Topic
U-statistic
About: U-statistic is a research topic. Over the lifetime, 1209 publications have been published within this topic receiving 32898 citations.
Papers published on a yearly basis
Papers
More filters
••
TL;DR: In this article, Kruskal and Blough give an unbiased estimator of the squared multiple correlation, which is a strictly increasing function of the usual estimator differing from it only by terms of order l/n and consequently having the same asymptotic distribution.
Abstract: 1. Summary and introduction. This paper deals with the unbiased estimation of the correlation of two variates having a bivariate normal distribution (Sec. 2), and of the intraclass correlation, i.e., the common correlation coefficient of a p-variate normal distribution with equal variances and equal covariances (Sec. 3). In both cases, the estimator has the following properties. It is a function of acomplete sufficient statistic and is therefore the unique (except for sets of probability zero) minimum variance unbiased estimator. Its range is the region of possible values of the estimated quantity. It is a strictly increasing function of the usual estimator differing from it only by terms of order l/n and consequently having the same asymptotic distribution. Since the unbiased estimators are cumbersome in form in that they are expressed as series or integrals, tables are included giving the unbiased estimators as functions of the usual estimators. In Sec. 4 we give an unbiased estimator of the squared multiple correlation. It has the properties mentioned in the second paragraph except that it may be negative, which the squared multiple correlation cannot. In each case the estimator is obtained by inverting a Laplace transform. We are grateful to W. H. Kruskal and L. J. Savage for very helpful comments and suggestions, and to R. R. Blough for his able computations.
407 citations
••
01 Jan 2011TL;DR: The phenomenon of self-organized criticality (SOC) can be identified from many observations in the universe, by sampling statistical distributions of physical parameters, such as the distributions of time scales, spatial scales, or energies, for a set of events.
Abstract: The phenomenon of self-organized criticality (SOC) can be identified from many observations in the universe, by sampling statistical distributions of physical parameters, such as the distributions of time scales, spatial scales, or energies, for a set of events. SOC manifests itself in the statistics of nonlinear processes.
382 citations
••
TL;DR: In this article, a linear model in the form, where is an unknown parameter and ξ is a hypothetical random variable with a given dispersion structure but containing unknown parameters called variance and covariance components.
Abstract: We write a linear model in the form , where is an unknown parameter and ξ is a hypothetical random variable with a given dispersion structure but containing unknown parameters called variance and covariance components. A new method of estimation called MINQUE (Minimum Norm Quadratic Unbiased Estimation) developed in a previous article [5] is extended for the estimation of variance and covariance components.
348 citations
••
TL;DR: In this article, the authors considered the problem of unbiased estimation, restricted only by the postulate of section 2, and derived necessary and sufficient conditions for the existence of only one unbiased estimate with finite central moment.
Abstract: The problem of unbiased estimation, restricted only by the postulate of section 2, is considered here. For a chosen number $s > 1$, an unbiased estimate of a function $g$ on the parameter space, is said to be best at the parameter point $\theta_0$ if its $s$th absolute central moment at $\theta_0$ is finite and not greater than that for any other unbiased estimate. A necessary and sufficient condition is obtained for the existence of an unbiased estimate of $g$. When one exists, the best one is unique. A necessary and sufficient condition is given for the existence of only one unbiased estimate with finite $s$th absolute central moment. The $s$th absolute central moment at $\theta_0$ of the best unbiased estimate (if it exists) is given explicitly in terms of only the function $g$ and the probability densities. It is, to be more precise, specified as the l.u.b. of certain set $\mathcal{a}$ of numbers. The best estimate is then constructed (as a limit of a sequence of functions) with the use of only the data (relating to $g$ and the densities) associated with any particular sequence in $\mathcal{a}$ which converges to the l.u.b. of $\mathcal{a}$. The case $s = \infty$ is considered apart. The case $s = 2$ is studied in greater detail. Previous results of several authors are discussed in the light of the present theory. Generalizations of some of these results are deduced. Some examples are given to illustrate the applications of the theory.
347 citations
••
TL;DR: A computing procedure is given for obtainingpermutational expectations and variances so that departure from the permutational distribuLtion can be judged.
Abstract: On each of n individuals p + q variables are observed. A nonnegative distance or closeness measure between any two individuals on any one variable can be based on ranks or tied ranks for orderable variables (continuous, discrete, or categorical); for nonorderable categorical variables the distance measure reflects whether the two individuals belong to the same category. Let Xi and Yi, represent weighted sums over the p and the q variables, respectively, of the distance measures between individuals i and j; there will be 'n(n + 1) such pairs of weighted sulms. A test statistic for judging whether closeness in the set of p variables is related to closeness in the set of q variables is given by Z = E E Xt iYi. The permutational distribution of the statistic is defined by the random pairing of p-variable and q-variable observation vectors. Being a U statistic, this measuLre is asymptotically normally distributed. A computing procedure is given for obtaining permutational expectations and variances so that departure from the permutational distribuLtion can be judged.
340 citations