Residual variance estimation using a nearest neighbor statistic

In this paper we consider the problem of estimating E[(Y−E[Y∣X])²] based on a finite sample of independent, but not necessarily identically distributed, random variables . We analyze the theoretical properties of a recently developed estimator. It is shown that the estimator has many theoretically interesting properties, while the practical implementation is simple.     

Minimum variance quadratic unbiased estimators as a tool to identify compound normal distributions

We derive the minimum variance quadratic unbiased estimator (MIVQUE) of the variance of the components of a random vector having a compound normal distribution (CND). We show that the MIVQUE converges in probability to a random variable whose distribution is essentially the mixing distribution characterising the CND. This fact is very important, because the MIVQUE allows us to make out the signature of a particular CND, and notably allows us to check if an hypothesis of normality for multivariate observations y₁,…,y_M is plausible.     

Asymptotically minimum variance unbiased estimation for a class of power series distributions

Summary The problem of finding an asymptotically minimum variance unbiased estimator (A.M.V.U.E.) for the parameter of certain truncated power series distributions, is discussed, when the generating function of their coefficients are i) polynomials of binomial type ii) generalized ascending factorials iii) polynomials with coefficients the well known Eulerian numbers.     

Uniformly minimum variance nonnegative quadratic unbiased estimation in a generalized growth curve model

Consider the generalized growth curve model subject to R(X_m)⊆?⊆R(X₁), where B_i are the matrices of unknown regression coefficients, and E=(ε₁,…,ε_s)^′ and are independent and identically distributed with the same first four moments as a random vector normally distributed with mean zero and covariance matrix Σ. We derive the necessary and sufficient conditions under which the uniformly minimum variance nonnegative quadratic unbiased estimator (UMVNNQUE) of the parametric function with C≥0 exists. The necessary and sufficient conditions for a nonnegative quadratic unbiased estimator with of to be the UMVNNQUE are obtained as well.     

Kolmogorov statistic in the case of a piecewise-continuous distribution function

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 40, No. 2, pp. 145–149, March–April, 1988.     

When the degree sequence is a sufficient statistic

There is a uniquely defined random graph model with independent adjacencies in which the degree sequence is a sufficient statistic. The model was recently discovered independently by several authors. Here we join to the statistical investigation of the model, proving that if the degree sequence is in the interior of the polytope defined by the Erd?s–Gallai conditions, then a unique maximum likelihood estimate exists.     

Bounds for the variance of Mann-Whitney statistic

Summary Lower and upper bounds for the variance of Mann-Whitney statistic have been obtained by Birnbaum and Klose recently. The main problem in their paper reduces to that of obtaining bounds of an integral of a convex function of a cumulative distribution function satisfying certain side restrictions. Using the methods and results of Rustagi and Harris, bounds for such an integral are obtained and bounds for the variance of the Mann-Whitney statistic are derived therefrom.     

Nonparametric variance function estimation with missing data

In this paper, a fixed design regression model where the errors follow a strictly stationary process is considered. In this model the conditional mean function and the conditional variance function are unknown curves. Correlated errors when observations are missing in the response variable are assumed. Four nonparametric estimators of the conditional variance function based on local polynomial fitting are proposed. Expressions of the asymptotic bias and variance of these estimators are obtained. A simulation study illustrates the behavior of the proposed estimators.     

Quadratic error of unbiased estimator of density of joint distribution of sufficient statistics of multidimensional normal distribution

Quadratic errors or unbiased density estimators are obtained for a number of distributions and the cases of normal distribution and Wishart distribution are considered.Translated from Staticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 4–10, 1988.     

Minimum variance of a random integer value

The statement on a variance minimum of a random integer value with fixed mathematical expectation is proved.     

Quadratic error of an unbiased estimate of the probability density of sufficient statistics of the normal distribution

Translated from Statisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 41–48, 1984.     

Minimax estimation of a variance

The nonparametric problem of estimating a variance based on a sample of sizen from a univariate distribution which has a known bounded range but is otherwise arbitrary is treated. For squared error loss, a certain linear function of the sample variance is seen to be minimax for eachn from 2 through 13, exceptn=4. For squared error loss weighted by the reciprocal of the variance, a constant multiple of the sample variance is minimax for eachn from 2 through 11. The least favorable distribution for these cases gives probability one to the Bernoulli distributions.     

Methodology for the invariant estimation of a continuous distribution function

Consider both the calssical and some more general invariant decision problems of estimating a continuous distribution function, with the loss function {ie503-1} and a sample of sizen fromF. It is proved that any nonrandomized estimator can be approximated in Lebesgue measure by the more general invariant estimators. Some methods for investigating the finite sample problem are discussed. As an application, a proof that the best invariant estimator is minimax when the sample size is 1 is given.