On estimation of the L r norm of a regression function

f be observed with noise. In the present paper we study the problem of nonparametric estimation of certain nonsmooth functionals of f, specifically, L _r norms ||f|| _r of f. Known from the literature results on functional estimation deal mostly with two extreme cases: estimating a smooth (differentiable in L ₂ ) functional or estimating a singular functional like the value of f at certain point or the maximum of f. In the first case, the convergence rate typically is n ^−1/2, n being the number of observations. In the second case, the rate of convergence coincides with the one of estimating the function f itself in the corresponding norm. We show that the case of estimating ||f|| _r is in some sense intermediate between the above extremes. The optimal rate of convergence is worse than n ^−1/2 but is better than the rate of convergence of nonparametric estimates of f. The results depend on the value of r. For r even integer, the rate occurs to be n ^{−β/(2β+1−1/r)} where β is the degree of smoothness. If r is not an even integer, then the nonparametric rate n ^{−β/(2β+1)} can be improved, but only by a logarithmic in n factor. Received: 6 February 1996hinspaceairsp/Revised version: 10 June 1998     

Estimation of the regression operator from functional fixed-design with correlated errors

We consider the estimation of the regression operator r in the functional model: Y=r(x)+ε, where the explanatory variable x is of functional fixed-design type, the response Y is a real random variable and the error process ε is a second order stationary process. We construct the kernel type estimate of r from functional data curves and correlated errors. Then we study their performances in terms of the mean square convergence and the convergence in probability. In particular, we consider the cases of short and long range error processes. When the errors are negatively correlated or come from a short memory process, the asymptotic normality of this estimate is derived. Finally, some simulation studies are conducted for a fractional autoregressive integrated moving average and for an Ornstein-Uhlenbeck error processes.     

Estimation in generalized linear models for functional data via penalized likelihood

We analyze in a regression setting the link between a scalar response and a functional predictor by means of a Functional Generalized Linear Model. We first give a theoretical framework and then discuss identifiability of the model. The functional coefficient of the model is estimated via penalized likelihood with spline approximation. The L² rate of convergence of this estimator is given under smoothness assumption on the functional coefficient. Heuristic arguments show how these rates may be improved for some particular frameworks.     

Asymptotic bias and variance for a general class of varying bandwidth density estimators

Summary We consider a general class of varying bandwidth estimators of a probability density function. The class includes the Abramson estimator, transformation kernel density estimator (TKDE), Jones transformation kernel density estimator (JTKDE), nearest neighbour type estimator (NN), Jones-Linton-Nielsen estimator (JLN), Taylor series approximations of TKDE (TTKDE) and Simpson's formula approximations of TKDE (STKDE). Each of these estimators needs a pilot estimator. Starting with an ordinary kernel estimator , it is possible to iterate and compute a sequence of estimates , using each estimate as a pilot estimator in the next step. The first main result is a formula for the bias order. If the bandwidths used in different steps have a common orderh=h(n), the bias of is of orderh ^2km ,k=1, ...,t. Hereh ^m is the bias order of the ideal estimator (defined by using the unknownf as pilot). The second main result is a recursive formula for the leading bias and stochastic terms in an asymptotic expansion of the density estimates. Ifm<, it is possible to make asymptotically equivalent to the ideal estimator.     

Order restricted inference for sequential k-out-of-n systems

Sequential order statistics have been introduced to model sequential k-out-of-n systems which, as an extension of k-out-of-n systems, allow the failure of some components of the system to influence the remaining ones. Based on an independent sample of vectors of sequential order statistics, the maximum likelihood estimators of the model parameters of a sequential k-out-of-n system are derived under order restrictions. Special attention is paid to the simultaneous maximum likelihood estimation of the model parameters and the distribution parameters for a flexible location-scale family. Furthermore, order restricted hypothesis tests are considered for making the decision whether the usual k-out-of-n model or the general sequential k-out-of-n model is appropriate for a given data.     

AC 0-collocation-like method for two-point boundary value problems

Summary A method of a collocation type based onC ⁰-piecewise polynomial spaces is presented for a two-point boundary value problem of the second order. The method has an optimal order of convergence under smoothness requirements on the exact solution which are weaker than forC ¹-collocation methods. If the differential operator is symmetric, a modification of this method leads to a symmetric system of linear equations. It is shown that if the collocation solution is a piecewise polynomial of degree not greater thanr, the method is stable and convergent with orderh ^r inH ¹-norm. A similar symmetric modification forC ⁰-colloction-finite element method [7] is also obtained. Superconvergence at the nodes is established.     

Second-order accuracy of depth-based bootstrap confidence regions

We consider the problem of setting bootstrap confidence regions for multivariate parameters based on data depth functions. We prove, under mild regularity conditions, that depth-based bootstrap confidence regions are second-order accurate in the sense that their coverage error is of order n⁻¹, given a random sample of size n. The results hold in general for depth functions of types A and D, which cover as special cases the Tukey depth, the majority depth, and the simplicial depth. A simulation study is also provided to investigate empirically the bootstrap confidence regions constructed using these three depth functions.     

Fixed-gain estimation in continuous time

We prove that the parameter estimation error of continuous-time linear stochastic systems that is obtained in connection with a fixed-gain estimation method can be written as a stochastic integral plus a residual term, the moments of which are of order+o(1) where is the forgetting factor.     

Testing model assumptions in functional regression models

In the functional regression model where the responses are curves, new tests for the functional form of the regression and the variance function are proposed, which are based on a stochastic process estimating L²-distances. Our approach avoids the explicit estimation of the functional regression and it is shown that normalized versions of the proposed test statistics converge weakly. The finite sample properties of the tests are illustrated by means of a small simulation study. It is also demonstrated that for small samples, bootstrap versions of the tests improve the quality of the approximation of the nominal level.     

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.     

Representations of best linear unbiased estimators in the Gauss-Markoff model with a singular dispersion matrix

In the general Gauss-Markoff model (Y, Xβ, σ²V), when V is singular, there exist linear functions of Y which vanish with probability 1 imposing some restrictions on Y as well as on the unknown β. In all earlier work on linear estimation, representations of best-linear unbiased estimators (BLUE's) are obtained under the assumption: “L′Y is unbiased for Xβ ? L′X = X.” Such a condition is not, however, necessary. The present paper provides all possible representations of the BLUE's some of which violate the condition L′X = X. Representations of X for given classes of BLUE's are also obtained.     

Propagation-Separation Approach for Local Likelihood Estimation

The paper presents a unified approach to local likelihood estimation for a broad class of nonparametric models, including e.g. the regression, density, Poisson and binary response model. The method extends the adaptive weights smoothing (AWS) procedure introduced in Polzehl and Spokoiny (2000) in context of image denoising. The main idea of the method is to describe a greatest possible local neighborhood of every design point X_i in which the local parametric assumption is justified by the data. The method is especially powerful for model functions having large homogeneous regions and sharp discontinuities. The performance of the proposed procedure is illustrated by numerical examples for density estimation and classification. We also establish some remarkable theoretical nonasymptotic results on properties of the new algorithm. This includes the ``propagation' property which particularly yields the root-n consistency of the resulting estimate in the homogeneous case. We also state an ``oracle' result which implies rate optimality of the estimate under usual smoothness conditions and a ``separation' result which explains the sensitivity of the method to structural changes.     

Estimation of density functions by order statistics

The properties of the empirical density function,f _n(x) = k/n( _j ⁺ – _j-1 ⁺ ) if _j-1 ⁺ < x ⁺ where _j-1 ⁺ and _j ⁺ are sample elements and there are exactlyk – 1 sample elements between them, are studied in that practical point of view how to choose a suitablek for a good estimation. A bound is given for the expected value of the absolute value of difference between the empirical and theoretical density functions.     

Change point estimators by local polynomial fits under a dependence assumption

We study a random design regression model generated by dependent observations, when the regression function itself (or its ν-th derivative) may have a change or discontinuity point. A method based on the local polynomial fits with one-sided kernels to estimate the location and the jump size of the change point is applied in this paper. When the jump location is known, a central limit theorem for the estimator of the jump size is established; when the jump location is unknown, we first obtain a functional limit theorem for a local dilated-rescaled version estimator of the jump size and then give the asymptotic distributions for the estimators of the location and the jump size of the change point. The asymptotic results obtained in this paper can be viewed as extensions of corresponding results for independent observations. Furthermore, a simulated example is given to show that our theory and method perform well in practice.     

Translation-Invariant Bilinear Operators with Positive Kernels

We study L ^r (or L ^{r, ∞}) boundedness for bilinear translation-invariant operators with nonnegative kernels acting on functions on \mathbb Rⁿ{\mathbb {R}^n}. We prove that if such operators are bounded on some products of Lebesgue spaces, then their kernels must necessarily be integrable functions on \mathbb R²ⁿ{\mathbb R^{2n}}, while via a counterexample we show that the converse statement is not valid. We provide certain necessary and some sufficient conditions on nonnegative kernels yielding boundedness for the corresponding operators on products of Lebesgue spaces. We also prove that, unlike the linear case where boundedness from L ¹ to L ¹ and from L ¹ to L ^{1, ∞} are equivalent properties, boundedness from L ¹ × L ¹ to L ^1/2 and from L ¹ × L ¹ to L ^{1/2, ∞} may not be equivalent properties for bilinear translation-invariant operators with nonnegative kernels.     

A new ideal metric with applications to multivariate stable limit theorems

Summary A new ideal metric of orderr>1 is introduced on ^k and a thorough analysis of its metric properties is given. In comparison to the known ideal metric of Zolotarev this new metric allows estimates from above by pseudo difference moments and thus allows applications to stable limit theorems. As applications we give the right order Berry-Esséen type result in the stable case, obtain the limiting behaviour of multivariate summability methods and discuss the approximation problem by compound Poisson distributions.Research supported by NATO GRANT CRG 900 798 and by a DFG Grant     

Linear combination of concomitants of order statistics with application to testing and estimation

Summary Let (X ₁,Y ₁), (X ₂,Y ₂),…, (X _n,Y _n) be i.i.d. as (X, Y). TheY-variate paired with therth orderedX-variateX _rn is denoted byY _rn and terms the concomitant of therth order statistic. Statistics of the form are considered. The asymptotic normality ofT _n is established. The asymptotic results are used to test univariate and bivariate normality, to test independence and linearity ofX andY, and to estimate regression coefficient based on complete and censored samples.     

Structural test in regression on functional variables

Many papers deal with structural testing procedures in multivariate regression. More recently, various estimators have been proposed for regression models involving functional explanatory variables. Thanks to these new estimators, we propose a theoretical framework for structural testing procedures adapted to functional regression. The procedures introduced in this paper are innovative and make the link between former works on functional regression and others on structural testing procedures in multivariate regression. We prove asymptotic properties of the level and the power of our procedures under general assumptions that cover a large scope of possible applications: tests for no effect, linearity, dimension reduction, …     

Optimal rates of convergence in the Weibull model based on kernel-type estimators

Let F be a distribution function in the maximal domain of attraction of the Gumbel distribution such that −log(1−F(x))=x^1/θL(x) for a positive real number θ, called the Weibull tail index, and a slowly varying function L. It is well known that the estimators of θ have a very slow rate of convergence. We establish here a sharp optimality result in the minimax sense, that is when L is treated as an infinite dimensional nuisance parameter belonging to some functional class. We also establish the rate optimal asymptotic property of a data-driven choice of the sample fraction that is used for estimation.