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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TL;DR: For degraded memoryless stationary wiretap channels, it is shown that an orthogonal coding scheme with randomized pulse position and constant pulse shape achieves the secrecy capacity per unit cost with a zero-cost input letter.
Abstract: The concept of channel capacity per unit cost was introduced by Verdu in 1990 to study the limits of cost-efficient wide-band communication. It was shown that orthogonal signaling can achieve the channel capacity per unit cost of memoryless stationary channels with a zero-cost input letter. This paper introduces a concept of secrecy capacity per unit cost to study cost-efficient wide-band secrecy communication. For degraded memoryless stationary wiretap channels, it is shown that an orthogonal coding scheme with randomized pulse position and constant pulse shape achieves the secrecy capacity per unit cost with a zero-cost input letter. For general memoryless stationary wiretap channels, the performance of orthogonal codes is studied, and the benefit of further randomizing the pulse shape is demonstrated via a simple example.
54 citations
Cites background or methods from "A Measure of Asymptotic Efficiency ..."
...By the Chernoff-Stein lemma [1] a decision rule can be found such that β l → 0 in the limit as k → ∞ and...
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...By the Chernoff-Stein lemma [1], since β i < /2, we can achieve...
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TL;DR: A set of sufficient conditions and a set of necessary conditions for recovery, which are asymptotically tight with sharp constants are derived, showing the condition for the voting procedure to succeed is almost necessary for exact recovery.
Abstract: We study the problem of recovering a hidden community of cardinality $K$ from an $n \times n$ symmetric data matrix $A$ , where for distinct indices $i,j$ , $A_{ij} \sim P$ if $i, j$ both belong to the community and $A_{ij} \sim Q$ otherwise, for two known probability distributions $P$ and $Q$ depending on $n$ . If $P={\mathrm{ Bern}}(p)$ and $Q={\mathrm{ Bern}}(q)$ with $p>q$ , it reduces to the problem of finding a densely connected $K$ -subgraph planted in a large Erdos–Renyi graph; if $P=\mathcal {N}(\mu ,1)$ and $Q=\mathcal {N}(0,1)$ with $\mu >0$ , it corresponds to the problem of locating a $K \times K$ principal submatrix of elevated means in a large Gaussian random matrix. We focus on two types of asymptotic recovery guarantees as $n \to \infty $ : 1) weak recovery: expected number of classification errors is $o(K)$ and 2) exact recovery: probability of classifying all indices correctly converges to one. Under mild assumptions on $P$ and $Q$ , and allowing the community size to scale sublinearly with $n$ , we derive a set of sufficient conditions and a set of necessary conditions for recovery, which are asymptotically tight with sharp constants. The results hold, in particular, for the Gaussian case, and for the case of bounded log likelihood ratio, including the Bernoulli case whenever $({p}/{q})$ and $({1-p})/({1-q})$ are bounded away from zero and infinity. Previous work has shown that if weak recovery is achievable; then, exact recovery is achievable in linear additional time by a simple voting procedure. We provide a converse, showing the condition for the voting procedure to succeed is almost necessary for exact recovery.
54 citations
Cites methods from "A Measure of Asymptotic Efficiency ..."
...In the special case of linear community size, i.e., K = Θ(n), (13) and (14) can be simplified by replacing E Q 1 K log n K by the Chernoff index [10]: C(P, Q) sup 0≤λ≤1 − log dP dQ λ dQ....
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TL;DR: Using mutational signatures, it is predicted that majority of the mutations in blood originated in hematopoietic stem and early progenitor cells, and tissues carried signatures of distinct mutagenic processes such as oxidative DNA damage and transcription-coupled repair.
Abstract: Mutations acquired during development and aging lead to inter- and intra-tissue genetic variations. Evidence linking such mutations to complex traits and diseases is rising. We detected somatic mutations in protein-coding regions in 140 benign tissue samples representing nine tissue-types (bladder, breast, liver, lung, prostate, stomach, thyroid, head and neck) and paired blood from 70 donors. A total of 80% of the samples had 2-39 mutations detectable at tissue-level resolution. Factors such as age and smoking were associated with increased burden of detectable mutations, and tissues carried signatures of distinct mutagenic processes such as oxidative DNA damage and transcription-coupled repair. Using mutational signatures, we predicted that majority of the mutations in blood originated in hematopoietic stem and early progenitor cells. Missense to silent mutations ratio and the persistence of potentially damaging mutations in expressed genes carried signatures of relaxed purifying selection. Our findings have relevance for etiology, diagnosis and treatment of diseases including cancer.
53 citations
Additional excerpts
...As shown by Bansal (2), the probability that the sum X = X1 + X2 +···+Xn of n independent Bernoulli random variables deviate from its mean μ = p! ! can be approximated using the Chernoff bound (3):...
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