
Wolfgang WefelmeyerUniversity of Cologne | UOC · Department of Mathematics
Wolfgang Wefelmeyer
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133
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Introduction
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Publications
Publications (133)
When we observe a stationary time series with observations missing at periodic time points, we can still estimate its marginal distribution well, but the dependence structure of the time series may not be recoverable at all, or the usual estimators may have much larger variance than in the fully observed case. We show how non‐parametric estimators...
We construct an efficient estimator for the error distribution function of the nonparametric regression model Y = r(Z) + e. Our estimator is a kernel smoothed empirical distribution function based on residuals from an under-smoothed local quadratic smoother for the regression function.
We consider estimation of the drift function of a stationary diffusion process when we observe high-frequency data with microstructure noise over a long time interval. We propose to estimate the drift function at a point by a Nadaraya–Watson estimator that uses observations that have been pre-averaged to reduce the noise. We give conditions under w...
We consider nonparametric regression models in which the regression function is a step function, and construct a convolution estimator for the response density that has the same bias as the usual estimators based on the responses, but a smaller asymptotic variance.
We consider a partially linear regression model with multivariate covariates and with responses that are allowed to be missing at random. This covers the usual settings with fully observed data and the nonparametric regression model as special cases. We first develop a test for additivity of the nonparametric part in the complete data model. The te...
Suppose we want to estimate a density at a point where we know the values of its first or higher order derivatives. In this case a given kernel estimator of the density can be modified by adding appropriately weighted kernel estimators of these derivatives. We give conditions under which the modified estimators are asymptotically normal. We also de...
We consider a nonparametric regression model $Y=r(X)+\varepsilon$ with a
random covariate $X$ that is independent of the error $\varepsilon$. Then the
density of the response $Y$ is a convolution of the densities of $\varepsilon$
and $r(X)$. It can therefore be estimated by a convolution of kernel estimators
for these two densities, or more general...
Densities of functions of two or more independent random variables can be estimated by local U-statistics. Frees (1994) gave conditions under which they converge pointwise at the parametric root-n rate. Uniform convergence at this rate was established by Schick and Wefelmeyer (2004b) for sums of random variables. Giné and Mason (2007) gave conditio...
Densities of functions of independent and identically distributed random observations can be estimated by using a local UU-statistic. Under an appropriate integrability condition, this estimator behaves asymptotically like an empirical estimator. In particular, it converges at the parametric rate. The integrability condition is rather restrictive....
We consider semiparametric additive regression models with a linear parametric part and a nonparametric part, both involving multivariate covariates. For the nonparametric part we assume two models. In the first, the regression function is unspecified and smooth; in the second, the regression function is additive with smooth components. Depending o...
The usual estimator for the expectation of a function of a random vector is the empirical estimator. Assume that some of the components of the random vector are conditionally independent given the other components. We construct a plug-in estimator for the expectation that uses this information, prove a central limit theorem for the estimator, and s...
We consider nonlinear and heteroscedastic autoregressive models whose residuals are martingale increments with conditional distributions that fulfil certain constraints. We treat two classes of constraints: residuals depending on the past through some function of the past observations only, and residuals that are invariant under some finite group o...
We consider a stationary Markov renewal process whose inter-arrival time density depends multiplicatively on the distance between the past and present state of the embedded chain. This is appropriate when the jump size is governed by influences that accumulate over time. Then we can construct an estimator for the inter-arrival time density that has...
Recent results show that densities of convolutions can be estimated by local U-statistics at the root-n rate in various norms. Motivated by this and the fact that convolutions of normal densities are normal, we introduce new tests for normality which use as test statistics weighted L1-distances between the standard normal density and local U-statis...
For sufficiently non-regular distributions with bounded support, the extreme observations converge to the boundary points at a faster rate than the square root of the sample size. In a nonparametric regression model with such a non-regular error distribution, this fact can be used to construct an estimator for the regression function that converges...
arguments. At points in the support of the squared random variable, the rate of the estimator slows down by a logarithmic factor and is inde- pendent of the bandwidth, but the asymptotic variance depends on the rate of the bandwidth, and otherwise only on the density of the squared random variable at this point and at zero. A functional cen- tral l...
The stationary density of a centered invertible linear process can be represented as a convolution of innovation-based densities, and it can be estimated at the parametric rate by plugging residual-based kernel estimators into the convolution representation. We have shown elsewhere that a functional central limit theorem holds both in the space of...
We prove a Bahadur representation for a residual-based estimator of the innovation distribution function in a nonparametric
autoregressive model. The residuals are based on a local linear smoother for the autoregression function. Our result implies
a functional central limit theorem for the residual-based estimator.
We consider nonparametric regression models with multivariate covariates and estimate the regression curve by an undersmoothed local polynomial smoother. The resulting residual-based empirical distribution function is shown to differ from the error-based empirical distribution function by the density times the average of the errors, up to a uniform...
Densities of functions of independent and identically distributed random observations can be estimated by a local U-statistic. It has been shown recently that, under an appropriate integrability condition, this estimator behaves asymptotically
like an empirical estimator. In particular, it converges at the parametric rate. The integrability conditi...
We obtain root-n consistency and functional central limit theorems in weighted L 1 -spaces for plug-in estimators of the two-step transition density in the classical stationary linear autoregressive model of order one, assuming essentially only that the innovation density has bounded variation. We also show that plugging in a properly weighted resi...
Suppose we observe a time series that alternates between different nonlinear autoregressive processes. We give conditions under which the model is locally asymptotically normal, derive a characterization of efficient estimators for differentiable functionals of the model, and use it to construct efficient estimators for the autoregression parameter...
In this paper we describe the historical development of some parts of semiparametric statistics. The emphasis is on efficient estimation. We understand semiparametric model in the general sense of a model that is neither parametric nor nonparametric. We restrict attention to models with independent and identically distributed observations and to ti...
For the stationary invertible moving average process of order one with unknown innovation distribution F, we construct root-n consistent plug-in estimators of conditional expectations E(h(Xn+1)|X1,…,Xn). More specifically, we give weak conditions under which such estimators admit Bahadur-type representations, assuming some smoothness of h or of F....
The stationary density of an invertible linear processes can be estimated at the parametric rate by a convolution of residual-based
kernel estimators. We have shown elsewhere that the convergence is uniform and that a functional central limit theorem holds
in the space of continuous functions vanishing at infinity. Here we show that analogous resul...
Suppose we observe a geometrically ergodic semi-Markov process and have a parametric model for the transition distribution of the embedded Markov chain, for the conditional distribution of the inter-arrival times, or for both. The first two models for the process are semiparametric, and the parameters can be estimated by conditional maximum likelih...
Suppose we observe a discrete-time Markov chain only at certain periodic or random time points. Which observation patterns
allow us to identify the transition distribution? In the case where we can identify it, how can we construct (good) estimators?
We discuss these questions both for nonparametric models and for linear autoregression.
Pointwise and uniform convergence rates for kernel estimators of the stationary density of a linear process have been obtained by several authors. Here we obtain rates in weighted L1 spaces. In particular, if infinitely many coefficients of the process are non-zero and the innovation density has bounded variation, then nearly parametric rates are a...
Suppose we observe a time series that alternates between different autoregressive processes. We give conditions under which it has a stationary version, derive a characterization of efficient estimators for differenthble functionals of Llie model, and use it to construct efficient estilrrators for the autoregressioir paramneters and the innovation...
Convergence rates of kernel density estimators for stationary time series are well studied. For invertible linear processes, we construct a new density estimator that converges, in the supremum norm, at the better, parametric, rate $n^{-1/2}$. Our estimator is a convolution of two different residual-based kernel estimators. We obtain in particular...
We construct root-n consistent plug-in estimators for conditional expectations of the form E(h(Xn+1,..., Xn+m)lX(1,)...,X-n) in invertible linear processes. More specifically, we prove a Bahadur-type representation for such estimators, uniformly over certain classes of not necessarily bounded functions h. We obtain in particular a uniformly root-n...
It is known that the convolution of a smooth density with itself can be estimated at the root-n rate by a convolution of an appropriate density estimator with itself. We show that this remains true even for discontinuous densities as long as they are of bounded variation. The assumption of bounded variation can be relaxed. We consider convergence i...
Conditional expectations given past observations in stationary time se- ries are usually estimated directly by kernel estimators, or by plugging in kernel estimators for transition densities. We show that, for linear and non- linear autoregressive models driven by independent innovations, appropriate smoothed and weighted von Mises statistics of re...
We consider semiparametric additive regression models with a linear parametric part and a nonparametric part, both involving multivariate covariates. For the nonparametric part we assume two models. In the first, the regression function is unspecified and smooth; in the second, the regression function is additive with smooth components. Depending o...
We consider estimation of linear functionals of the joint law of regression models in which responses are missing at random.
The usual approach is to work with the fully observed data, and to replace unobserved quantities by estimators of appropriate
conditional expectations. Another approach is to replace all quantities by such estimators. We show...
Convergence rates and central limit theorems for kernel estimators of the stationary density of a linear process have been obtained under the assumption that the innovation density is smooth (Lipschitz). We show that smoothness is not required. For example, it suffices that the innovation density has bounded variation.
We illustrate several recent results on efficient estimation for semiparametric time series models with a simple class of models: first-order nonlinear autoregression with independent innovations. We consider in particular estimation of the autoregression parameter, the innovation distribution, conditional expectations, the stationary distribution,...
This paper considers residual-based and randomly weighted kernel estimators for innovation densities of nonlinear autoregressive models. The weights are chosen to make use of the information that the innovations have mean zero. Rates of convergence are obtained in weighted L 1-norms. These estimators give rise to smoothed and weighted empirical dis...
We give new results, under mild assumptions, on convergence rates in L<sub>1</sub> and L<sub>2</sub> for residual-based kernel estimators of the innovation density of moving average processes. Exploiting the convolution representation of the stationary density of moving average processes, these estimators can be used to obtain n<sup>1/2</sup>-consi...
The density of a sum of independent random variables can be estimated by the convolution of kernel estimators for the marginal densities. We show under mild conditions that the resulting estimator is n -consistent and converges in distribution in the spaces C0 (R) and L1 to a centered Gaussian process.
Suppose we observe an invertible linear process with independent mean-zero innovations and with coefficients depending on a finite-dimensional parameter, and we want to estimate the expectation of some function under the stationary distribution of the process. The usual estimator would be the empirical estimator. It can be improved using the fact t...
Suppose we observe a geometrically ergodic Markov chain with a parametric model for the marginal, but no (further) information about the transition distribution. Then the empirical estimator for a linear functional of the joint law of two successive observations is no longer efficient. We construct an improved estimator and show that it is efficien...
This paper addresses estimation of linear functionals of the error distribution in nonparametric regression models. It derives an i.i.d. representation for the empirical estimator based on residuals, using undersmoothed estimators for the regression curve. Asymptotic efficiency of the estimator is proved. Estimation of the error variance is discuss...
We consider semiparametric models of semi-Markov processes with arbitrary state space. Assuming that the process is geometrically ergodic, we characterize efficient estimators, in the sense of Hájek and Le Cam, for arbitrary real-valued smooth functionals of the distribution of the embedded Markov renewal process. We construct efficient estimators...
We illustrate several recent results on efficient estimation for semiparametric time series models with two types of AR(1) models: having independent and centered innovations, and having general and conditionally centered innovations. We consider in particular estimation of the autoregression parameter, the stationary distribution, the innovation d...
Suppose we want to calculate the expectation of a function f under a distribution on some space E. If E is of high dimension, or if is defined indirectly, it may be difficult to calculate the expectation f = E f = R (dx)f(x) analytically or even by numerical integration. The classical Monte Carlo method generates i.i.d. realizations X 0 ; : : : ; X...
We consider estimation of linear functionals of the error distribution for two regression models: parametric and nonparametric, and for two types of errors: independent of the covariate and centered (type I), and conditionally centered given the covariate (type II). We show that the residual-based empirical estimators for the nonparametric type I m...
We consider estimation of linear functionals of the joint law of regression models in which responses are missing at random. The usual approach is to work with the fully observed data, and to replace unobserved quantities by estimators of appropriate conditional expectations. Another approach is to replace all quantities by such estimators. We show...
For nonparametric regression models with fixed and random design, two classes of estimators for the error variance have been introduced: second sample moments based on residuals from a nonparametric fit, and difference-based estimators. The former are asymptotically optimal but require estimating the regression function; the latter are simple but h...
The marginal density of a rst order moving average process can be written as convolution of two innovation densities. Saavedra and Cao (2000) propose to estimate the marginal density by plugging in kernel density estimators for the innovation densities, based on estimated innovations. They obtain that for an appropriate choice of bandwidth the vari...
A subthreshold signal may be detected if noise is added to the data. The noisy signal must be strong enough to exceed the threshold at least occasionally; but very strong noise tends to drown out the signal. There is an optimal noise level, called stochastic resonance. We explore the detectability of different signals, using statistical detectabili...
This paper addresses estimation of linear functionals of the error distribution in nonparametric regression models. It derives an i.i.d. representation for the empirical estimator based on residuals, using undersmoothed estimators for the regression curve. Asymptotic eciency of the estimator is proved. Estimation of the error variance is discussed...
Suppose we have independent observations from a distribution which we know to fulfill a finite-dimensional linear constraint involving an unknown finite-dimensional parameter. We construct efficient estimators for finite-dimensional functionals of the distribution. The estimators are obtained by first constructing an efficient estimator for the fun...
Suppose we observe an ergodic Markov chain and know that the stationary law of one or two successive observations fulfills a linear constraint. We show how to improve the given estimators exploiting this knowledge, and prove that the best of these estimators is efficient.
Suppose we have independent observations from a distribution which we know to fulll a nite-dimensional linear constraint involving an unknown nite-dimensional parameter. We construct ecient estimators for nite-dimensional functionals of the distribution. The estimators are obtained by rst constructing an ecient estimator for the functional when the...
If we have a parametric model for the invariant distribution of a Markov chain but cannot or do not want to use any information about the transition distribution (except, perhaps, that the chain is reversible) -- what, then, is the best use we can make of the observations? It is not optimal to proceed as if the observations were i.i.d. We determine...
Consider a locally asymptotically normal semiparametric model with a real parameter # and a possibly innite-dimensional parameter F . We are interested in estimating a real-valued functional a(F ). If ^ a # estimates a(F ) for known #, and ^ # estimates #, then the plug-in estimator ^ a ^ # estimates a(F ) if # is unknown. We show that ^ a ^ # is a...
. Suppose we observe an invertible linear process with independent mean zero innovations, and with coecients depending on a nite-dimensional parameter, and we want to estimate the expectation of some function under the stationary distribution of the process. The usual estimator would be the empirical estimator. It can be improved using that the inn...
The usual estimator for the expectation of a function under the innovation distribution of a nonlinear autoregressive model is the empirical estimator based on estimated innovations. It can be improved by exploiting that the innovation distribution has mean zero. We show that the resulting estimator is ecient if the innovations are estimated with a...
Introduction Stochastic resonance is a nonlinear cooperative eect in which large-scale stochastic uctuations (e.g., oise") are entrained by an independent, often but not necessarily periodic, weak uctuation (or signal"), with the result that the weaker signal uctuations are amplied (see Gammaitoni et al. (1998) for a review). The term stochastic re...
Consider a locally asymptotically normal semiparametric model with a real parameter # and a possibly innite-dimensional parameter F . We are interested in estimating a real-valued functional a(F ). If ^ a # estimates a(F ) for known #, and ^ # estimates #, then the plug-in estimator ^ a ^ # estimates a(F ) if # is unknown. We show that ^ a ^ # is a...
We consider d-order Markov chains satisfying a conditional constraint E(a#(Xi 1;Xi) j Xi 1) = 0, where Xi 1 = (Xi 1;:::;Xi d). These com- prise quasi-likelihood models and nonlinear and conditionally heteroscedastic autoregressive models with martingale innovations. Estimators for # can be obtained from estimating equations Pn i=1W#(Xi 1) >a#(Xi 1;...
A subthreshold signal may be detected if noise is added to the data. We study a simple model, consisting of a constant signal to which at uniformly spaced times independent and identically distributed noise variables with known distribution are added. A detector records the times at which the noisy signal exceeds a threshold. There is an optimal no...
The expectation of a local function on a stationary random eld can be estimated from observations in a large window by the empirical estimator, i.e., the average of the function over all shifts within the window. Under appropriate conditions, the estimator is consistent and asymptotically normal. Suppose that the eld is a Gibbs eld with known nite...
This is called a quasi-likelihood Markovchain model. Although it is similar to the #rst model, it is much larger: In a certain sense, it is close to nonparametric, and e#cient estimators can be constructed more easily. To estimate #,we use the fact that X i , m # #X i,1 # are martingale increments. Hence # can be estimated by solutions of martingal...
We characterize efficient estimators for the expectation of a function under the invariant distribution of a Markov chain and outline ways of constructing such estimators. We consider two models. The first is described by a parametric family of constraints on the transition distribution; the second is the heteroscedastic nonlinear autoregressive mo...
. Suppose we observe a stationary Markov chain with unknown transition distribution. The empirical estimator for the expectation of a function of two successive observations is known to be efficient. For reversible Markov chains, an appropriate symmetrization is efficient. For functions of more than two arguments, these estimators cease to be effic...
If we have a parametric model for the invariant distribution of a Markov chain but cannot or do not want to use any information about the transition distribution (except, perhaps, that the chain is reversible) --- what, then, is the best use we can make of the observations? It is not optimal to proceed as if the observations were i.i.d. We determin...
The expectation of a local function on a stationary random field can be estimated from observations in a large window by the empirical estimator, that is, the average of the function over all shifts within the window. Under appropriate conditions, the estimator is consistent and asymptotically normal. Suppose that the field is a Gibbs field with kn...
Random field models in image analysis and spatial statistics usually have local interactions. They can be simulated by Markov chains which update a single site at a time. The updating rules typically condition on only a few neighboring sites. If we want to approximate the expectation of a bounded function, can we make better use of the simulations...
We have shown elsewhere that the empirical estimator for the expectation of a local function on a Markov field over a lattice is efficient if and only if the function is a sum of functions each of which depends only on the values of the field on a clique of sites. For countable state space, the estimation of such expectations reduces to the estimat...
The expectation of a function can be estimated by the empirical estimator based on the output of a Markov chain Monte Carlo method. We review results on the asymptotic variance of the empirical estimator, and on improving the estimator by exploiting knowledge of the underlying distribution or of the transition distribution of the Markov chain.
If we wish to efficiently estimate the expectation of an arbitrary function on the basis of the output of a Gibbs sampler, which is better: deterministic or random sweep? In each case we calculate the asymptotic variance of the empirical estimator, the average of the function over the output, and determine the minimal asymptotic variance for estima...
Cox showed that the likelihood of regression models for discrete-time processes factors into a partial likelihood and a product of conditional laws for the covariates, given the history. Jacod constructed a partial likelihood for continuous-time regression models in terms of the predictable characteristics of the response process. Here we prove a f...
Suppose we want to estimate the expectation of a function of two arguments under the stationary distribution of two successive observations of a reversible Markov chain. Then the usual empirical estimator can be improved by symmetrizing. We show that the symmetrized estimator is efficient. We point out applications to discretely observed continuous...
Consider a regression model for discrete-time stochastic processes, with a (partially specified) model for the conditional distribution of the response given the covariate and the past observations. Suppose we also have some knowledge about how the parameter of interest affects the conditional distribution of the covariate given the past. We assume...
We outline the theory of efficient estimation for semiparametric Markov chain models, and illustrate in a number of simple cases how the theory can be used to determine lower bounds for the asymptotic variance of estimators and to construct efficient estimators. In particular, we consider estimation of stationary distributions of Markov chains, of...
We consider regression models in which covariates and responses jointly form a higher order Markov chain. A quasi-likelihood model specifies parametric models for the conditional means and variances of the responses given the past observations. A simple estimator for the parameter is the maximum quasi-likelihood estimator. We show that it does not...
Given a Markov chain sampling scheme, does the standard empirical estimator make best use of the data? We show that this is not so and construct better estimators. We restrict attention to nearest-neighbor random fields and to Gibbs samplers with deterministic sweep, but our approach applies to any sampler that uses reversible variable-at-a-time up...
We introduce a form of Rao--Blackwellization for Markov chains which uses the transition distribution for conditioning. We show that for reversible Markov chains, this form of Rao--Blackwellization always reduces the asymptotic variance, and derive two explicit forms of the variance reduction obtained through repeated Rao--Blackwellization. The res...
Suppose we observe an ergodic Markov chain on the real line, with a parametric model for the autoregression function, i.e. the conditional mean of the transition distribution. If one specifies, in addition, a parametric model for the conditional variance, one can define a simple estimator for the parameter, the maximum quasi-likelihood estimator. I...
Suppose we have specified a parametric model for the transition distribution of a Markov chain, but that the true transition distribution does not belong to the model. Then the maximum likelihood estimator estimates the parameter which maximizes the Kullback--Leibler information between the true transition distribution and the model. We prove that...
A semi-Markov process stays in state x for a time s and then jumps to state y according to a transition distribution Q(x; dy; ds). A statistical model is described by a family of such transition distributions. We give conditions for a nonparametric version of local asymptotic normality as the observation time tends to infinity. Then we introduce `e...
Consider an ergodic Markov chain on the real line, with parametric models for the conditional mean and variance of the transition distribution. Such a setting is an instance of a quasi-likelihood model. The customary estimator for the parameter is the maximum quasi-likelihood estimator. It is not efficient, but as good as the best estimator that ig...
A concept of asymptotically efficient estimation is presented when a misspecified parametric time series model is fitted to a stationary process. Efficiency of several minimum distance estimates is proved and the behavior of the Gaussian maximum likelihood estimate is studied. Furthermore, the behavior of estimates that minimize the h-step predicti...
Suppose we observe a uniformly ergodic Markov chain with unknown transition distribution. The empirical estimator for a linear functional of the (invariant) joint distribution of two successive observations is defined using the pairs of successive observations. Its efficiency is proved using a martingale approximation. As a corollary we show effici...
The distribution of a homogeneous, continuous-time Markov step process with values in an arbitrary state space is determined by the transition distribution and the mean holding time, which may depend on the state. We suppose that both are unknown, introduce a class of functionals which determines the transition distribution and the mean holding tim...
A multivariate point process is a random jump measure in time and space. Its distribution is determined by the compensator of the jump measure. By an empirical estimator we understand a linear functional of the jump measure. We give conditions for a nonparametric version of local asymptotic normality of the model as the observation time tends to in...
Consider a stationary first-order autoregressive process, with i.i.d. residuals following an unknown mean zero distribution. The customary estimator for the expectation of a bounded function under the residual distribution is the empirical estimator based on the estimated residuals. We show that this estimator is not efficient, and construct a simp...
A quasi-likelihood model for a stochastic process is defined by parametric models for the conditional mean and variance processes given the past. The customary estimator for the parameter is the maximum quasi-likelihood estimator. We discuss some ways of improving this estimator. For simplicity we restrict attention to a Markov chain with condition...
We give two local asymptotic minimax bounds for models which admit a local quadratic approximation at every parameter point, but are not necessarily locally asymptotically normal or mixed normal. Such parameter points appear as critical points for stochastic process models exhibiting both stationary and explosive behavior. The first result shows th...