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.

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.

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.

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.

scholarly journals Stochastic Order and Generalized Weighted Mean Invariance

Entropy ◽  
2021 ◽  
Vol 23 (6) ◽  
pp. 662
Author(s):  
Mateu Sbert ◽  
Jordi Poch ◽  
Shuning Chen ◽  
Víctor Elvira
Keyword(s):  
Monte Carlo ◽  
Probability Distributions ◽  
Stochastic Order ◽  
Arithmetic Mean ◽  
Stochastic Orders ◽  
Arithmetic Means ◽  
Increasing Convex Order ◽  
Monte Carlo Techniques ◽  
Multiple Importance Sampling ◽  
Invariance Properties

In this paper, we present order invariance theoretical results for weighted quasi-arithmetic means of a monotonic series of numbers. The quasi-arithmetic mean, or Kolmogorov–Nagumo mean, generalizes the classical mean and appears in many disciplines, from information theory to physics, from economics to traffic flow. Stochastic orders are defined on weights (or equivalently, discrete probability distributions). They were introduced to study risk in economics and decision theory, and recently have found utility in Monte Carlo techniques and in image processing. We show in this paper that, if two distributions of weights are ordered under first stochastic order, then for any monotonic series of numbers their weighted quasi-arithmetic means share the same order. This means for instance that arithmetic and harmonic mean for two different distributions of weights always have to be aligned if the weights are stochastically ordered, this is, either both means increase or both decrease. We explore the invariance properties when convex (concave) functions define both the quasi-arithmetic mean and the series of numbers, we show its relationship with increasing concave order and increasing convex order, and we observe the important role played by a new defined mirror property of stochastic orders. We also give some applications to entropy and cross-entropy and present an example of multiple importance sampling Monte Carlo technique that illustrates the usefulness and transversality of our approach. Invariance theorems are useful when a system is represented by a set of quasi-arithmetic means and we want to change the distribution of weights so that all means evolve in the same direction.

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 Multiperiod conditional distribution functions for conditionally normal GARCH(1, 1) models

2005 ◽  
Vol 42 (02) ◽  
pp. 426-445
Author(s):  
Raymond Brummelhuis ◽  
Dominique Guégan
Keyword(s):  
Conditional Probability ◽  
Conditional Distribution ◽  
Probability Distributions ◽  
Random Variables ◽  
Distribution Functions ◽  
Tail Behavior ◽  
Asymptotic Tail

We study the asymptotic tail behavior of the conditional probability distributions of r t+k and r t+1+⋯+r t+k when (r t ) t∈ℕ is a GARCH(1, 1) process. As an application, we examine the relation between the extreme lower quantiles of these random variables.

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