\documentclass[runningheads]{smtda2010}

\usepackage{graphicx} 
\usepackage{amsmath,amssymb}

\def\hp{\hat p}
\def\hr{\hat r}
\def\hg{\hat g}
\def\ha{\hat a}
\def\ve{\varepsilon}
\def\hve{\hat\varepsilon}
\def\vf{\varphi}
\def\hvf{\hat\varphi}
\def\vt{\vartheta}
\def\hvt{\hat\vartheta}
\def\tp{\tilde p}
\def\tg{\tilde g}
\def\hf{\hat f}
\def\si{\sum_{i=1}^n}
\def\avi{\dfrac{1}{n}\sum_{i=1}^n}
\def\sj{\sum_{j=1}^n}
\def\avj{\dfrac{1}{n}\sum_{j=1}^n}
\def\op{o_P(1)}
\def\opn{o_P(n^{-1/2})}
\def\1{\mathbf{1}}
\def\R{\mathbb{R}}
\def\var{\mathop{\rm Var}\nolimits}
\def\endproof{\hfill $\square \;$}

\def\ben{\begin{eqnarray}}
\def\non{\end{eqnarray}}
\def\bea{\[ \begin{aligned}}
\def\ea{\end{aligned} \]}

\newcommand\bel[1]{\begin{equation}\label{#1}}
\def\ee{\end{equation}}
\newcommand{\Ref}[1]{(\ref{#1})}

\begin{document}

\title*{Non-Standard Behavior of Density Estimators 
for Functions of Independent Observations}

\toctitle{Non-Standard Behavior of Density Estimators 
for Functions of Independent Observations}

\titlerunning{Non-Standard Behavior of Density Estimators}

\author{
U. U. M\"uller\inst{1}\thanks{Supported by NSF Grant DMS 0907014.}
\and
A. Schick\inst{2}\thanks{Supported by NSF Grant DMS 0906551.}
\and
W. Wefelmeyer\inst{3}
}

\index{M\"uller, U. U.}
\index{Schick, A.}
\index{Wefelmeyer, W.}

\authorrunning{U. U. M\"uller, A. Schick and W. Wefelmeyer}

\institute{
Department of Statistics,
Texas A\&M University \\
College Station, TX 77843-3143, USA \\
(email: \texttt{uschi@stat.tamu.edu})
\and
Department of Mathematical Sciences,
Binghamton University \\ 
Binghamton, NY 13902-6000, USA \\
(email: \texttt{anton@math.binghamton.edu})
\and
Mathematical Institute,
University of Cologne \\
50931 Cologne, Germany \\
(email: \texttt{wefelm@math.uni-koeln.de})
}

\maketitle

\begin{abstract}
Densities of functions of two or more independent random variables
can be estimated by local U-statistics. Frees (1994) gives conditions 
under which they converge pointwise at the parametric root-$n$ rate.
Gin\'e and Mason (2007) give conditions under which this rate also holds
in $L_p$-norms. We present several natural applications in which 
the parametric rate fails to hold in $L_p$ or even pointwise.

1. The density estimator of a sum of squares of independent observations
typically slows down by a logarithmic factor. For exponents greater than two,
the estimator behaves like a classical density estimator. 

2. The density estimator of a product of two independent observations 
typically has the root-$n$ rate pointwise, but not in $L_p$-norms.
An application is given to semi-Markov processes and estimation 
of an inter-arrival density that depends multiplicatively
on the jump size.

3. The stationary density of a nonlinear or nonparametric 
autoregressive time series driven by independent innovations
can be estimated by a local U-statistic (now based on dependent observations
and involving additional parameters), but the root-$n$ rate can fail
if the derivative of the autoregression function vanishes at some point.
\keyword{Density estimator}
\keyword{Local U-statistic}
\keyword{Local von Mises statistic}
\keyword{Convergence rate}
\keyword{Autoregressive time series}
\keyword{Semi-Markov process}
\end{abstract}


\section{Introduction}


It is often of interest to estimate densities of known or unknown functions
of independent observations. Consider for example a regression model 
$Y=r(X)+\ve$ with independent error $\ve$ and covariate $X$.
If we have independent observations $(X_i,Y_i)$, $i=1,\dots,n$,
then the density of the response $Y$ could be estimated by a kernel estimator
based on $Y_1,\dots,Y_n$. However, a much better estimator is obtained
if we exploit the independence of $\ve$ and $X$ and write $Y$ 
as a sum $r(X)+\ve$ of independent random variables. Then the density $p$
of $Y$ can be estimated by a local von Mises statistic
\[
\hp(z) = \frac{1}{n^2}\sum_{i,j=1}^n k_b(z-\hr(X_i)-\hve_j).
\]
Here $k_b(z)=k(z/b)/b$ with kernel $k$ and bandwidth $b$,
$\hr$ is some estimator of the regression function $r$, 
and $\hve_j=Y_j-\hr_j(X_j)$ are the corresponding residuals.
Under appropriate conditions, the estimator $\hp(z)$ converges
at the parametric rate $n^{1/2}$; 
see St\o ve and Tj\o stheim, 2011~\cite{Stove},
Escanciano and Jacho-Ch\'avez, 2011~\cite{Escanciano}, and, 
for nonlinear regression and with responses missing at random, 
M\"uller, 2010~\cite{Mueller10}.
It is the purpose of this review to indicate why such rates are possible,
and to illustrate when they fail.

The most straightforward version of the problem is the following.
Let $X_1,\dots,X_n$ be independent real-valued observations with density $f$.
We want to estimate the density $p$ of some transformation $T(X_1,\dots,X_m)$
of $m$ of these observations, with $m$ at least 2.
Frees, 1994~\cite{Frees} proposed as an estimator of $p(z)$
the local U-statistic
\[
\hp(z) = \frac{1}{\binom{n}{m}}\sum_{1\leq i_1<\dots<i_m\leq n}
k_b(z-T(X_{i_1},\dots,X_{i_m}))
\]
with $k_b(x)=k(x/b)/b$ for a kernel $k$ and a bandwidth $b$.
He showed that this estimator can be pointwise $n^{1/2}$-consistent
under some assumptions on $f$ and $T$. 
Saavedra and Cao, 2000~\cite{Saavedra} consider
the function $T(X_1,X_2)=X_1+\vf X_2$. 
It is even possible to obtain $n^{1/2}$-consistency in various norms,
together with functional central limit theorems in the corresponding spaces.
Schick and Wefelmeyer, 2004~\cite{Schick04}, 2007~\cite{Schick07} 
prove such results for transformations
of the form $T(X_1,\dots,X_m)=T_1(X_1)+\dots+T_m(X_m)$ 
and $T(X_1,X_2)=X_1+X_2$ in the sup-norm and in $L_1$-norms.
Gin\'e and Mason, 2007~\cite{Gine} consider general transformations 
$T(X_1,\dots,X_m)$ and obtain such results in the $L_p$-norms. 
Their results hold locally uniformly in the bandwidth. 
More general results applicable here are in Nickl, 2007~\cite{Nickl07}
and Nickl, 2009~\cite{Nickl09}.

These results are less generally valid than appears at first sight.
In Section 2 we restrict attention to $m=2$ and to transformations 
of the special form $T(X_1,X_2)=T_1(X_1)+T_2(X_2)$
and explain under which conditions the local U-statistic $\hp(z)$
is asymptotically linear, $n^{1/2}$-consistent, and asymptotically normal.
The rate is typically slower when, say,
$T_1(y)=T_1(x)+c(y-x)^\nu+o(|y-x|^\nu)$ for $y$ to the left or right 
of some point $x$, with $\nu\geq 2$. Then the density of $T_1(X)$
has a strong peak. Specifically, we consider
$T_1(x)=T_2(x)=x^\nu$ and describe the rates of the local U-statistic. 
Then we discuss the two-sample case and applications to regression, 
to time series driven by independent innovations, 
and to renewal processes with multiplicative waiting times.


\section{Results and Applications}


Let $X_1,\dots,X_n$ be independent real-valued observations with density $f$.
An estimator for the density $p$ of a transformation of the form
$T(X_1,X_2)=T_1(X_1)+T_2(X_2)$ is the local U-statistic
\[
\hp(z) = \frac{2}{n(n-1)}\sum_{1\leq i<j\leq n} k_b(z-T_1(X_i)-T_2(X_j)),
\]
where $k_b(z)=k(z/b)/b$ for a kernel $k$ and a bandwidth $b$.
Suppose that $T_1(X)$ and $T_2(X)$ have densities $g_1$ and $g_2$.
The estimator $\hp(z)$ has the Hoeffding decomposition
\bea
\hp(z) = p*k_b(z) + \avi\big(&g_1*k_b(z-T_2(X_i))-p*k_b(z) \\
&+g_2*k_b(z-T_1(X_i))-p*k_b(z)\big) 
+ U(z),
\ea
where
\bea
U(z) = \frac{2}{n(n-1)}\sum_{1\leq i<j\leq n}
\big(&k_b(z-T_1(X_i)-T_2(X_i))-g_1*k_b(z-T_2(X_i)) \\
&-g_2*k_b(z-T_1(X_i))+p*k_b(z)\big)
\ea
is a degenerate local U-statistic. We have 
\[
n(n-1)E[U^2(z)] \leq 2E[k_b^2(z-T_1(X_1)-T_2(X_2))] = 2p*k_b^2(z)
\]
and
\[
p*k_b^2(z) = \frac{1}{b}\int p(z-bu)k^2(u)\,du 
\leq \frac{\|p\|_\infty}{b}\int k^2(u)\,du.
\]
If $p$ is bounded and $\int k^2(u)\,du$ is finite, we obtain
$U(z)=O_P(1/(nb^{1/2}))$, which is of order $\opn$ if $nb\to\infty$.
The Hoeffding decomposition then says that the centered local U-statistic
$\hp(z)-p*k_b(z)$ is approximated by a sum of two centered and smoothed
empirical ``estimators'' of $p(z)$ (that involve the unknown densities
$g_1$ and $g_2$). Under mild assumptions one can remove the smoothing;
see e.g.\ Schick and Wefelmeyer, 2004~\cite{Schick04}.
If $p$ is H\"older with exponent $\alpha$, then the bias
$p*k_b(z)-p(z)$ is of order $o(n^{-1/2})$ if $nb^{2\alpha}\to 0$.
This implies that $\hp(z)$ is asymptotically linear,
\bel{1}
\hp(z) = p(z) + \avi\big(g_1(z-T_2(X_i))+g_2(z-T_1(X_i))-2p(z)\big) + \opn.
\ee
If $E[g_1^2(z-T_2(X_2))]$ and $E[g_2^2(z-T_1(X_1))]$ are finite,
then $\hp(z)$ is $n^{1/2}$-consistent and asymptotically normal.

\begin{remark}\label{rem.1}
(Convolution of density estimators.)
The density $p$ has the convolution representation
\[
p(z) = \int g_2(z-y)g_1(y)\,dy.
\]
Therefore, it can also be estimated by a convolution of density estimators
\[
\hg_{conv}(z) = \int \hg_2(z-y)\hg_1(y)\,dy
\]
with kernel estimator for $g_1(y)$ based on $T_1(X_1),\dots,T_1(X_n)$,
\[
\hg_1(y) = \avi k_b(y-T_1(X_i)),
\]
and, correspondingly,
\[
\hg_2(y) = \avi k_b(y-T_2(X_i)).
\]
The estimator $\hg_{conv}$ is asymptotically equivalent to $\hg$. \endproof
\end{remark}

\begin{remark}\label{rem.2}
(Transform density estimator or transform observations.)
Suppose that $T_1$, say, is strictly increasing and differentiable. 
Then the density of $T_1(X)$ at $y$ is
\[
g_1(y) = \frac{f(T_1^{-1}(y))}{T_1'(T_1^{-1}(y))}.
\]
We obtain an alternative estimator of $g_1(y)$ by plugging in 
a kernel estimator for $f$,
\[
\hf(x) = \avi k_b(x-X_i).
\]
We expect that it depends on $T_1$ whether $\hg_1(y)$ is better than
\[
\tg_1(y) = \frac{\hf(T_1^{-1}(y))}{T_1'(T_1^{-1}(y))}.
\]
In the convolution representation $p(z)=\int g_2(z-y)g_1(y)\,dy$
we can use $\hg_1$ or $\tg_1$.
If $T_2$ is also strictly increasing and differentiable,
we can combine $\hg_1$ or $\tg_1$ with $\hg_2$ or $\tg_2$. \endproof
\end{remark}

We now discuss cases in which $\hp(z)$ is not $n^{1/2}$-consistent.

\begin{remark}\label{rem.3}
(Piecewise constant transformations.)
The distribution of $T_1(X)$ does not always have a density.
Suppose that $T_1$ is piecewise constant,
\[
T_1(X) = \sum_{s=1}^t c_s\1[X\in I_s],
\]
with $c_s\in\R$, and $I_s$, $s=1,\dots,t$, a partition of $\R$.
If $T_2(X)$ has a density $g_2$, then $T_1(X_1)+T_2(X_2)$
has a density $p$ that is a finite mixture of shifts of $g_2$,
\[
p(z) = \sum_{s=1}^m a_s g_2(z-c_s)
\]
with weights $a_s=P(X\in I_s)$. As soon as each interval $I_s$
contains at least one observation, the constants $c_s$ can be observed,
and $p(z)$ can be estimated by
\[
\hp(z) = \sum_{s=1}^t \ha_s \hg_2(z-c_s),
\]
where $\ha_s=(1/n)\si\1[X_i=c_s]$. The rate of $\hp(z)$
equals the pointwise rate of $\hg_2$. \endproof
\end{remark}

Even if $T_1$ and $T_2$ are not constant on any interval,
$\hp(z)$ can fail to be $n^{1/2}$-consistent.
In the following we describe a situation in which $T_1(X)$ and $T_2(X)$
have densities, but $g_1(z-T_2(X))$ does not necessarily have finite variance.
For notational simplicity, assume that $f$ is supported on $(0,\infty)$,
and set $T_1(x)=T_2(x)=x^\nu$ for some $\nu>0$.
Then $g_1=g_2=g$ with
\[
g(y) = \frac{1}{\nu}y^{1/\nu-1}f(y^{1/\nu}),
\]
and the stochastic expansion \Ref{1} of $\hp(z)$ specializes to
\bel{2}
\hp(z) = p(z) + \frac{2}{n}\si g(z-X_i^\nu) + \opn.
\ee
In the theorems below, we take the kernel $k$ 
to be continuously differentiable with support $[-1,1]$. 
We also assume that $f$ is bounded.
First let $\nu<2$. Then $g$ is square-integrable, and $g(z-X^\nu)$
has finite variance. By the arguments of Schick and Wefelmeyer, 
2004~\cite{Schick04} and 2009~\cite{Schick-squared} or 
Gin\'e and Mason, 2007~\cite{Gine}, we have the following result.

\begin{theorem}\label{thm.1}
Let $\nu<2$.
Suppose the density $f$ is of bounded variation and $f(0+)$ is positive.
Let $b\sim(\log n)^{1/2}/n$. 
Then $\hp(z)$ has the stochastic expansion \Ref{2}, and
\[
n^{1/2}\big(\hp(z)-p(z)\big) 
\Rightarrow N\big(0,4\var(g(z-X^{\nu}))\big).
\]
\end{theorem}

For $\nu=2$, square-integrability of $g$ fails just barely,
resulting in a rate for $\hp(z)$ that is only slightly worse than $n^{-1/2}$.
More precisely, Schick and Wefelmeyer, 2009~\cite{Schick-powers}
prove the following result.

\begin{theorem}\label{thm.2}
Let $\nu=2$.
Suppose $f$ is of bounded variation, and $f(0+)$ and $g(z-)$ are positive.
Let $b\sim(\log n)^{1/2}/n$. Then 
\[
\Big(\frac{n}{\log n}\Big)^{1/2}\big(\hp(z)-p(z)\big) 
\Rightarrow N(0,f^2(0+)g(z-)).
\]
\end{theorem}

For $\nu>2$, the rate of $\hp(z)$ is of order $n^{-1/\nu}$
if $f$ is of bounded variation and $f(0+)$ and $g(z-)$ are positive.
Faster rates are possible under additional smoothness assumptions 
on $p$ at $z$.

\begin{theorem}\label{thm.3}
Let $\nu>2$.
Suppose $f$ is of bounded variation, and $f(0+)$ and $g(z-)$ are positive.
Let $b\sim 1/n$.  Then 
\[
\hp_b(z)-p(z)= O_P(n^{-\beta}).
\]
\end{theorem}

Even in the case $\nu\geq 2$, the estimator $\hp(z)$ can be 
$n^{1/2}$-consistent if $g(z-)=0$ since this works against 
the peak of $g$ at 0 in the representation $p(z)=g*g(z)$.
For details we refer to Schick and Wefelmeyer, 2009~\cite{Schick-powers}
and 2009~\cite{Schick-squared}.

We will now briefly discuss possible applications of the above results.

\begin{remark}\label{rem.4}
(Several samples.)
The above results carry over to $m$-sample cases. We restrict attention 
to $m=2$. Suppose $X_1,\dots,X_n$ and $Z_1,\dots,Z_n$ are real-valued
and independent with densities $f_1$ and $f_2$, respectively.
An estimator for the density $p$ of a transformation $T_1(X)+T_2(Z)$
is the local von Mises statistic
\[
\hp(z) = \frac{1}{n^2}\sum_{i,j=1}^n k_b(z-T_1(X_i)-T_2(Z_j)).
\]
Let $g_1$ and $g_2$ denote the densities of $T_1(X)$ and $T_2(Z)$.
As in the one-sample case \Ref{1} we obtain a stochastic expansion
\[
\hp(z) = p(z) + \avi\big(g_1(z-T_2(Z_i))+g_2(z-T_1(X_i))-2p(z)\big) + \opn.
\]
Appropriate versions of Theorems \ref{thm.1}--\ref{thm.3}
continue to hold. \endproof
\end{remark}

\begin{remark}\label{rem.5}
(Regression.)
Two-sample results can be applied to regression models $Y=r(X)+\ve$
with $\ve$ independent of $X$. If we have independent observations
$(X_i,Y_i)$, $i=1,\dots,n$, then the density $p$ of $Y$ can be estimated
by the local von Mises statistic
\[
\hp(z) = \frac{1}{n^2}\sum_{i,j=1}^n k_b(z-\hr(X_i)-\hve_j)
\]
based on some estimator $\hr$ of the regression function $r$,
and on residuals $\hve_j=Y_j-\hr(X_j)$. Note that the ``pseudo-observations''
$\hr(X_i)$ and $\hve_j$ are only approximately independent, so we are  
close to the two-sample case with $Z=\ve$, $T_1(X)=r(X)$, and $T_2(\ve)=\ve$.
As seen above, we can expect a rate $n^{-1/2}$ for $\hp(z)$
if $r$ has a derivative that is bounded away from 0.

Suppose $r$ is only piecewise monotone and continuously differentiable,
and there are points $x$ with 
\[
r(y) = c(y-x)^\nu + o(|y-x|^\nu)
\]
for $y$ to the left or right of $x$.
Then the convergence rate of $\hp(z)$ will be determined 
by the largest such $\nu$. \endproof
\end{remark}

\begin{remark}\label{rem.6}
(Time series.)
Results for regression carry over to time series driven by independent 
innovations. Consider a first-order moving average process
$X_i=\ve_i+\vf\ve_{i-1}$, with independent innovations $\ve_i$
that have mean 0, finite variance, and density $f$.
If $\vf\neq 0$, the stationary density $p$ can be estimated by 
a local von Mises statistic
\[
\hp(z) = \frac{1}{n^2}\sum_{i,j=1}^n k_b(z-\hve_i-\hvf\hve_j)
\]
with $\hvf$ an estimator of $\vf$. Saavedra and Cao, 1999~\cite{Saavedra99}
obtain $n^{1/2}$-consistency; see also 
Schick and Wefelmeyer, 2004~\cite{Schick04a}.
Functional results for higher-order moving average
processes and general linear processes are obtained in
Schick and Wefelmeyer, 2004~\cite{Schick04c},
2007~\cite{Schick07a}, 2008~\cite{Schick08} and 2009~\cite{Schick09}.
Nonlinear and nonparametric time series can also be treated. \endproof
\end{remark}


\begin{remark}\label{rem.7}
(Renewal processes.)
Here is a two-sample case where $T(X,Z)$ is a product rather than a sum of
functions $T_1(X)$ and $T_2(Z)$.
Let $(X_i,T_i)$, $i=0,\dots,n$ be observations of a Markov renewal process 
with real state space. Assume that the embedded Markov chain is stationary.
We make the structural assumption that the waiting times depend
multiplicatively on some power of the distance between the previous
and the present state of the embedded Markov chain,
\[
T_i-T_{i-1} = |X_i-X_{i-1}|^\nu W_i,
\]
where $\nu>0$ and the $W_i$ are independent with density $g$ and independent
of the embedded Markov chain. Note that $W_i$ is observable as a function 
of the observations $(X_{i-1},T_{i-1})$ and $(X_i,T_i)$. 
We can estimate the waiting time density $p$ of $T_i-T_{i-1}$
by the local von Mises statistic
\[
\hp(z) = \frac{1}{n^2}\sum_{i,j=1}^n k_b(z-|X_i-X_{i-1}|^\nu W_j).
\]
Greenwood et al., 2011~\cite{Greenwood} give conditions under which $\hp(z)$
has rate $n^{-1/2}$ and is asymptotically linear and asymptotically normal.
\endproof
\end{remark}

\begin{thebibliography}{20}

\bibitem{Escanciano}
Escanciano, J. C. and Jacho-Ch\'avez, D. T. 
``$\sqrt{n}$-Uniformly Consistent Density Estimation 
in Nonparametric Regression Models''. 
To appear in \emph{J. Econometrics} (2011).

\bibitem{Frees}
Frees, E. W.
``Estimating Densities of Functions of Observations''.
\emph{J. Amer.\ Statist.\ Assoc.} 89:517--525 (1994).

\bibitem{Gine}
Gin\'e, E. and Mason, D.
``On Local $U$-Statistic Processes and the Estimation of Densities 
of Functions of Several Sample Variables''.
\emph{Ann.\ Statist.} 35:1105--1145 (2007).

\bibitem{Greenwood}
Greenwood, P. E., Schick, A. and Wefelmeyer, W.
``Estimating the Inter-Arrival Time Density of Semi-Markov Processes
under Structural Assumptions on the Transition Distribution''.
\emph{Statist.\ Probab.\ Lett.} 81:277--282 (2011).

\bibitem{Mueller10}
M\"uller, U. U.
``Estimating the Density of a Possibly Missing Response Variable
in Nonlinear Regression''.
Submitted (2010).

\bibitem{Nickl07}
Nickl, R.
``Donsker-Type Theorems for Nonparametric Maximum Likelihood Estimators''.  
\emph{Probab.\ Theory Related Fields} 138:411--449 (2007). 
Erratum 141:331--332 (2008).

\bibitem{Nickl09}
Nickl, R.
``On Convergence and Convolutions of Random Signed Measures''.
\emph{J. Theoret.\ Probab.} 22:38--56 (2009).

\bibitem{Saavedra99}
Saavedra, A. and Cao, R.
``Rate of Convergence of a Convolution-Type Estimator
of the Marginal Density of an MA(1) process''.
\emph{Stochastic Process.\ Appl.} 80:129--155 (1999).

\bibitem{Saavedra}
Saavedra, A. and Cao, R.
``On the Estimation of the Marginal Density of a Moving Average Process''.
\emph{Canad.\ J. Statist.} 28:799--815 (2000).

\bibitem{Schick04a}
Schick, A. and Wefelmeyer, W. 
``Root $n$ Consistent and Optimal Density Estimators for
Moving Average Processes''.
\emph{Scand.\ J. Statist.} 31:63--78 (2004).

\bibitem{Schick04}
Schick, A. and Wefelmeyer, W.
``Root $n$ Consistent Density Estimators for Sums 
of Independent Random Variables''.
\emph{J. Nonparametr.\ Statist.} 16:925--935 (2004).

\bibitem{Schick04c}
Schick, A. and Wefelmeyer, W.
``Functional Convergence and Optimality of Plug-in Estimators
for Stationary Densities of Moving Average Processes''.
\emph{Bernoulli}, \textbf{10}, 889--917 (2004).

\bibitem{Schick07}
Schick, A. and Wefelmeyer, W.
``Root-$n$ Consistent Density Estimators of Convolutions in 
Weighted $L_1$-Norms''.
\emph{J. Statist.\ Plann.\ Inference} 37:1765--1774 (2007).

\bibitem{Schick07a}
Schick, A. and Wefelmeyer, W.
``Uniformly Root-$n$ Consistent Density Estimators for
Weakly Dependent Invertible Linear Processes''.
\emph{Ann.\ Statist.} 35:815--843 (2007).

\bibitem{Schick08}
Schick, A. and Wefelmeyer, W. 
``Root-$n$ Consistency in Weighted $L_1$-Spaces for Density Estimators 
of Invertible Linear Processes''.
\emph{Stat.\ Inference Stoch.\ Process.}, 11:281--310 (2008).

\bibitem{Schick-powers}
Schick, A. and Wefelmeyer, W.
``Convergence Rates of Density Estimators for Sums of Powers 
of Observations''.
\emph{Metrika} 69:249--264 (2009).

\bibitem{Schick-squared}
Schick, A. and Wefelmeyer, W.
``Non-Standard Behavior of Density Estimators for Sums 
of Squared Observations''. 
\emph{Statist.\ Decisions} 27:55-73 (2009).

\bibitem{Schick09}
Schick, A. and Wefelmeyer, W. 
``Improved Density Estimators for Invertible Linear Processes''.
\emph{Comm.\ Statist.\ Theory Methods} 38:3123-3147 (2009).

\bibitem{Stove}
St\o ve, B. and Tj\o stheim, D.
``A Convolution Estimator for the Density of Nonlinear Regression 
Observations''.
To appear in \emph{Scand.\ J. Statist.} (2011).

\end{thebibliography}

\end{document}
