Bounds for coverage probabilities with applications to sequential coverage problems

1974 ◽  
Vol 11 (2) ◽  
pp. 281-293 ◽  
Author(s):  
Peter J. Cooke
Keyword(s):  
Asymptotic Behaviour ◽  
Stopping Rules ◽  
Geometrical Probability ◽  
Coverage Probabilities ◽  
Coverage Problems

This paper discusses general bounds for coverage probabilities and moments of stopping rules for sequential coverage problems in geometrical probability. An approach to the study of the asymptotic behaviour of these moments is also presented.

Bounds for coverage probabilities with applications to sequential coverage problems

1974 ◽  
Vol 11 (02) ◽  
pp. 281-293 ◽  
Author(s):  
Peter J. Cooke
Keyword(s):  
Asymptotic Behaviour ◽  
Stopping Rules ◽  
Geometrical Probability ◽  
Coverage Probabilities ◽  
Coverage Problems

This paper discusses general bounds for coverage probabilities and moments of stopping rules for sequential coverage problems in geometrical probability. An approach to the study of the asymptotic behaviour of these moments is also presented.

scholarly journals Two-choice optimal stopping

2004 ◽  
Vol 36 (04) ◽  
pp. 1116-1147 ◽  
Author(s):  
David Assaf ◽  
Larry Goldstein ◽  
Ester Samuel-Cahn
Keyword(s):  
Distribution Function ◽  
Asymptotic Behaviour ◽  
Large Class ◽  
Optimal Stopping ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Indicator Function ◽  
Stopping Rules ◽  
A Value ◽  
Independent Identically Distributed

Let X n ,…,X 1 be independent, identically distributed (i.i.d.) random variables with distribution function F. A statistician, knowing F, observes the X values sequentially and is given two chances to choose Xs using stopping rules. The statistician's goal is to stop at a value of X as small as possible. Let equal the expectation of the smaller of the two values chosen by the statistician when proceeding optimally. We obtain the asymptotic behaviour of the sequence for a large class of Fs belonging to the domain of attraction (for the minimum) 𝒟(G α), where G α(x) = [1 - exp(-x α)]1(x ≥ 0) (with 1(·) the indicator function). The results are compared with those for the asymptotic behaviour of the classical one-choice value sequence , as well as with the ‘prophet value’ sequence

Random space filling and moments of coverage in geometrical probability

1978 ◽  
Vol 15 (2) ◽  
pp. 340-355 ◽  
Author(s):  
Andrew F. Siegel
Keyword(s):  
Coverage Probability ◽  
Arc Length ◽  
Random Set ◽  
Space Filling ◽  
Geometrical Probability ◽  
Almost Everywhere ◽  
Coverage Probabilities ◽  
Special Cases ◽  
Fixed Set ◽  
General Conditions

The moments of the random proportion of a fixed set that is covered by a random set (moments of coverage) are shown to converge under very general conditions to the probability that the fixed set is almost everywhere covered by the random set. Moments and coverage probabilities are calculated for several cases of random arcs of random sizes on the circle. When comparing arc length distributions having the same expectation, it is conjectured that if one concentrates more mass near that expectation, the corresponding coverage probability will be smaller. Support for this conjecture is provided in special cases.

scholarly journals Two-choice optimal stopping

2004 ◽  
Vol 36 (4) ◽  
pp. 1116-1147 ◽  
Author(s):  
David Assaf ◽  
Larry Goldstein ◽  
Ester Samuel-Cahn
Keyword(s):  
Distribution Function ◽  
Asymptotic Behaviour ◽  
Large Class ◽  
Optimal Stopping ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Indicator Function ◽  
Stopping Rules ◽  
A Value ◽  
Independent Identically Distributed

Let Xn,…,X1 be independent, identically distributed (i.i.d.) random variables with distribution function F. A statistician, knowing F, observes the X values sequentially and is given two chances to choose Xs using stopping rules. The statistician's goal is to stop at a value of X as small as possible. Let equal the expectation of the smaller of the two values chosen by the statistician when proceeding optimally. We obtain the asymptotic behaviour of the sequence for a large class of Fs belonging to the domain of attraction (for the minimum) 𝒟(Gα), where Gα(x) = [1 - exp(-xα)]1(x ≥ 0) (with 1(·) the indicator function). The results are compared with those for the asymptotic behaviour of the classical one-choice value sequence , as well as with the ‘prophet value’ sequence

Random space filling and moments of coverage in geometrical probability

1978 ◽  
Vol 15 (02) ◽  
pp. 340-355 ◽  
Author(s):  
Andrew F. Siegel
Keyword(s):  
Coverage Probability ◽  
Arc Length ◽  
Random Set ◽  
Space Filling ◽  
Geometrical Probability ◽  
Almost Everywhere ◽  
Coverage Probabilities ◽  
Special Cases ◽  
Fixed Set ◽  
General Conditions

The moments of the random proportion of a fixed set that is covered by a random set (moments of coverage) are shown to converge under very general conditions to the probability that the fixed set is almost everywhere covered by the random set. Moments and coverage probabilities are calculated for several cases of random arcs of random sizes on the circle. When comparing arc length distributions having the same expectation, it is conjectured that if one concentrates more mass near that expectation, the corresponding coverage probability will be smaller. Support for this conjecture is provided in special cases.

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