A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations
TLDR
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.read more
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Springer Texts in Statistics
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Proceedings ArticleDOI
SCHISM: a new approach for interesting subspace mining
TL;DR: SCHISM is presented, a new algorithm for mining interesting subspaces, using the notions of support and Chernoff-Hoeffding bounds, and uses a vertical representation of the dataset, and use a depth-first search with backtracking to find maximal interesting subSpaces.
Proceedings ArticleDOI
Path Sampling: A Fast and Provable Method for Estimating 4-Vertex Subgraph Counts
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A generalization of quantum Stein's Lemma
TL;DR: This work presents a generalization of quantum Stein’s Lemma to the situation in which the alternative hypothesis is formed by a family of states, which implies that whenever a multipartite state can be asymptotically converted into another entangled state by local operations and classical communication, the rate of conversion must be non-zero.
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Large triangle-free subgraphs in graphs without K 4
TL;DR: It is shown that for arbitrary positiveε there exists a graph withoutK4 and so that all its subgraphs containing more than 1/2 +ε portion of its edges contain a triangle (Theorem 2), and it is proved that such graphs have necessarily low edge density.
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