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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 survey of various concepts in quantum information is given in this article, with a main emphasis on the distinguishability of quantum states and quantum correlations, including generalized and least square measurements, state discrimination, quantum relative entropies, quantum Fisher information, the quantum Chernoff bound and quantum discord.
Abstract: A survey of various concepts in quantum information is given, with a main emphasis on the distinguishability of quantum states and quantum correlations. Covered topics include generalized and least square measurements, state discrimination, quantum relative entropies, the Bures distance on the set of quantum states, the quantum Fisher information, the quantum Chernoff bound, bipartite entanglement, the quantum discord, and geometrical measures of quantum correlations. The article is intended both for physicists interested not only by collections of results but also by the mathematical methods justifying them, and for mathematicians looking for an up-to-date introductory course on these subjects, which are mainly developed in the physics literature.

61 citations


Additional excerpts

  • ...One can show that the error probability decays exponentially like P opt err,N ∼ e−Nξ(p1,p2), with an exponent given by the Chernoff bound [40] ξ(p1,p2) = − lim N→∞ 1 N lnP opt err,N({p (N) i , ηi}) = − inf α∈(0,1) { ln (∫...

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Journal ArticleDOI
TL;DR: In this article, it was shown that the number of odd numbers in the first x rows of Pascal's triangle can be computed in the form of a digital sum of the form $x ∈ {( \log x ) / ( {2\log 2} )} + O( ∈ x( √ log x ) √ √ k - 1} √ 2.
Abstract: Let $F( x )$ be the number of odd numbers in the first x rows of Pascal’s triangle. Let $\theta = {( \log 3 ) / (\log 2 )} $. Let $\alpha = \lim \sup x^{-\theta} F( x )$ and $\beta = \lim \inf x^{-\theta} F( x )$. Then $0.72 \leqq \beta \leqq ( {\tfrac{9}{7}} )( {\tfrac{3} {4}} )^\theta \leqslant 0.815$ and $1 \leqq \alpha \leqq 1.052$. If $x = 2^{e_1 } + 2^{e_2 } + \cdots + 2^{e_r } $ where the $e_i $ are strictly decreasing, then $\sum olimits_{i = 1}^r {2^{i - 1} } 3^{e_i } $. These results are obtained from the known result that $F( x ) = \sum olimits_{n = 0}^{x - 1} {2^{B( n )} } $, where $B( n )$ is the number of ones in the binary expansion of n. The related sums $\sum olimits_{n \leqq x} {B^k ( n )} $ are shown to be of the form $x\{ {( \log x ) / ( {2\log 2} )} \}^k + O\{ x( \log x )^{k - 1} \}$; this is best possible. This curious history of digital sums and their estimates is briefly sketched.

61 citations

Journal ArticleDOI
TL;DR: In this article, explicit and almost tight bounds on the smooth entropies of n-fold product distributions, Pn, are derived in terms of the Shannon entropy of a single distribution, P.
Abstract: Smooth entropies characterize basic information-theoretic properties of random variables, such as the number of bits required to store them or the amount of uniform randomness that can be extracted from them (possibly with respect to side information). In this paper, explicit and almost tight bounds on the smooth entropies of n-fold product distributions, Pn, are derived. These bounds are expressed in terms of the Shannon entropy of a single distribution, P . The results can be seen as an extension of the asymptotic equipartition property (AEP).

60 citations

Journal ArticleDOI
TL;DR: Tight bounds are given on the average complexity of various problems of a bidirectional ring of n processors, where processors are anonymous, i.e., are indistinguishable.

60 citations

Proceedings ArticleDOI
23 Jan 1994
TL;DR: It is shown that a cut of weight within a (1 + 6) multiplicative factor of the minimum cut in a graph can be found in O(m + n(log3 n)/e*) time; thus any constant factor approximation can be achieved in d(m) time.
Abstract: We introduce the concept of randomized sparsification of a weighted, undirected graph. Randomized sparsification yields a sparse unweighted graph which closely approximates the minimum cut structure of the original graph. As a consequence, we show that a cut of weight within a (1 + 6) multiplicative factor of the minimum cut in a graph can be found in O(m + n(log3 n)/e*) time; thus any constant factor approximation can be achieved in d(m) time. Similarly, we show that a cut within a multiplicative factor of (Y of the minimum can be found in RNC using m + n21a processors. We also investigate a parametric version of our randomized sparsification approach. Using it, we show that for a graph undergoing a series of edge insertions and deletions, an O(dm)-approximation to the minimum cut value can be maintained at a cost of o(n6+1/2) time per insertion or deletion. If only insertions are allowed, the approximation can be maintained at a cost of O(ne) time per insertion.

60 citations

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