scispace - formally typeset
Search or ask a question
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
More filters
Journal ArticleDOI
TL;DR: Kullback-Leibler discrimination information and the Chernoff information measure are developed for the multivariate non-Gaussian case for discrimination between different classes of multivariate time series.
Abstract: Minimum discrimination information provides a useful generalization of likelihood methodology for classification and clustering of multivariate time series. Discrimination between different classes of multivariate time series that can be characterized by differing covariance or spectral structures is of importance in applications occurring in the analysis of geophysical and medical time series data. For discrimination between such multivariate series, Kullback-Leibler discrimination information and the Chernoff information measure are developed for the multivariate non-Gaussian case. Asymptotic error rates and limiting distributions are given for a generalized spectral disparity measure that includes the foregoing criteria as special cases. Applications to problems of clustering and classifying earthquakes and mining explosions are given.

285 citations

Journal ArticleDOI
TL;DR: A new class of statistical problems is introduced, involving the presence of communication constraints on remotely collected data, when the statistician has direct access to Y data but can be informed about X data only at a preseribed finite rate R.
Abstract: A new class of statistical problems is introduced, involving the presence of communication constraints on remotely collected data. Bivariate hypothesis testing, H_{0}: P_{XY} against H_{1}: P_{\={XY}} , is considered when the statistician has direct access to Y data but can be informed about X data only at a preseribed finite rate R . For any fixed R the smallest achievable probability of an error of type 2 with the probability of an error of type 1 being at most \epsilon is shown to go to zero with an exponential rate not depending on \epsilon as the sample size goes to infinity. A single-letter formula for the exponent is given when P_{\={XY}} = P_{X} \times P_{Y} (test against independence), and partial results are obtained for general P_{\={XY}} . An application to a search problem of Chernoff is also given.

272 citations

Book ChapterDOI
01 Jan 1980

272 citations

Journal ArticleDOI
01 Aug 1994
TL;DR: A model for evaluating scheduling strategies for single and multi-processor systems is developed and it takes into account various issues such as release times, execution time, preemption cost, and the interdependence between jobs.

269 citations

Journal ArticleDOI
TL;DR: The inversion of characteristic functions, a trapezoidal rule for numerical integration, and the sampling theorem in the frequency domain are related to, and interpreted in terms of, the results.
Abstract: The properties of the series are studied for both bounded and unbounded random variables. The technique is used to find efficient series for computation of the distributions of sums of uniform random variables and sums of Rayleigh random variables. A useful closed-form expression for the characteristic function of a Rayleigh random variable is presented, and an efficient method for computing a confluent hypergeometric function is given. An infinite series for the probability density function of a sum of independent random variables is also derived. The inversion of characteristic functions, a trapezoidal rule for numerical integration, and the sampling theorem in the frequency domain are related to, and interpreted in terms of, the results. >

267 citations

References
More filters