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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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01 Jan 1961

173 citations


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

  • ...In a number of problems of varying degrees of generality the following procedure has been found to yield asymptotically optimal solutions [6], [7]....

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Journal ArticleDOI
TL;DR: The use of randomized algorithms to solve some problems in control system designs that are perceived to be "difficult" is presented to show that the randomized approach can be quite successful in tackling a practical problem.
Abstract: The topic of the present article is the use of randomized algorithms to solve some problems in control system designs that are perceived to be "difficult". A brief introduction is given to the notions of computational complexity that are pertinent to the present discussion, and then some problems in control system analysis and synthesis that are difficult in a complexity-theoretic sense are described. Some of the elements of statistical learning theory, which forms the basis of the randomized approach, are briefly described. Finally, these two sets of ideas are brought together to show that it is possible to construct efficient randomized algorithms for each of the difficult problems discussed by using the ideas of statistical learning theory. A real-life design example of synthesizing a first-order controller for the longitudinal stabilization of an unstable fighter aircraft is then presented to show that the randomized approach can be quite successful in tackling a practical problem.

173 citations


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

  • ...The best result available at present is the so-called Chernoff bound [ 5 ] which states that this probability is no larger than 2e-2mEz, This means that, after tossing the coin m times, we can state with a confidence of at least 1 - 2e-ZmE2 that the empirical probability p, is no more than E different from the true but unknown probability p In fact, it is possible to “split” the bound into two parts....

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Journal ArticleDOI
TL;DR: The fault-tolerance properties of quorum systems, and their implications on quorum based distributed protocols, are studied.
Abstract: A quorum system is a collection of sets (quorums) every two of which have a nonempty intersection. Quorum systems have been used for a number of applications in the area of distributed systems. In this paper we study the fault-tolerance properties of quorum systems, and their implications on quorum based distributed protocols.

172 citations

Journal ArticleDOI
TL;DR: In the paper, randomized algorithms for stability and performance of linear time invariant uncertain systems described by a general M-/spl Delta/ configuration are studied and efficient polynomial-time algorithms for uncertainty structures /splDelta/ consisting of an arbitrary number of full complex blocks and uncertain parameters are developed.
Abstract: There has been a growing interest in developing randomized algorithms for probabilistic robustness of uncertain control systems. Unlike classical worst case methods, these algorithms provide probabilistic estimates assessing, for instance, if a certain design specification is met with a given probability. One of the advantages of this approach is that the robustness margins can be often increased by a considerable amount, at the expense of a small risk. In this sense, randomized algorithms may be used by the control engineer together with standard worst case methods to obtain additional useful information. The applicability of these probabilistic methods to robust control is presently limited by the fact that the sample generation is feasible only in very special cases which include systems affected by real parametric uncertainty bounded in rectangles or spheres. Sampling in more general uncertainty sets is generally performed through overbounding, at the expense of an exponential rejection rate. In the paper, randomized algorithms for stability and performance of linear time invariant uncertain systems described by a general M-/spl Delta/ configuration are studied. In particular, efficient polynomial-time algorithms for uncertainty structures /spl Delta/ consisting of an arbitrary number of full complex blocks and uncertain parameters are developed.

171 citations


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

  • ...We recall that the Chernoff bound [14] gives the sample size so that the empirical probability is “close” to with accuracy and confidence....

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Journal ArticleDOI
TL;DR: This article studies the differential-geometrical structure of a manifold induced by a divergence function, which consists of a Riemannian metric, and a pair of dually coupled affine connections, which are studied in information geometry.
Abstract: Measures of divergence between two points play a key role in many engineering problems. One such measure is a distance function, but there are many important measures which do not satisfy the properties of the distance. The Bregman divergence, KullbackLeibler divergence and f -divergence are such measures. In the present article, we study the differential-geometrical structure of a manifold induced by a divergence function. It consists of a Riemannian metric, and a pair of dually coupled affine connections, which are studied in information geometry. The class of Bregman divergences are characterized by a dually flat structure, which is originated from the Legendre duality. A dually flat space admits a generalized Pythagorean theorem. The class of f -divergences, defined on a manifold of probability distributions, is characterized by information monotonicity, and the Kullback-Leibler divergence belongs to the intersection of both classes. The f -divergence always gives the α-geometry, which consists of the Fisher information metric and a dual pair of ±α-connections. The α-divergence is a special class of f -divergences. This is unique, sitting at the intersection of the f -divergence and Bregman divergence classes in a manifold of positive measures. The geometry derived from the Tsallis q-entropy and related divergences are also addressed.

169 citations

References
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