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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.
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Journal ArticleDOI
01 Feb 1995
TL;DR: In this article, it was shown that Revesz's conjecture is true but the conclusion is not valid for general i.i.d. random variables with finite moment generating function.
Abstract: Let {Xn, n > 1} be i.i.d. random variables with P(Xi = +1) = 2 . Revesz (1990) proved 1 < lim inf max max (2k log n) -1/2(Sj+k Si) -noo O

32 citations


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

  • ...[1]), we have p((l+e)a(d¡))<e-x/dl for any 0 < / < (cx -c2)/(nc2), and hence, we can take a ô > 0 such that (2....

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  • ...Applying the well-known Ottaviani maximum inequality, the Chernoff theorem in [1], and (2....

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Journal ArticleDOI
TL;DR: A lower bound on the relative entropy between two finite-dimensional states is proved in terms of their entropy difference and the dimension of the underlying space, and the inequality is tight in the sense that equality can be attained for any prescribed value of the entropy difference.
Abstract: We prove a lower bound on the relative entropy between two finite-dimensional states in terms of their entropy difference and the dimension of the underlying space. The inequality is tight in the sense that equality can be attained for any prescribed value of the entropy difference, both for quantum and classical systems. We outline implications for information theory and thermodynamics, such as a necessary condition for a process to be close to thermodynamic reversibility, or an easily computable lower bound on the classical channel capacity. Furthermore, we derive a tight upper bound, uniform for all states of a given dimension, on the variance of the surprisal, whose thermodynamic meaning is that of heat capacity.

32 citations


Additional excerpts

  • ...Finally, in symmetric hypothesis testing between two classical (commuting) probability distributions ρ1, ρ2, the optimal error decay rate is given by the Chernoff information ξ(ρ1, ρ2) = − log min0≤s≤1[ρs 1ρ 2 ] [10], [11], which has the property that there exists a distribution σ (from the Hellinger arc between ρ1 and ρ2) satisfying ξ(ρ1, ρ2) = D(σ‖ρ1) = D(σ‖ρ2)....

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Journal ArticleDOI
TL;DR: In this paper, the authors apply the notions of reversed time for Markov processes and large deviations for jump-Markov systems to general jump-markov systems and show that the way a constant coefficient process approaches a rare event is roughly by following the path of another constant-coefficient process.
Abstract: When a subsystem goes into an infrequent state, how does the remainder of the system behave? We show how to calculate the relevant distributions using the notions of reversed time for Markov processes and large deviations. For ease of exposition, most of the work deals with a specific queueing model due to Flatto, Hahn, and Wright. However, we show how the theorems may be applied to much more general jump-Markov systems. We also show how the tools of time-reversal and large deviations complement each other to yield general theorems. We show that the way a constant coefficient process approaches a rare event is roughly by following the path of another constant coefficient process. We also obtain some properties, including a priori bounds, for the change of measure associated with some large deviations functionals; these are useful for accelerating simulations.

32 citations


Additional excerpts

  • ...Then by Chernoff's bound [3], [4], [17],...

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Journal ArticleDOI
TL;DR: In this article, an optimisation-based framework for analysing the robustness of advanced flight control laws for reusable launch vehicles is proposed and tested on an industrial-standard simulation model of a reusable launch vehicle equipped with a full authority nonlinear dynamic inversion-based flight control law.

32 citations


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

  • ...Unfortunately, the computational effort involved in the resulting clearance assessment increases exponentially with the number of uncertain parameters that are to be considered (gridding of extreme points) (Fielding et al., 2002), or with the desired statistical confidence levels for the clearance results (Monte Carlo simulation) ( Chernoff, 1952; Vidyasagar, 1998), and this constraint can severely limit the reliability of the analysis in ......

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  • ...The numbers of Monte Carlo trials required to achieve various levels of estimation uncertainty with known probability were calculated using the Chebyshev inequality and central limit theorem in Williams (2001) and are reproduced here in Table 7. Alternatively, using the well-known Chernoff (1952) bound (Vidyasagar, 1998) to estimate the number of simulations required, the numbers are as shown in Table 8 .I n both cases, the exponential ......

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Journal ArticleDOI
TL;DR: The idea of Markov chain lifting is studied to accelerate the convergence of distributed consensus, and two general pseudoalgorithms are presented that achieve the same scaling law in averaging time as the centralized scheme in wireless networks for all r satisfying the connectivity requirement.
Abstract: Existing works on distributed consensus explore linear iterations based on reversible Markov chains, which contribute to the slow convergence of the algorithms. It has been observed that by overcoming the diffusive behavior of reversible chains, certain nonreversible chains lifted from reversible ones mix substantially faster than the original chains. In this paper, we investigate the idea of accelerating distributed consensus via lifting Markov chains, and propose a class of Location-Aided Distributed Averaging (LADA) algorithms for wireless networks, where nodes' coarse location information is used to construct nonreversible chains that facilitate distributed computing and cooperative processing. First, two general pseudo-algorithms are presented to illustrate the notion of distributed averaging through chain-lifting. These pseudo-algorithms are then respectively instantiated through one LADA algorithm on grid networks, and one on general wireless networks. For a $k\times k$ grid network, the proposed LADA algorithm achieves an $\epsilon$-averaging time of $O(k\log(\epsilon^{-1}))$. Based on this algorithm, in a wireless network with transmission range $r$, an $\epsilon$-averaging time of $O(r^{-1}\log(\epsilon^{-1}))$ can be attained through a centralized algorithm. Subsequently, we present a fully-distributed LADA algorithm for wireless networks, which utilizes only the direction information of neighbors to construct nonreversible chains. It is shown that this distributed LADA algorithm achieves the same scaling law in averaging time as the centralized scheme. Finally, we propose a cluster-based LADA (C-LADA) algorithm, which, requiring no central coordination, provides the additional benefit of reduced message complexity compared with the distributed LADA algorithm.

32 citations


Additional excerpts

  • ...Applying the Chernoff bound [17], it can be shown that when r = Ω (√ log n n ) , the number of nodes in the shadowed area is upper bounded by 2rn w....

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References
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