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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: In this article, the problem of finding a 2-spanner in a given graph, with minimum maximum degree, is shown to be at least as hard as set cover and a randomized approximation algorithm is provided for this problem, with approximation ratio of $\tilde O(Delta^{1/4})$.
Abstract: A k-spanner of a connected (undirected unweighted) graph G=(V,E) is a subgraph G' consisting of all the vertices of V and a subset of the edges, with the additional property that the distance between any two vertices in G' is larger than that distance in G by no more than a factor of k. This paper is concerned with approximating the problem of finding a 2-spanner in a given graph, with minimum maximum degree. We first show that the problem is at least as hard to approximate as set cover. Then a randomized approximation algorithm is provided for this problem, with approximation ratio of $\tilde O(\Delta^{1/4})$. We then present a probabilistic algorithm that is more efficient for sparse graphs. Our algorithms are converted into deterministic ones using derandomization.

41 citations

Journal ArticleDOI
TL;DR: This work considers the task of verifying entangled states using one-way and two-way classical communication and completely characterize the optimal strategies via convex optimization, and finds that they can be constructed for any bipartite pure state.
Abstract: The efficient and reliable verification of quantum states plays a crucial role in various quantum information processing tasks. We consider the task of verifying entangled states using one-way and two-way classical communication and completely characterize the optimal strategies via convex optimization. We solve these optimization problems using both analytical and numerical methods, and the optimal strategies can be constructed for any bipartite pure state. Compared with the nonadaptive approach, our adaptive strategies significantly improve the efficiency of quantum state verification. Moreover, these strategies are experimentally feasible, as only few local projective measurements are required.

41 citations

Journal ArticleDOI
TL;DR: In this article, the relative performance of two statistical tests of a hypothesis for large sample sizes is investigated by means of asymptotic relative efficiencies in the sense of Pitman or Bahadur Pitman efficiencies are directed at local alternatives, Bahhadur efficiencies at fixed (non-local) alternatives.
Abstract: The relative performance of two statistical tests of a hypothesis for large sample sizes is often investigated by means of asymptotic relative efficiencies in the sense of Pitman or Bahadur Pitman efficiencies are directed at local alternatives, Bahadur efficiencies at fixed (non-local) alternatives The present paper reviews some methods and results on Bahadur efficiency with special attention to probabilities of large deviations, which play a key role in the computation of Bahadur efficiencies

41 citations

Journal ArticleDOI
TL;DR: In this article, the standard name of a black-box group G is computed by a Monte Carlo algorithm, and the running time is polynomial in the input length and in the time requirement for the group operations in G.
Abstract: Given a black-box group G isomorphic to some finite simple group of Lie type and the characteristic of G, we compute the standard name of G by a Monte Carlo algorithm. The running time is polynomial in the input length and in the time requirement for the group operations in G. The algorithm chooses a relatively small number of (nearly) uniformly distributed random elements of G, and examines the divisibility of the orders of these elements by certain primitive prime divisors. We show that the divisibility statistics determine G, except that we cannot distinguish the groups PWð2mþ 1; qÞ and PSpð2m; qÞ in this manner when q is odd and md 3. These two groups can, however, be distinguished by using an algorithm of Altseimer and

41 citations


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

  • ...The method is based on Cherno¤ ’s bound [15]....

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Journal ArticleDOI
V. Prabhu1
TL;DR: An upper bound to the probability of error of m -ary CPSK systems is given when an ideal CPSK signal is passed through a linear time-invariant noisy filter.
Abstract: In order to optimize the design of a coherent phase-shift-keyed (CPSK) system, it is necessary to estimate the amount of degradation produced by intersymbol interference. In this paper, an upper bound to the probability of error of m -ary CPSK systems is given when an ideal CPSK signal is passed through a linear time-invariant noisy filter. This bound can be used to estimate the maximum amount of deterioration produced by intersymbol interference, and hence to choose the filter and channel parameters appropriately. It is assumed that all m symbols have equal a priori probabilities and that the noise is Gaussian.

41 citations

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