On Excess Over the Boundary

Lorden, G.

doi:10.1214/aoms/1177697092

Cited by 190 publications

(93 citation statements)

References 5 publications

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“…√ log a . Using Theorem 4 of Lorden [14], for some K > 0, P X τ − a ≥ ǫa (1−δ)/2 ≤ KE X; X ≥ ǫa Similarly, applying Theorem 1.1 to two dimensional random walks where the second coordinates are projections of W n , for some K > 0,…”

mentioning

confidence: 99%

Multivariate sequential analysis with linear boundaries

Keener¹

2006

Institute of Mathematical Statistics Lecture Notes - Monograph Series

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Let {Sn = (Xn, Wn)} n≥0 be a random walk with Xn ∈ R and Wn ∈ R m . Let τ = τa = inf{n : Xn > a}. The main results presented are two term asymptotic expansions for the joint distribution of Sτ and τ and the marginal distribution of h(Sτ /a, τ /a) in the limit a → ∞. These results are used to study the distribution of t-statistics in sequential experiments with sample size τ , and to remove bias from confidence intervals based on Anscombe's theorem.

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mentioning

confidence: 99%

Multivariate sequential analysis with linear boundaries

Keener¹

2006

Institute of Mathematical Statistics Lecture Notes - Monograph Series

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“…The paper by J. T. Chang [1] established moment inequalities and almost sure upper bounds for the residual lifetime of a usual renewal process. This paper also contains bounds for the tail P{ζ t > x} and, in general, improves the results obtained earlier by G. Lorden [2]. Fundamental works by C. D. Fuh and T. L. Lai [3,4] deal with Markov random walks.…”

Section: Set (3)supporting

confidence: 76%

“…for p > 0 (see [2], [1, p. 1224]). Therefore these estimates also hold for the process generated by the sequence {X n }.…”

Section: The Variance Of the Counting Processmentioning

confidence: 99%

The counting process and summation of a random number of random variables

Sugakova

Theor. Probability and Math. Statist.

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Abstract. The behavior of the tail of the sum of a random number of random variablesEstimates of the convergence of F (x) to the limit function are constructed in terms of renewal theory. The estimates are based on the variance Var ν(t) of the counting process ν(t) = min n :A survey of bounds for Var ν(t) is given for different sequences {ξ i }, in particular, for the case where the terms of the sequence {ξ i } are not identically distributed.

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“…Due to Lorden's inequality [31], we have 13 Assuming that the first two moments of log-likelihood ration are finite, then we have…”

Section: Appendix a Proof Of One Target Detectionmentioning

confidence: 99%

Active hypothesis testing on a tree: Anomaly detection under hierarchical observations

Wang

Cohen

Zhao

2017

2017 IEEE International Symposium on Information Theory (ISIT)

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Abstract-The problem of detecting a few anomalous processes among a large number of data streams is considered. At each time, aggregated observations can be taken from a chosen subset of processes, where the chosen subset conforms to a given binary tree structure. The random observations are drawn from a general distribution that may depend on the size of the chosen subset and the number of anomalous processes in the subset. We propose a sequential search strategy by devising an informationdirected random walk (IRW) on the tree-structured observation hierarchy. The sample complexity of the IRW policy is shown to be asymptotically optimal in terms of detection accuracy and order optimal in terms of the number of data streams.

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On Excess Over the Boundary

Cited by 190 publications

References 5 publications

Multivariate sequential analysis with linear boundaries

Multivariate sequential analysis with linear boundaries

The counting process and summation of a random number of random variables

Active hypothesis testing on a tree: Anomaly detection under hierarchical observations

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