total positivity (TP2) and multivariate monotone likelihood ratio (MLR) for probability measures on Rn These TP and MLR concepts are intimately related to supermodularity as discussed in Topkis (1968), (1978) and the FKG inequality of Fortuin, Kasteleyn and Ginibre (1971). This note points out connections between these concepts and uniform conditional stochastic order (ucso) as defined in Whitt (1980). ucso holds for two probability distributions if there is ordinary stochastic order for the corresponding conditional probability distributions obtained by conditioning on subsets from a specified class. The appropriate subsets to condition on for ucso appear to be the sublattices of Rn . Then MLR implies ucso, with the two orderings being equivalent when at least one of the probability measures is TP2.

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 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.

scholarly journals Antisymmetry of the Stochastical Order on all Ordered Topological Spaces

2019 ◽  
Vol 7 (1) ◽  
pp. 250-252 ◽  
Author(s):  
Tobias Fritz
Keyword(s):  
Topological Space ◽  
General Result ◽  
Elementary Proof ◽  
Short Note ◽  
Stochastic Order ◽  
Topological Spaces ◽  
Probability Measures ◽  
Ordered Topological Space ◽  
Special Cases

Abstract In this short note, we prove that the stochastic order of Radon probability measures on any ordered topological space is antisymmetric. This has been known before in various special cases. We give a simple and elementary proof of the general result.

scholarly journals A note on the idempotent measures on countable semigroups

1970 ◽  
Vol 11 (4) ◽  
pp. 417-420
Author(s):  
Tze-Chien Sun ◽  
N. A. Tserpes
Keyword(s):  
Probability Measure ◽  
Continuous Mapping ◽  
Compact Semigroup ◽  
Probability Measures ◽  
Left Zero ◽  
Product Topology ◽  
Locally Compact ◽  
Locally Compact Semigroup ◽  
Image Position ◽  
Completely Simple

In [6] we announced the following Conjecture: Let S be a locally compact semigroup and let μ be an idempotent regular probability measure on S with support F. Then(a) F is a closed completely simple subsemigroup.(b) F is isomorphic both algebraically and topologically to a paragroup ([2], p.46) X × G × Y where X and Y are locally compact left-zero and right-zero semi-groups respectively and G is a compact group. In X × G × Y the topology is the product topology and the multiplication of any two elements is defined by , x where [y, x′] is continuous mapping from Y × X → G.(c) The induced μ on X × G × Y can be decomposed as a product measure μX × μG× μY where μX and μY are two regular probability measures on X and Y respectively and μG is the normed Haar measure on G.

Central Limit Theorems for Interchangeable Processes

1958 ◽  
Vol 10 ◽  
pp. 222-229 ◽  
Author(s):  
J. R. Blum ◽  
H. Chernoff ◽  
M. Rosenblatt ◽  
H. Teicher
Keyword(s):  
Stochastic Process ◽  
Probability Measure ◽  
Central Limit ◽  
Joint Distribution ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Probability Measures ◽  
Image Position ◽  
Positive Integers

Let {Xn} (n = 1, 2 , …) be a stochastic process. The random variables comprising it or the process itself will be said to be interchangeable if, for any choice of distinct positive integers i 1, i 2, H 3 … , ik, the joint distribution of depends merely on k and is independent of the integers i 1, i 2, … , i k. It was shown by De Finetti (3) that the probability measure for any interchangeable process is a mixture of probability measures of processes each consisting of independent and identically distributed random variables.

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.

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