Estimating the reduced moments of a random measure

1996 ◽  
Vol 28 (02) ◽  
pp. 335-336
Author(s):  
Kiên Kiêu ◽  
Marianne Mora
Keyword(s):  
Transformation Group ◽  
Smooth Boundary ◽  
Random Measure ◽  
Random Sets ◽  
Random Set ◽  
Counting Measure ◽  
Geometrical Properties ◽  
Measure Decomposition ◽  
Moment Measures ◽  
Curvature Measures

Random measures are commonly used to describe geometrical properties of random sets. Examples are given by the counting measure associated with a point process, and the curvature measures associated with a random set with a smooth boundary. We consider a random measure with an invariant distribution under the action of a standard transformation group (translatioris, rigid motions, translations along a given direction and so on). In the framework of the theory of invariant measure decomposition, the reduced moments of the random measure are obtained by decomposing the related moment measures.

Estimating the reduced moments of a random measure

1996 ◽  
Vol 28 (2) ◽  
pp. 335-336 ◽  
Author(s):  
Kiên Kiêu ◽  
Marianne Mora
Keyword(s):  
Transformation Group ◽  
Smooth Boundary ◽  
Random Measure ◽  
Random Sets ◽  
Random Set ◽  
Counting Measure ◽  
Geometrical Properties ◽  
Measure Decomposition ◽  
Moment Measures ◽  
Curvature Measures

Random measures are commonly used to describe geometrical properties of random sets. Examples are given by the counting measure associated with a point process, and the curvature measures associated with a random set with a smooth boundary. We consider a random measure with an invariant distribution under the action of a standard transformation group (translatioris, rigid motions, translations along a given direction and so on). In the framework of the theory of invariant measure decomposition, the reduced moments of the random measure are obtained by decomposing the related moment measures.

Estimating the reduced moments of a random measure

1999 ◽  
Vol 31 (1) ◽  
pp. 48-62 ◽  
Author(s):  
Kiên Kiêu ◽  
Marianne Mora
Keyword(s):  
Invariant Measure ◽  
Stochastic Geometry ◽  
Transformation Group ◽  
Random Measure ◽  
Mixing Condition ◽  
Asymptotic Results ◽  
Unbiased Estimators ◽  
Measure Decomposition ◽  
General Method

We consider a random measure for which distribution is invariant under the action of a standard transformation group. The reduced moments are defined by applying classical theorems on invariant measure decomposition. We present a general method for constructing unbiased estimators of reduced moments. Several asymptotic results are established under an extension of the Brillinger mixing condition. Examples related to stochastic geometry are given.

Estimating the reduced moments of a random measure

1999 ◽  
Vol 31 (01) ◽  
pp. 48-62 ◽  
Author(s):  
Kiên Kiêu ◽  
Marianne Mora
Keyword(s):  
Invariant Measure ◽  
Stochastic Geometry ◽  
Transformation Group ◽  
Random Measure ◽  
Mixing Condition ◽  
Asymptotic Results ◽  
Unbiased Estimators ◽  
Measure Decomposition ◽  
General Method

We consider a random measure for which distribution is invariant under the action of a standard transformation group. The reduced moments are defined by applying classical theorems on invariant measure decomposition. We present a general method for constructing unbiased estimators of reduced moments. Several asymptotic results are established under an extension of the Brillinger mixing condition. Examples related to stochastic geometry are given.

MULTIVARIATE MODELS AND VARIABILITY INTERVALS: A LOCAL RANDOM SET APPROACH

2011 ◽  
Vol 19 (05) ◽  
pp. 799-823 ◽  
Author(s):  
THOMAS FETZ
Keyword(s):  
Confidence Level ◽  
Structural Mechanics ◽  
Random Sets ◽  
Confidence Limits ◽  
Random Set ◽  
Multivariate Functions ◽  
Multivariate Models ◽  
Multivariate Case ◽  
Fréchet Bounds

This article is devoted to the propagation of families of variability intervals through multivariate functions comprising the semantics of confidence limits. At fixed confidence level, local random sets are defined whose aggregation admits the calculation of upper probabilities of events. In the multivariate case, a number of ways of combination is highlighted to encompass independence and unknown interaction using random set independence and Fréchet bounds. For all cases we derive formulas for the corresponding upper probabilities and elaborate how they relate. An example from structural mechanics is used to exemplify the method.

scholarly journals Stationary countable dense random sets

2000 ◽  
Vol 32 (01) ◽  
pp. 86-100 ◽  
Author(s):  
Wilfrid S. Kendall
Keyword(s):  
Probability Theory ◽  
Polish Space ◽  
Measurable Subset ◽  
Locally Compact Group ◽  
Random Sets ◽  
Random Set ◽  
Locally Compact ◽  
Polish Spaces ◽  
Probability Zero ◽  
Group Of Symmetries

We study the probability theory of countable dense random subsets of (uncountably infinite) Polish spaces. It is shown that if such a set is stationary with respect to a transitive (locally compact) group of symmetries then any event which concerns the random set itself (rather than accidental details of its construction) must have probability zero or one. Indeed the result requires only quasi-stationarity (null-events stay null under the group action). In passing, it is noted that the property of being countable does not correspond to a measurable subset of the space of subsets of an uncountably infinite Polish space.

Law of small numbers as concentration inequalities for sums of independent random setsand random set valued mappings

2012 ◽  
Author(s):  
Vladimir I. Norkin ◽  
Roger J-B Wets
Keyword(s):  
Hilbert Space ◽  
Discrete Distributions ◽  
Random Sets ◽  
Concentration Inequalities ◽  
Random Set ◽  
Infinite Dimensional ◽  
Random Functions ◽  
The Monte Carlo Method ◽  
Exponential Bounds ◽  
Set Valued Mappings

In the paper we study concentration of sample averages (Minkowski's sums) of independent bounded random sets and set valued mappings around their expectations. Sets and mappings are considered in a Hilbert space. Concentration is formulated in the form of exponential bounds on probabilities of normalized large deviations. In a sense, concentration phenomenon reflects the law of small numbers, describing non-asymptotic behavior of the sample averages. We sequentially consider concentration inequalities for bounded random variables, functions, vectors, sets and mappings, deriving next inequalities from preceding cases. Thus we derive concentration inequalities with explicit constants for random sets and mappings from the sharpest available (Talagrand type) inequalities for random functions and vectors. The most explicit inequalities are obtained in case of discrete distributions. The obtained results contribute to substantiation of the Monte Carlo method in infinite dimensional spaces.

Curvature Measures and Random Sets, I

1984 ◽  
Vol 119 (1) ◽  
pp. 327-339 ◽  
Author(s):  
M. Zähle
Keyword(s):  
Random Sets ◽  
Curvature Measures

scholarly journals Random set and coverage measure

1991 ◽  
Vol 23 (04) ◽  
pp. 972-974 ◽  
Author(s):  
Guillermo Ayala ◽  
Juan Ferrandiz ◽  
Francisco Montes
Keyword(s):  
Random Measure ◽  
Sufficient Condition ◽  
Random Set ◽  
Necessary And Sufficient Condition ◽  
Equivalent Formulation ◽  
Coverage Measure ◽  
Necessary And Sufficient ◽  
Reverse Implication

It is well known that a random set determines its random coverage measure. The paper gives a necessary and sufficient condition for the reverse implication. An equivalent formulation of the condition constitutes a first step in the search for a way to recognize a random measure as being the random coverage measure of a random set.

Computation of Reliability with Both Aleatory and Epistemic Uncertainty Based on Random Sets Theory

2010 ◽  
Vol 29-32 ◽  
pp. 1252-1257 ◽  
Author(s):  
Hui Xin Guo ◽  
Lei Chen ◽  
Hang Min He ◽  
Dai Yong Lin
Keyword(s):  
Epistemic Uncertainty ◽  
Limit State ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Random Sets ◽  
Extension Principle ◽  
State Function ◽  
Random Set ◽  
Cumulative Distribution Functions ◽  
Aleatory And Epistemic Uncertainty

A method to handle hybrid uncertainties including aleatory and epistemic uncertainty is proposed for the computation of reliability. The aleatory uncertainty is modeled as random variable and the epistemic uncertainty is modeled with evidence theory. The two types of uncertainty are firstly transformed into random set, and the limit-state function of a product is mapped into a random set by using the extension principle of random set. Then, the belief function and the plausibility function of safety event are determined. The two functions are viewed as the lower and the upper cumulative distribution functions of reliability, respectively. The reliability of a product will be bounded by two cumulative distribution functions, and then an interval estimation of reliability can be obtained. The proposed method is demonstrated with an example.

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