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Journal ArticleDOI

A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations

01 Dec 1952-Annals of Mathematical Statistics (Institute of Mathematical Statistics)-Vol. 23, Iss: 4, pp 493-507
TL;DR: In this paper, it was shown that the likelihood ratio test for fixed sample size can be reduced to this form, and that for large samples, a sample of size $n$ with the first test will give about the same probabilities of error as a sample with the second test.
Abstract: In many cases an optimum or computationally convenient test of a simple hypothesis $H_0$ against a simple alternative $H_1$ may be given in the following form. Reject $H_0$ if $S_n = \sum^n_{j=1} X_j \leqq k,$ where $X_1, X_2, \cdots, X_n$ are $n$ independent observations of a chance variable $X$ whose distribution depends on the true hypothesis and where $k$ is some appropriate number. In particular the likelihood ratio test for fixed sample size can be reduced to this form. It is shown that with each test of the above form there is associated an index $\rho$. If $\rho_1$ and $\rho_2$ are the indices corresponding to two alternative tests $e = \log \rho_1/\log \rho_2$ measures the relative efficiency of these tests in the following sense. For large samples, a sample of size $n$ with the first test will give about the same probabilities of error as a sample of size $en$ with the second test. To obtain the above result, use is made of the fact that $P(S_n \leqq na)$ behaves roughly like $m^n$ where $m$ is the minimum value assumed by the moment generating function of $X - a$. It is shown that if $H_0$ and $H_1$ specify probability distributions of $X$ which are very close to each other, one may approximate $\rho$ by assuming that $X$ is normally distributed.
Citations
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Journal ArticleDOI
TL;DR: A class of algorithms that use randomized signatures to compare remotely located file copies is presented and a comparison of these techniques is discussed.
Abstract: A class of algorithms that use randomized signatures to compare remotely located file copies is presented. A simple technique that sends on the order of 4/sup f/log(n) bits, where f is the number of differing pages that are to be diagnosed and n is the number of pages in the file, is described. A method to improve the bound in the number of bits sent, making them grow with f as flog(f) and with n as log(n)log(log(n)), and a class of algorithms in which the number of signatures grows with f as fr/sup f/, where r can be made to approach 1, are also presented. A comparison of these techniques is discussed. >

41 citations

Journal ArticleDOI
TL;DR: It is found that under a wide range of conditions the performance of the Apriori Algorithm is almost as bad as is permitted under sophisticated worst-case analyses.
Abstract: The failure rate of the Apriori Algorithm is studied analytically for the case of random shoppers. The time needed by the Apriori Algorithm is determined by the number of item sets that are output (successes: item sets that occur in at least k baskets) and the number of item sets that are counted but not output (failures: item sets where all subsets of the item set occur in at least k baskets but the full set occurs in less than k baskets). The number of successes is a property of the data; no algorithm that is required to output each success can avoid doing work associated with the successes. The number of failures is a property of both the algorithm and the data.We find that under a wide range of conditions the performance of the Apriori Algorithm is almost as bad as is permitted under sophisticated worst-case analyses. In particular, there is usually a bad level with two properties: (1) it is the level where nearly all of the work is done, and (2) nearly all item sets counted are failures. Let l be the...

41 citations


Cites background or methods from "A Measure of Asymptotic Efficiency ..."

  • ...They do not have closed forms (implied by [15]), but, as we show below, Chernoff techniques [8] lead to useful approximations....

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  • ...Chernoff [8] noticed that useful bounds for partial binomial sums result when one uses L(i) = x−k+i with x ≤ 1 and U(i) = x−k+i with x ≥ 1 (22)...

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Journal ArticleDOI
TL;DR: In this paper, a decision rule for selecting a maximum number of the Xi's subject to the following constraints: (1) the sum of the elements selected must not exceed a given constant c > 0, and (2) the items must be inspected in strict sequence with the decision to accept or reject an element being final at the time it is inspected.
Abstract: We model a selection process arising in certain storage problems. A sequence (X1, - - -, X,) of non-negative, independent and identically distributed random variables is given. F(x) denotes the common distribution of the Xi's. With F(x) given we seek a decision rule for selecting a maximum number of the Xi's subject to the following constraints: (1) the sum of the elements selected must not exceed a given constant c >0, and (2) the Xi's must be inspected in strict sequence with the decision to accept or reject an element being final at the time it is inspected. We prove first that there exists such a rule of threshold type, i.e. the ith element inspected is accepted if and only if it is no larger than a threshold which depends only on i and the sum of the elements already accepted. Next, we prove that if F(x) - Ax ' as x -- 0 for some A, a > 0, then for fixed c the expected number, En(c), selected by an optimal threshold is characterized by

41 citations


Additional excerpts

  • ...The proof of the general result is based chiefly on Chernoff estimates [5]....

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Proceedings ArticleDOI
01 Mar 1989
TL;DR: This work shows that several bounded-degree derivatives of the hypercube, including the deBruijn, ShuffleExchange, Butterfly, and Cube-Connected Cycles networks, have such a tolerance to processor faults with a slowdown factor doubly logarithmic in the size of the network.
Abstract: We present algbri~hms which reconfigure faulty deBruijn and Butterfly networks. We show that, with high probability, an N-node deBruijn (resp., Butterfly) network with uniformly distributed faulty processors can simulate a fault-free N-node deBrnijn (resp., Butterfly) network with a slowdown factor of O( loglogN). Our configuration algorithm is deterministic, operates in a distributed fashion, uses only local control, and takes time O (log 2 N) . 1. I N T R O D U C T I O N In any VLSI implementation of a large-scale computer network, some positive fraction of the processing elements are likely to contain faults. l~ecent work of Hastad, Leighton, and Newman [6] has shown that the hypercube has a strong fault-tolerance property: an N-node hypercube with a constant fraction of randomly distributed processor failures can still, with probability 1 O(1/N), simulate the computation of a fault-free N-node hypercube, with only a constant factor slowdown in total proPermission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to r-epublish, requires a fee and/or specific permission. © 1989 ACM 0-89791-323-X/89/0006/0179 $1.50 cessing time. However, it remained in question whether any bounded-degree networks shared this fault-tolerance property. Our results show that several bounded-degree derivatives of the hypercube, including the deBruijn, ShuffleExchange, Butterfly, and Cube-Connected Cycles networks, have such a tolerance to processor faults with a slowdown factor doubly logarithmic in the size of the network. The hypercube has been shown to have tremendous advantages as an interconnection network in terms of communication power and robustness. However, a large scale implementation of the hypercube is hampered by the fact that the node-degree 1 grows with the size of the network. 2 Networks of bounded degree have been proposed as "approximations" to the hypercube; these include the deBruijn, ShuffleExchange, Butterfly, and Cube-Connected Cycles networks. We call these approximations to the hypercube the HC-derivative networks. Indeed, in certain significant computational tasks these networks perform with efficiency equalling that of the hypercube. As in [6], we assume that we are presented with an N-node network in which processors have a constant probability p < 1 of failure, 1 T h e degree of a n e t w o r k is t h e m a x i m u m n u m b e r of c o m m u n i c a t i u n l inks (or edges ) i nc iden t to a node . 2Each node of an N n o d e h y p e r c u b e h a s degree log 2 N .

41 citations


Cites background from "A Measure of Asymptotic Efficiency ..."

  • ...H. Chernoff (1952): A measure of asymp- totic efficiency for testing of a hypothesis based on the sum of observations....

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Journal ArticleDOI
TL;DR: This review focuses on the use of robustness analysis methods for the validation or invalidation of models of biological systems, and provides a comprehensive overview of the tools and methods available for this task.
Abstract: Robustness, the ability of a system to function correctly in the presence of both internal and external uncertainty, has emerged as a key organising principle in many biological systems. Biological robustness has thus become a major focus of research in Systems Biology, particularly on the engineering-biology interface, since the concept of robustness was first rigorously defined in the context of engineering control systems. This review focuses on one particularly important aspect of robustness in Systems Biology, that is, the use of robustness analysis methods for the validation or invalidation of models of biological systems. With the explosive growth in quantitative modelling brought about by Systems Biology, the problem of validating, invalidating and discriminating between competing models of a biological system has become an increasingly important one. In this review, the authors provide a comprehensive overview of the tools and methods that are available for this task, and illustrate the wide range of biological systems to which this approach has been successfully applied.

41 citations

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