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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: It is concluded that (under the model) any asynchronous algorithm with good time complexity will also have good communication complexity, on the average, under a very general model of distributed computation.
Abstract: We study the communication complexity of asynchronous distributed algorithms. Such algorithms can generate excessively many messages in the worst case. Nevertheless, we show that, under certain probabilistic assumptions, the expected number of messages generated per time unit is bounded by a polynomial function of the number of processors under a very general model of distributed computation. Furthermore, for constant-degree processor graphs, the expected number of generated messages is only O(nT), where n is the number of processors and T is the running time. We conclude that (under our model) any asynchronous algorithm with good time complexity will also have good communication complexity, on the average.

29 citations

Book ChapterDOI
15 Aug 1999
TL;DR: In this paper, the authors considered the broadcast exclusion problem, where a message is transmitted over a broadcast channel shared by n = 2n users so that all but some specified coalition of k excluded users can understand the contents of the message.
Abstract: We consider the broadcast exclusion problem: how to transmit a message over a broadcast channel shared by N = 2n users so that all but some specified coalition of k excluded users can understand the contents of the message. Using error-correcting codes, and avoiding any computational assumptions in our constructions, we construct natural schemes that completely avoid any dependence on n in the transmission overhead. Specifically, we construct: (i) (for illustrative purposes,) a randomized scheme where the server's storage is exponential (in n), but the transmission overhead is O(k), and each user's storage is O(kn); (ii) a scheme based on polynomials where the transmission overhead is O(kn) and each user's storage is O(kn); and (iii) a scheme using algebraic-geometric codes where the transmission overhead is O(k2) and each user is required to store O(kn) keys. In the process of proving these results, we show how to construct very good cover-free set systems and combinatorial designs based on algebraic-geometric codes, which may be of independent interest and application. Our approach also naturally extends to solve the problem in the case where the broadcast channel may introduce errors or lose information.

29 citations

Journal ArticleDOI
TL;DR: Three rejection-based algorithms for the generation of a nonuniform discrete random variate are presented, and it is shown that, under fairly unrestrictive conditions, the long-run expected effort is O(1).
Abstract: One of the most fundamental operations when simulating a stochastic discrete-event dynamic system is the generation of a nonuniform discrete random variate. The simplest form of this operation can be stated as follows: Generate a random variable X that is distributed over the integers 1,2,…,n such that P(X=i) = ai/(a1 +…+an), where ai's are fixed nonnegative numbers. The well-known “alias algorithm” is available to accomplish this task in O(1) time. A more difficult problem is to generate variates for X when the ai's are changing with time. We present three rejection-based algorithms for this task, and for each algorithm we characterize the performance in terms of acceptance probability and the expected effort to generate a variate. We show that, under fairly unrestrictive conditions, the long-run expected effort is O(1). Applications to Markovian queuing networks are discussed. We also compare the three algorithms with competing schemes appearing in the literature.

29 citations

Journal ArticleDOI
TL;DR: In this article, the authors consider spherical regression with two sets of unit-length vectors when the set of vectors is the same size as the input vectors, and apply it to data integration, language translation, bioinformatics, and computer vision.
Abstract: Motivated by a series of applications in data integration, language translation, bioinformatics, and computer vision, we consider spherical regression with two sets of unit-length vectors when the ...

29 citations

References
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