Minimax risk overl p -balls forl p -error

Summary Consider estimating the mean vector from dataN _n (, ² I) withl _q norm loss,q1, when is known to lie in ann-dimensionall _p ball,p(0, ). For largen, the ratio of minimaxlinear risk to minimax risk can bearbitrarily large ifp. Obvious exceptions aside, the limiting ratio equals 1 only ifp=q=2. Our arguments are mostly indirect, involving a reduction to a univariate Bayes minimax problem. Whenp, simple non-linear co-ordinatewise threshold rules are asymptotically minimax at small signal-to-noise ratios, and within a bounded factor of asymptotic minimaxity in general. We also give asymptotic evaluations of the minimax linear risk. Our results are basic to a theory of estimation in Besov spaces using wavelet bases (to appear elsewhere).     

Efficient nonparametric estimation of distribution density in the basis of algebraic polynomials

A nonparametric estimatef ^* of an unknown distribution densityf W is called locally minimax iff it is minimax for all not too small neighborhoodsW _g ,g W, simultaneously, whereW is some dense subset ofW. Radaviius and Rudzkis proved the existence of such an estimate under some general conditions. However, the construction of the estimate is rather complicated. In this paper, a new estimate is proposed. This estimate is locally minimax under some additional assumptions which usually hold for orthobases of algebraic polynomial and is almost as simple as the linear projective estimate. Thus, it takes a form convenient for the construction of an adaptive estimator, which does not usea-priori information about the smoothness of the density. The adaptive estimation problem is briefly discussed and an unknown density fitting by Jacobi polynomials is investigated more explicitly.     

Nonparametric estimation of location parameter after a preliminary test on regression in the multivariate case

For a simple multivariate regression model, nonparametric estimation of the (vector of) intercept following a preliminary test on the regression vector is considered. Along with the asymptotic distribution of these estimators, their asymptotic bias and dispersion matrices are studied and allied efficiency results are presented.     

Properties of nonparametric estimators of autocovariance for stationary random fields

Summary We introduce nonparametric estimators of the autocovariance of a stationary random field. One of our estimators has the property that it is itself an autocovatiance. This feature enables the estimator to be used as the basis of simulation studies such as those which are necessary when constructing bootstrap confidence intervals for unknown parameters. Unlike estimators proposed recently by other authors, our own do not require assumptions such as isotropy or monotonicity. Indeed, like nonparametric function estimators considered more widely in the context of curve estimation, our approach demands only smoothness and tail conditions on the underlying curve or surface (here, the autocovariance), and moment and mixing conditions on the random field. We show that by imposing the condition that the estimator be a covariance function we actually reduce the numerical value of integrated squared error.     

Sieve maximum likelihood estimation for doubly semiparametric zero-inflated Poisson models

For nonnegative measurements such as income or sick days, zero counts often have special status. Furthermore, the incidence of zero counts is often greater than expected for the Poisson model. This article considers a doubly semiparametric zero-inflated Poisson model to fit data of this type, which assumes two partially linear link functions in both the mean of the Poisson component and the probability of zero. We study a sieve maximum likelihood estimator for both the regression parameters and the nonparametric functions. We show, under routine conditions, that the estimators are strongly consistent. Moreover, the parameter estimators are asymptotically normal and first order efficient, while the nonparametric components achieve the optimal convergence rates. Simulation studies suggest that the extra flexibility inherent from the doubly semiparametric model is gained with little loss in statistical efficiency. We also illustrate our approach with a dataset from a public health study.     

A Bahadur efficiency comparison between one and two sample rank statistics and their sequential rank statistic analogues

One and two sample rank statistics are shown in general to be more efficient in the Bahadur sense than their sequential rank statistic analogues as defined by Mason (1981, Ann. Statist.9 424–436) and Lombard (1981, South African Statist. J.15 129–152), even though the two families of statistics (those based on full ranks and those based on sequential ranks) have the same Pitman efficiency against local alternatives. In the process, general results on large deviation probabilities and laws of large numbers for statistics based on sequential ranks are obtained.     

Posterior contraction rates for the Bayesian approach to linear ill-posed inverse problems

We consider a Bayesian nonparametric approach to a family of linear inverse problems in a separable Hilbert space setting with Gaussian noise. We assume Gaussian priors, which are conjugate to the model, and present a method of identifying the posterior using its precision operator. Working with the unbounded precision operator enables us to use partial differential equations (PDE) methodology to obtain rates of contraction of the posterior distribution to a Dirac measure centered on the true solution. Our methods assume a relatively weak relation between the prior covariance, noise covariance and forward operator, allowing for a wide range of applications.     

Asymptotic statistical equivalence for scalar ergodic diffusions

For scalar diffusion models with unknown drift function asymptotic equivalence in the sense of Le Cam's deficiency between statistical experiments is considered under long-time asymptotics. A local asymptotic equivalence result is established with an accompanying sequence of simple Gaussian shift experiments. Corresponding globally asymptotically equivalent experiments are obtained as compound experiments. The results are extended in several directions including time discretisation. An explicit transformation of decision functions from the Gaussian to the diffusion experiment is constructed. The authors acknowledge the financial support provided through the European Community's Human Potential Programme under contract HPRN-CT-2000-00100, DYNSTOCH     

Asymptotic equivalence for nonparametric generalized linear models

Summary. We establish that a non-Gaussian nonparametric regression model is asymptotically equivalent to a regression model with Gaussian noise. The approximation is in the sense of Le Cam's deficiency distance Δ; the models are then asymptotically equivalent for all purposes of statistical decision with bounded loss. Our result concerns a sequence of independent but not identically distributed observations with each distribution in the same real-indexed exponential family. The canonical parameter is a value f(t _i ) of a regression function f at a grid point t _i (nonparametric GLM). When f is in a H?lder ball with exponent we establish global asymptotic equivalence to observations of a signal Γ(f(t)) in Gaussian white noise, where Γ is related to a variance stabilizing transformation in the exponential family. The result is a regression analog of the recently established Gaussian approximation for the i.i.d. model. The proof is based on a functional version of the Hungarian construction for the partial sum process. Received: 4 February 1997     

Minimum distance estimation in linear regression with unknown error distributions

This paper discusses minimum distance (m.d.) estimators of the paramter vector in the multiple linear regression model when the distributions of errors are unknown. These estimators are defined in terms of L₂-distances involving certain weighted empirical processes. Their finite sample properties and asymptotic behavior under heteroscedastic, symmetric and asymmetric errors are discussed. Some robustness properties of these estimators are also studied.