Herman Chernoff

Bio: Herman Chernoff is an academic researcher from Harvard University. The author has contributed to research in topics: Decision theory & Likelihood-ratio test. The author has an hindex of 36, co-authored 88 publications receiving 12277 citations. Previous affiliations of Herman Chernoff include Massachusetts Institute of Technology & University of California.

An extension of a result of Liapounoff on the range of a vector measure

Abstract: Liapounoff2 established in 1940 that the range of a countably additive finite measure with values in a finite-dimensional real vector space is bounded and closed and in the nonatomic case convex. A simplified proof of this result was given by Halmos' in 1948. The aim of the present paper is to extend this result to the following case. Let pit, 1 0 there exist a a>0 such that g*(E)

A note on the inversion of power series

Abstract: At such times there exists a choice between two long methods. The computer who is faced with this problem very often may derive explicit formulae for the desired coefficients and substitute directly. This method has the drawback usually encountered in substituting in formulae, namely that the computations are usually unsystematic and therefore become tedious and subject to many errors. Furthermore the formulae become extremely long and complicated for coefficients of appreciable order.1 The chief purpose of this paper lies in presenting a method which will be especially useful to the person who is unwilling to derive complicated formulae or undergo the ordeal of substituting in them. The method which will be presented will enable the computer to obtain « coefficients of the inverse power series using only one page of computations with approximately J(n + l)2 numbers. Besides being compact, this method has the advantage of being systematic. Furthermore similar methods can be easily obtained for most formal calculations with power series.

A Property of some Type a Regions

Abstract: In a test of an hypothesis one may regard a sample in the critical region as evidence that the hypothesis is false Let us assume that for some reason it is desired to increase the critical size of the test, ie, to make rejection of the hypothesis more probable Then one may expect that an observation which led to rejection in the first test should still lead to rejection in the new test In other words, one should expect $W_\alpha \supset W_{\alpha'}$ if $\alpha > \alpha'$, where $W_\alpha$ is the critical region for the test of size $\alpha$ An example is given where regions of type $A$ are uniquely specified except for sets of measure zero, but fail to have this property

Numerical Methods for Bayes Sequential Decision Problems

Abstract: : The general approach to sequential decision-theoretic problems where sums of successive observations are approximated by a continuous time Wiener process has a number of fundamental advantages. Simple numerical techniques which can be employed to obtain explicit descriptions of the solutions of the resulting continuous time optimal stopping problems are described. The techniques are not well adapted for very precise results, but are surprisingly effective for reasonably accurate approximations. Special features of particular problems can be exploited to reduce the necessary computational effort. The techniques are illustrated in a number of problems thereby clearly indicating their properties.

Visualizing Data using t-SNE

Abstract: We present a new technique called “t-SNE” that visualizes high-dimensional data by giving each datapoint a location in a two or three-dimensional map. The technique is a variation of Stochastic Neighbor Embedding (Hinton and Roweis, 2002) that is much easier to optimize, and produces significantly better visualizations by reducing the tendency to crowd points together in the center of the map. t-SNE is better than existing techniques at creating a single map that reveals structure at many different scales. This is particularly important for high-dimensional data that lie on several different, but related, low-dimensional manifolds, such as images of objects from multiple classes seen from multiple viewpoints. For visualizing the structure of very large datasets, we show how t-SNE can use random walks on neighborhood graphs to allow the implicit structure of all of the data to influence the way in which a subset of the data is displayed. We illustrate the performance of t-SNE on a wide variety of datasets and compare it with many other non-parametric visualization techniques, including Sammon mapping, Isomap, and Locally Linear Embedding. The visualizations produced by t-SNE are significantly better than those produced by the other techniques on almost all of the datasets.

Regression Models and Life-Tables

Abstract: The analysis of censored failure times is considered. It is assumed that on each individual arc available values of one or more explanatory variables. The hazard function (age-specific failure rate) is taken to be a function of the explanatory variables and unknown regression coefficients multiplied by an arbitrary and unknown function of time. A conditional likelihood is obtained, leading to inferences about the unknown regression coefficients. Some generalizations are outlined.

Experimental Design and Data Analysis for Biologists

Abstract: An essential textbook for any student or researcher in biology needing to design experiments, sample programs or analyse the resulting data The text begins with a revision of estimation and hypothesis testing methods, covering both classical and Bayesian philosophies, before advancing to the analysis of linear and generalized linear models Topics covered include linear and logistic regression, simple and complex ANOVA models (for factorial, nested, block, split-plot and repeated measures and covariance designs), and log-linear models Multivariate techniques, including classification and ordination, are then introduced Special emphasis is placed on checking assumptions, exploratory data analysis and presentation of results The main analyses are illustrated with many examples from published papers and there is an extensive reference list to both the statistical and biological literature The book is supported by a website that provides all data sets, questions for each chapter and links to software