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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\title[Testing for additivity]{Testing for additivity in partially
linear regression \\
with possibly missing responses}
%\footnote{AMS 2000 subject classification: Primary 62G05, 62G08, 62G20, 62J02
%\newline
%Key words and Phrases: Local linear smoother, i.i.d.\ representation,
%Donsker class, efficiency.
%}}
\author{Ursula U. M\"uller, Anton Schick and Wolfgang Wefelmeyer}

\address{Ursula U. M\"uller,
%\thanks{Ursula U. M\"uller was supported by NSF Grant DMS 0907014.}, 
Department of Statistics, 
Texas A\&M University,
College Station, TX 77843-3143, USA}
\address{Anton Schick,
%\thanks{Anton Schick was supported by NSF Grant DMS 0906551.}
Department of Mathematical Sciences,
Binghamton University,
Binghamton, NY 13902-6000, USA}
%\texttt{anton@math.binghamton.edu} \\
%\texttt{http://math.binghamton.edu/anton/index.html}
\address{Wolfgang Wefelmeyer,
Mathematical Institute,
University of Cologne,
Weyertal 86--90,
50931 Cologne, Germany}
%\texttt{anton@math.binghamton.edu} \\
%\texttt{http://math.binghamton.edu/anton/index.html}
\thanks{Ursula U. M\"uller was supported by NSF Grant DMS0907014.
Anton Schick was supported by NSF Grant DMS 0906551.}


\begin{abstract}
We consider a partially linear regression model with 
multivariate covariates %both in the linear and in the nonparametric part
%of the regression function.
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 test statistic is based on the difference of two empirical 
estimators that estimate the errors in two different ways: 
the first uses a local polynomial smoother for the nonparametric part; 
the second estimates the additive components
by a marginal integration estimator derived from the local polynomial
smoother. We present a uniform stochastic expansion of the 
empirical estimator based on the marginal integration estimator, and 
we derive the asymptotic distribution of the test statistic.
The transfer principle of Koul, M\"uller and Schick (2012) then allows a direct
adaptation of the results to the case when responses are missing at 
random. We examine the performance of the tests in a small simulation 
study. 
\end{abstract}

\maketitle

\emph{Key words:} Partially linear regression, 
nonparametric regression, additive regression, local polynomial smoother, 
marginal integration estimator, uniform stochastic expansion, 
goodness of fit, transfer principle, responses missing at random.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Introduction} 
\label{sec1}


We consider the partially linear regression model
$$
Y=\vt ^{\top}U+\vr(X) + \ve,
$$
where the error $\ve$ has mean zero, finite variance $\sigma^2$ 
and a density $f$
and is independent of the covariate pair $(U,X)$, 
with $U$ a $p$-dimensional random vector 
and $X$ a $q$-dimensional random vector,
and where $\vt$ is an unknown vector in $\R^p$ and $\vr$ is an unknown smooth function.
In the ideal situation one observes the triplet $(U,X,Y)$.  
Here we are also interested in the case when some responses are missing,
which is a common problem in applications.
Then one observes $(\delta,U,X,\delta Y)$ with $\delta$ 
an indicator random variable,
with the interpretation that for $\delta=1$
one observes the full triplet $(U,X,Y)$,
while for $\delta=0$ one observes only the covariates $(U,X)$.  
We make the assumption that the responses are \emph{missing at random},
which means that the conditional distribution of $\delta$ given $(U,X,Y)$ 
depends only on the covariates $(U,X)$,
$$
P(\delta=1|U,X,Y)= P(\delta=1|U,X).
$$
Monographs on missing data are Little and Rubin (2002) and Tsiatis (2006).
We want to test the hypothesis that $\vr$ is \emph{additive}, i.e., 
$$
\vr(x) = \vr_1(x_1) + \cdots + \vr_q(x_q).
$$
%We will assume that the components $\vr_1, \cdots, \vr_q$ are smooth. 
It is important to have a diagnostic tool to assess additivity. 
As shown by Stone (1985), additive models avoid the curse 
of dimensionality and are easy to interpret.
Recently, Koul et al.\ (2012) have developed a method, 
which they call the \emph{transfer principle},
that makes it easy to derive procedures for missing data 
from those with fully observed data.  We will follow their approach here.
First we develop a test procedure for the model with fully observed data,
which we describe next. 

Assume that we observe $n$ independent copies 
$(U_1,X_1,Y_1),\dots,(U_n,X_n,Y_n)$ of $(U,X,Y)$.
% and 
%want to test the hypothesis that $\vr$ is additive, 
%i.e. $\vr(x) = \vr_1(x_1) + \cdots + \vr_q(x_q)$, with smooth
%components $\vr_1, \cdots, \vr_q$. 
Our test statistic for additivity will be of the form
$$
T = n^{1/2} \norm{\hemp-\temp} = n^{1/2}\; \sup_{t\in \R} |\hemp(t)-\temp(t)| 
$$
with two different residual-based empirical distribution functions 
$\hemp$ and $\temp$.
%$$
%\hemp(t)= \avj \1[\hve_j \leq t]
%\und 
%\temp(t)= \avj \1[\tve_j \leq t].	
%$$
The first uses residuals of the form
$\hve_j = Y_j - \hvt^\top U_j - \rhat(X_j)$ 
with $\rhat$ a local polynomial smoother based on
the covariates $X_j$ and the ``observations'' $Y_j-\hvt^\top U_j$.
The second exploits the additivity assumption and works with residuals 
of the form $\tve_j = Y_j - \hvt^\top U_j - \rtilde(X_j)$
with $\rtilde$ the marginal integration estimator derived from $\rhat$.
In both cases, $\hvt$ is some $\sqrt{n}$-consistent estimator of $\vt$.
Efficient estimators of $\vt$ for additive $\vr$
are constructed in Schick (1996b).
% and Yu et al.\ (2011).
Our test statistic $T$ is a variant of the test statistic in 
Neumeyer and Van Keilegom (2010) who test for additivity 
in a nonparametric regression model with heteroskedastic errors.
%$Y= m(X)+ \sigma(X)\epsilon$. 
These authors study a bootstrap test based on their test statistic. 
Here we use instead the asymptotic distribution to develop our test. 
We show that, under additivity, $T$ converges in distribution to $\kappa |Z|$, 
where $Z$ is standard normal and $\kappa$ is a constant depending 
on the underlying distribution. 
This leads us to the test $\1[T> \hat \kappa z_{\alpha/2}]$ 
which rejects the null hypothesis if $T$ exceeds $\hat \kappa z_{\alpha/2}$ 
with $z_{\alpha/2}$ the $(1-\alpha/2)$-quantile 
of the standard normal distribution 
and $\hat \kappa$ a consistent estimator of $\kappa$.

Our test for missing data uses the \emph{complete-case} version 
of the above test, which is constructed using only the observations 
with observed responses. More precisely, we reject the null hypothesis 
if $T_c$ exceeds $\hat \kappa_c z_{\alpha/2}$, where $T_c$ and $\hat \kappa_c$ 
are the complete-case versions of $T$ and $\hat \kappa$.  
The complete-case version of a statistic 
$S_n=s_n((U_1,X_1,Y_1),\dots,(U_n,X_n,Y_n))$ 
is of the form 
$S_c = s_N((U_{i_1},X_{i_1},Y_{i_1}),\dots, (U_{i_N},X_{i_N},Y_{i_N}))$,
where $(U_{i_1},X_{i_1},Y_{i_1}),\dots, (U_{i_N},X_{i_N},Y_{i_N})$ are 
the $N=\sj \delta_j$ observations with observed responses.
An implementation of the test is straightforward since
it suffices to write a program for the model with fully observed data.
This program then can be used for applications with responses
missing at random: just delete all cases that involve missing responses
and work with the the remaining $N$ cases that are complete.

The reason for using the marginal integration estimator in $\temp$ is that
the stochastic expansion of $\temp$ is then {\em different} from that of
$\hemp$ even under the hypothesis of additivity of $\vr$, 
as will be shown in Section \ref{sec2}. 
This is necessary for the test based on $T$
to have power under \emph{contiguous} alternatives of the form
$\vr(x)=\vr_1(x_1)+\cdots+\vr_q(x_q)+n^{-1/2}s(x)$.
The two stochastic expansions of $\hemp$ and $\temp$ 
imply in particular an expansion of our test statistic $T$
under the hypothesis of additivity. From this we obtain the
asymptotic distribution of $T$ and hence an asymptotic critical value
for the test. 

We note that the marginal integration estimator is not particularly 
suited for estimating the error distribution function $F$. 
A better estimator would be the series estimator studied in Section 4 
of M\"uller et al.\ (2012). 
The empirical distribution function of this estimator 
would however be stochastically equivalent to $\hemp$ and therefore 
lead to a test with local asymptotic power equal to the significance level.
The estimator $\hemp$ was studied in M\"uller et al.\ (2012), 
generalizing results 
of M\"uller et al.\ (2007) who  estimate the error distribution 
function in the \emph{partially linear regression model} 
but only for one-dimensional $X$.
%They use a local \emph{linear} smoother for the regression function. 
%Here we follow the approach of that paper, but, in order to handle the more
%complex case when $X$ is a random \emph{vector}, we use local 
%\emph{polynomial} smoothers as in 
The case $\vt=0$ was studied by M\"uller et al.\ (2009), 
and by Neumeyer and Van Keilegom (2010), who assume heteroskedastic 
errors.

%In the one-dimensional case both methods can be used.
%Which one to choose will depend on the smoothness assumptions 
%about the regression function and the error distribution one is willing to make.
%For more explanations we refer to M\"uller et al.\ (2009), in particular 
%to Remark 5. 

The components of the regression function in \emph{additive} regression models
can be estimated in several ways. 
Stone (1985) uses an additive spline estimator.
The backfitting method of
Breiman and Friedman (1985), and Buja et al.\ (1989), estimates
the additive components one by one and iterates this procedure.
Orthogonal series estimators for semiparametric regression
models are studied by Eubank et al.\ (1990),
Andrews (1991), Donald and Newey (1994), Eubank (1999),
Li (2000), and Delecroix and Protopopescu (2001); for partially linear
additive regression models see M\"uller et al.\ (2012).
We use the marginal integration method of Newey (1994),
Tj\o stheim and Auestad (1994), and Linton and Nielsen (1995). The method
starts with an estimator $\rhat$ for a multivariate nonparametric regression 
function and obtains estimators for the additive components by integrating 
out all but one of the variables, usually with empirical estimators based
on the remaining components of the covariates. Linton (1997) uses marginal
integration to provide an initial estimator, 
and then a single backfitting step.
See also Fan et al.\ (1998), and Mammen et al.\ (1999).
The estimators are compared by Sperlich et al.\ (1999),
Delecroix and Protopopescu (2000), and Dette et al.\ (2005).

Residual-based empirical distribution functions can be used
to test various other hypotheses about regression models.
Tests for parametric hypotheses about the regression function
are considered in nonparametric regression by Stute (1997),
Khmaladze and Koul (2004, 2009), and Stute et al.\ (2008).
Tests for a parametric regression function in heteroscedastic 
nonparametric regression are studied in Van Keilegom et al.\ (2008).

The paper is organized as follows. In Section \ref{sec2}
we derive a uniform stochastic expansion for $\temp$.
The proof is is Section \ref{sec6}. We apply the result to testing
$\vr(x) = \vr_1(x_1) + \cdots + \vr_q(x_q)$ in Section \ref{sec3}.
Section \ref{secmar} shows how the results carry over to
the situation with responses missing at random.
In Section \ref{secsim} we study the performance of the test 
with simulations. 


\section{Residual-based empirical distribution functions}
\label{sec2}


First we consider the general partially linear model
$Y = \vt^\top U + \vr(X) + \ve$, where the error $\ve$ has mean zero, 
finite variance and a density $f$,  and is independent 
of the covariate pair $(U,X)$, with $U$ a $p$-dimensional random vector 
and $X$ a $q$-dimensional random vector and $\vr$ is a smooth function.
We make the following standard assumptions on $U$ and $X$.

\begin{enumerate}

\item[(G)] The distribution $G$ of $X$ is quasi-uniform on $\cube=[0,1]^q$ 
in the sense that $G(\cube)=1$ and has a density $g$ that is bounded 
and bounded away from zero on $\cube$.

\item[(H)] The covariate vector $U$ satisfies $E[|U|^2] < \infty$ and 
the matrix 
$$
W= E[(U-E(U|X))(U-E(U|X))^{\top}]
$$ 
is positive definite.

\end{enumerate}


For a non-negative integer $m$ and a $\gamma \in (0,1]$ 
we introduce the H\"older space $\H_q(m,\gamma)$ as follows.
We say that a function $h$ from $\cube$ to $\R$ belongs to $\H_q(m,\gamma)$ 
if it has continuous partial derivatives up to order $m$ and the 
partial derivatives of order $m$ are H\"older with exponent $\gamma$.
We assume that the function $\vr$ belongs to $\H_q(m,\gamma)$,
and estimate it by a local polynomial smoother 
of degree $m$; see Stone (1980, 1982), and Ruppert and Wand (1994)
for general results on multivariate local polynomial smoothers. 
Such estimators were used in M\"uller et al.\ (2009) for estimating
the error distribution function in the case $\vt=0$,
i.e., for the nonparametric regression model. 
Since $\vt$ is not zero here, we need a
$\sqrt{n}$-consistent estimator $\hvt$ of $\vt$.
Such estimators exist, see e.g. Schick (1996a). 
%,b) and Yu et al.\ (2011).
Then we work with the difference $Y_j-\hvt^\top U_j$ instead of the 
response variable $Y_j$.

In order to define the local polynomial smoother, we introduce some notation. 
By a \emph{multi-index} we mean a $q$-dimensional vector $i=(i_1,\dots,i_q)$ 
whose components are non-negative integers. 
For a multi-index $i$ let $\psi_i$ denote the function on $\R^q$ defined by 
$$
\psi_i(x)= \dfrac{x_1^{i_1}}{i_1!}\cdots \dfrac{x_q^{i_q}}{i_q!}, 
\quad x=(x_1,\dots,x_q)\in \R^q.
$$
Set $\idot= i_1+\dots+i_q$.  Let $I(m)$ denote the set of multi-indices 
$i$ with $\idot\leq m$,
and $J(m)$ the set of multi-indices $i$ with $\idot=m$.
Now fix densities $w_1,\dots,w_q$ and set 
$$
w(x)= w_1(x_1)\cdots w_q(x_q), \quad x=(x_1,\dots,x_q)\in \R^q.
$$
Let $c_n$ be a bandwidth.
%Fix a non-negative integer $d$. 
Then the  \emph{local polynomial smoother $\rhat$ (of degree $m$)} 
is defined as follows.  For a fixed $x$ in $\cube$,
the estimator $\rhat(x)$ is the component 
$\hb_0(x)$ corresponding to the multi-index $0=(0,\dots,0)$ 
of a minimizer 
$$
\hb(x) =\argmin_{\beta=(\beta_i)_{i\in I(m)}} \sj 
w\Big(\dfrac{X_j-x}{c_n}\Big)
\Bigg( Y_j -\hvt^\top U_j 
- \sum_{i\in I(m)} \beta_i \psi_i\Big(\dfrac{X_j-x}{c_n}\Big)\Bigg)^2.
$$
We estimate the errors $\ve_j$ by the residuals 
$$
\hve_j = Y_j - \hvt^\top U_j - \rhat(X_j).
$$
The empirical distribution functions for $F$ based on the errors $\ve_j$ and 
on the residuals $\hve_j$, respectively, are denoted by
$$
\emp(t) = \frac1n \sj \1[\ve_j \leq t], \qquad
\hemp(t) = \frac1n \sj \1[\hve_j \leq t].
$$
Let us write 
$$
\mu(X)=E(U|X), \qquad
\tau(X)=E(|U|^2|X).
$$
M\"uller et al.\ (2012) have shown the following uniform 
stochastic expansion for $\hemp$.

%%%%%%%%%%%%%%%%%%%%
%   Theorem 1
%%%%%%%%%

\Thm{1}
Suppose (G) and (H) hold,  $\|U\|$ has a moment greater than 2, 
$\mu$ is continuous and $\tau g$ is bounded.
Suppose that $\vr$ belongs to $\H_q(m,\gamma)$ with $s=m+\gamma>3q/2$.
Let the error density $f$ have mean zero, 
a finite moment of order greater than $4s/(2s-q)$, 
and be H\"older with exponent greater than $q/(2s-q)$.
%Let $\hvt$ be a $\sqrt{n}$-consistent estimator of $\vt$.
Let the densities $w_1,\dots,w_q$ be $(q+2)$ times continuously
differentiable with compact support $[-1,1]$. 
Choose a bandwidth $c_n\sim (n \log n)^{-1/(2s)}$.
Then we have the uniform stochastic expansion
$$
\sup_{t\in \R} 
\Big| \hemp(t) - \emp(t) - f(t) \avj \ve_j \Big|
= \opn.
$$
\end{theorem}


The smoothness parameter $s = m + \gamma$ is assumed to be greater than
$3q/2$. This means that the higher the dimension $q$ of the covariate vector
$X$, the more partial derivatives for $\vr$ we need.
We also point out that  
$4s/(2s-q)<3$ and $q/(2s-q)<1/2$ if $s>3q/2$.
Thus the assumptions on the error density $f$ are satisfied 
if $f$ has mean zero, a finite third moment, and is 
H\"older with exponent $1/2$. 
The H\"older condition is met by all densities 
with finite Fisher information for location.

\bigskip
Suppose now that the regression function $\vr$ is \emph{additive}, 
$\vr(x) = \vr_1(x_1) + \cdots + \vr_q(x_q)$.
For this model we introduce an estimator for $\vr$
such that the corresponding empirical distribution function has a
stochastic expansion that is different from that of the empirical
distribution function $\hemp$ based on the above local polynomial smoother
$\rhat$, even in this submodel.
Specifically, we take the \emph{marginal integration estimator} $\rtilde$ 
of $\vr$,
$$
\tilde \vr(x)=(1-q) \bar Y_* + \sum_{l=1}^q \avj \hat \vr(X_{j,-l}(x_l)), 
$$
where the random vector $X_{j,-l}(x_l)$ is obtained from $X_j$ by replacing 
its $l$-th coordinate by $x_l$, and where $\bar Y_*$ is the average  
\bel{ybar}
\bar Y_* = \avj (Y_j- \hvt^{\top} U_j) = \avj \ve_j + \avj \vr(X_j) 
-(\hvt-\vt)^{\top} \avj U_j.
\ee
This leads to the residuals 
$\tilde \ve_j = Y_j -\hvt^{\top}U_j - \tilde \vr(X_j)$ %, $j=1,\dots,n$, 
and to the residual-based empirical distribution function
$$
\temp(t)= \avj \1[\tilde \ve_j \le t] %, \quad t\in \R,
$$
for $F$. The asymptotic behavior of $\hemp$ differs from that of $\temp$,
as shown next. 
To state the result, we need some notation. 

For $l=1,\dots,q$ let $g_l$ denote the density of the $l$-th coordinate of 
$X$, let $g_{-l}$ denote the density of the vector obtained from $X$ by 
deleting its $l$-th coordinate, and set
$$
g_{(l)} (x)= g_l(x_l) g_{-l}(x_1,\dots,x_{l-1}, x_{l+1},\dots,x_q),
\quad x=(x_1,\dots,x_q)\in \R^q.
$$
Note that $g_{(l)}$ is the density of $X_{1,-l}(X_{2,l})$, 
where $X_{2,l}$ is the $l$-th coordinate of $X_2$. 
Let us write
$$
h(x)= \sum_{l=1}^q \dfrac{g_{(l)}(x) - g(x)}{g(x)}, \quad x\in \cube,
$$
and
$$
%\mu(X)=E(U|X), \quad
%\tau(X)=E(|U|^2|X), \quad
\nu = \int \mu(x)h(x)g(x)\,dx.
$$


\Thm{2}
Suppose that the assumption of \thmref{1} are satisfied,
now for the partially linear additive model
with $\vr_1,\dots,\vr_q$ belonging to $\H_1(m,\gamma)$,
where $s=m+\gamma>3q/2$.
%Suppose (G) and (H) hold,  $\|U\|$ has a moment greater than 2, 
%$\mu$ is continuous and $\tau g$ is bounded.
%Suppose that $\vr$ belongs to $\H_q(m,\gamma)$ with $s=m+\gamma>3q/2$.
%Let the error density $f$ have mean zero, 
%a finite moment of order greater than $4s/(2s-q)$, 
%and be H\"older with exponent greater than $q/(2s-q)$.
%%Let $\hvt$ be a $\sqrt{n}$-consistent estimator of $\vt$.
%Let the densities $w_1,\dots,w_q$ be $(q+2)$ times continuously
%differentiable with compact support $[-1,1]$. 
%Choose a bandwidth $c_n\sim (n \log n)^{-1/(2s)}$.
Then we have the uniform stochastic expansion
$$
%\bel{texp}
\sup_{t\in \R} \Big|
\temp(t) -\emp(t)
-f(t) 
\Big(\avj \ve_j(1+h(X_j)) - (\hvt-\vt)^{\top}\nu \Big) \Big| = \opn.
$$
%\ee
\end{theorem}


The proof of \thmref{2} is in Section \ref{sec6}.


\section{Testing for additivity} 
\label{sec3} 


In this section we test the hypothesis 
$\vr(x) = \vr_1(x_1) + \cdots + \vr_q(x_q)$ in the partially linear 
regression model $Y = \vt^\top U + \vr(X) + \ve$.
As in Section \ref{sec2}, let $\hemp$ and $\temp$ denote the residual-based 
empirical distribution functions based on residuals 
$\hve_j = Y_j - \hvt^\top U_j - \rhat(X_j)$ and
$\tve_j = Y_j - \hvt^\top U_j - \rtilde(X_j)$, respectively.

It follows from Theorems \ref{thm.1} and \ref{thm.2} 
that, under the hypothesis of additivity, the test statistic
$$
T = n^{1/2} \norm{\hemp-\temp}
%\sup_{t\in \R} n^{1/2}| \hemp(t)-\temp(t)|
$$
satisfies the stochastic expansion
\bel{tex}
T = \norm{f} \Big|n^{-1/2} \sum_{j=1}^n \ve_j h(X_j) 
- n^{1/2}(\hvt-\vt)^{\top} \nu \Big| + \op.
\ee
Now assume that $\hvt$ satisfies the stochastic expansion 
\bel{hvt}
\hvt = \vt + \avj W^{-1} (U_j-\mu(X_j)) \ve_j + \opn.
\ee
Many authors have constructed such estimators; 
see e.g. Chen (1988) and Schick (1996a), and the references therein.
%, Bhattacharya and Zhao (1997).
For estimators satisfying \Ref{hvt} the expansion \Ref{tex}  becomes
$$
T = \norm{f} \; \Big|n^{-1/2} \sum_{j=1}^n \ve_j \Big(h(X_j) 
- \nu^{\top} W^{-1}(U_j-\mu(X_j))\Big) \Big| + \op,
$$
and the test statistic $T$ converges in distribution to 
$\norm{f} \sigma \gamma |Z|$, 
where $Z$ is a standard normal random variable and 
\bel{tau}
\gamma= \big(E[h^2(X)] + \nu^{\top} W^{-1}\nu\big)^{1/2}.
\ee
The implementation of this test requires estimators 
of $\norm{f}$, $\sigma$ and $\gamma$.
We estimate $\sigma$ by the sample standard deviation $\hat \sigma$ 
based on the residuals $\hve_1,\dots,\hve_n$.
An estimator for $\norm{f}$ is $\norm{\hat f}$, 
with $\hat f$ a kernel density estimator based on these residuals.
We estimate $\gamma$ by 
$$
\hat \gamma = \Big(\avj \hat h^2(X_j) 
+ \hat \nu^{\top} \hat W^{-1} \hat \nu\Big)^{1/2},
$$
where $\hat h$ is a plug-in estimator of $h$ 
using kernel estimators of $g_{l}$, $g_{-l}$ and $g$,  
and where
$$
\hat \nu= \avj \hat \mu(X_j) \hat h(X_j)  \und 
\hat W= \avj (U_j-\hat \mu(X_j))(U_j-\hat \mu(X_j))^{\top}
$$ 
with $\hat \mu$ a nonparametric estimator of $\mu$ 
such as a Nadaraya--Watson estimator. 
For properly chosen kernels and bandwidths, these estimators 
are consistent under the assumptions of \thmref{1}.
The resulting test is 
$\1[T> \norm{\hat f} \hat \sigma \hat \gamma z_{\alpha/2}]$.
If $\gamma$ is positive, it will have asymptotic size $\alpha$.

%Now assume also that $\hvt$ satisfies a stochastic expansion  of the
%form
%\bel{hvt}
%\hvt = \vt + \avj J(U_j,X_j,Y_j) + \opn
%\ee
%with $E[J(U,X,Y)]=0$ and $E[J(U,X,Y)J^{\top}(U,X,Y)]$ positive definite.
%Then the test statistic $T$ converges in distribution to 
%$\kappa |Z|$, where $Z$ is a standard normal random variable and 
%\bel{tau}
%\kappa= \sup_{t\in \R} f(t) v^{1/2}, 
%\quad v= E[(\ve h(X)- \nu^{\top}J(U,X,Y))^2].
%\ee
%Typical estimators of $\hvt$ have influence function $J$ of the form 
%$J(U,X,Y)= t(U,X)\ve$ with $E(t(U,X)|X)=0$ and 
%$\Lambda = E[t(U,X)t^{\top}(U,X)]$ positive definite.  
%In this case the variance 
%simplifies to $\sigma^2(E[h^2(X)]+ \nu^{\top} \Lambda \nu)$.
%The implementation of this test requires consistent estimators of 
%$\sup_{t\in \R} f(t)$ and the variance.

The local asymptotic power of our test can be derived under a local 
alternative with $\vr$ replaced by $\vr+n^{-1/2} \Delta$, 
where $\Delta$ belongs 
to $L_2(G)$ and is orthogonal to the subspace of additive functions.  
Let us briefly sketch this.
% for the case when $J(U,X,Y)= t(U,X)\ve$ with $t$ as above. 
Suppose that $f$ has finite Fisher information for 
location and set $\ell =-f'/f$.  
Under the local alternative, $T$ converges in distribution 
to $\norm{f} \big| \sigma \gamma Z+\int h \Delta\,dG \big|$.
This follows from Le\,Cam's third lemma which says that the shift
must be
$$
\begin{aligned}
E\big[\ve(h(X) &-\nu^{\top}W^{-1}(U-\mu(X)))\ell(\ve) \Delta(X)\big] 
\\ &= E\big[\ve \ell(\ve)\big] 
E[\big(h(X)-\nu^{\top}W^{-1}(U-\mu(X)))\Delta(X)\big] 
= \int h \Delta\,dG. 
\end{aligned}
$$
Here we used the property $E[\ve\ell(\ve)]=1$.
For the test to detect the local alternative, 
the shift $\int h \Delta\,dG$ must be non-zero. 
The shift is always zero if, for example, the covariate $X$ 
follows a uniform distribution on $\cube$. Then $g_{(l)}=g=\1_{\cube}$
and therefore $h=0$.

Our test is easily modified to cover the nonparametric regression model
$Y=\vr(X)+\ve$.
In this case we take $\hvt=\vt=0$ and obtain the expansion 
\bel{npexp}
\sup_{t\in\R}\Big|\hemp(t)-\temp(t)+f(t)\avj\ve_j h(X_j)\Big| = \opn
\ee
under the assumption that $\vr$ is additive. 
Now $\gamma$ simplifies to $\gamma_0= (E[(h^2(X)])^{1/2}$,
and we work with the test 
$$
\1\Big[T> \norm{\hat f} 
\hat\sigma \Big(\avj \hat h^2(X_j)\Big)^{1/2} z_{\alpha/2}\Big].
$$
Neumeyer and Van Keilegom (2010) consider testing for additivity 
in the heteroscedastic nonparametric regression model 
$Y=\vr(X)+s(X)\eta$ with $E[\eta]=0$ and $E[\eta^2]=1$. 
They study a bootstrap test based on an appropriate version 
of $n^{1/2}\norm{\hemp-\temp}$ using standardized residuals. 
Under the null hypothesis they obtain an expansion 
which coincides with \Ref{npexp} when specialized to the homoscedastic case, 
i.e. to the case $s(X)=\sigma$ and $\ve=\sigma\eta$.

%Under local alternatives of the form 
%$\vr(x)=\vr_1(x_1)+\cdots+\vr_q(x_q)+n^{-1/2}s(x)$ with $E[s(X)]=0$, 
%their Theorem 4.4 says that $Z$ is shifted by $b=E[s(X)]/\sigma$.
%Since this is always zero, their test would have no power,
%contradicting their simulations. By the third lemma of Le Cam,
%setting $\ell=-f'/f$, the shift should be 
%$$
%b=-E[\ve h(X)s(X)\ell(\ve)]=-E[\ve\ell(\ve)]E[h(X)s(X)]=-E[h(X)s(X)], 
%$$
%using $E[\ve\ell(\ve)]=1$ for the last equality.
%
%The test statistic degenerates if the covariate $X$ follows
%a uniform distribution on $\cube$. Then $g_{(l)}=g=\1_{\cube}$
%and therefore $h=0$ and $b=0$. This means that the simulations
%in Neumeyer and Van Keilegom (2010) do not illustrate the theoretical results, 
%but rather higher-order differences between $\hemp$ and $\temp$.


\section{Responses missing at random}
\label{secmar}


The results of the previous sections carry over to the situation 
in which responses are \emph{missing at random}. 
Then we observe i.i.d.\ copies
$(\delta_1,U_1,X_1,\delta_1 Y_1),\dots,(\delta_n,U_n,X_n,\delta_n Y_n)$
of $(\delta,U,X,\delta Y)$ where $\delta$ is an indicator variable
depending on the covariables $(U,X)$, but not on the response $Y$.
The simplest approach is the \emph{complete-case analysis}, 
which uses only the $N=\sj \delta_j$ completely
observed triplets $(U_{i_1},X_{i_1},Y_{i_1}),\dots,(U_{i_N},X_{i_N},Y_{i_N})$,
where $i_1,\dots,i_N$ are the indices $i_j$ for which $\delta_{i_j}=1$.
Koul et al.\ (2012) show how stochastic expansions carry over 
from a statistic $T_n=t_n(U_1,X_1,Y_1,\dots,U_n,X_n,Y_n)$
to the corresponding \emph{complete-case statistic} 
$T_c=t_N(U_{i_1},X_{i_1},Y_{i_1},\dots,U_{i_N},X_{i_N},Y_{i_N})$.
We apply this to our empirical distribution functions $\hemp$ and $\temp$.
For the conditional probability of $\delta=1$ given $(U,X,Y)$ we write
\[
\pi(U,X) = P(\delta=1|U,X) = P(\delta=1|U,X,Y).
\]
Let $R$ denote the joint law of the covariates $(U,X)$,
and $Q(U,X,dy)$ the conditional distribution of the response $Y$
given $(U,X)$. Then the joint law of $(\delta,U,X,\delta Y)$ is
\[
P(dz,du,dx,dy) = R(du,dx)B_{\pi(u,x)}(dz)\big(zQ(u,x,dy)+(1-z)\Delta_0(dy)\big),
\]
where $B_p$ denotes the Bernoulli distribution with parameter $p$,
and $\Delta_t$ is the Dirac measure at $t$. It follows that the conditional
distribution of $(U,X,Y)$ given $\delta=1$ is 
\[
P_c(du,dx,dy) = R(du,dx)\frac{\pi(u,x)}{E[\delta]}Q(u,x,dy).
\]
Let $\mu_c$, $g_c$, $h_c$ and $\nu_c$ be defined as in Section 2, 
but now with the distribution of $(U,X)$ replaced by 
the conditional distribution of $(U,X)$ given $\delta=1$. Explicitly,
$\mu_c(X)=E(U|X,\delta=1)$, and $g_c$ is the density of $X$ given $\delta=1$.
For $l=1,\dots,q$ let $g_{c,l}$ denote the conditional density 
of the $l$-th coordinate of $X$ given $\delta=1$, 
let $g_{c,-l}$ denote the conditional density of the vector obtained 
from $X$ by deleting its $l$-th coordinate, and set
\[
g_{c(l)} (x)= g_{c,l}(x_l) g_{c,-l}(x_1,\dots,x_{l-1},x_{l+1},\dots,x_q),
\quad x=(x_1,\dots,x_q)\in \R^q.
\]
Write
\[
h_c(x)= \sum_{l=1}^q \frac{g_{c(l)}(x)-g_c(x)}{g_c(x)}, \quad x\in \cube,
\]
and define $\nu_c = \int\mu_c(x) h_c(x) g_c(x)\,dx$.

We must of course assume that $E[\delta]>0$.
The assumptions (G) and (H) on $U$ and $X$ are now required to hold
under the conditional distribution of $(U,X)$ given $\delta=1$.
This means that the conditional distribution of $U$ given $\delta=1$
is quasi-uniform, $E(|U|^2|\delta=1)<\infty$, and 
$$
W_c= E\big((U-\mu_c(X))(U-\mu_c(X))^\top|\delta=1\big)
$$
is positive definite.
These assumptions are implied by (G) and (H) if $\pi$ is bounded away
from zero.

Assume that $\hvt$ has a stochastic expansion of the form \Ref{hvt}.
Let $\hvt_c$ denote the version of $\hvt$ based on the complete observations.
{From} the arguments of Koul et al.\ (2012) it follows that $\hvt_c$
has the stochastic expansion
\[
\hvt_c = \vt + \avj\frac{\delta_j}{E[\delta]} W_c^{-1}(U_j-\mu_c(X_j))\ve_j 
+ \opn.
\]
Define the local polynomial smoother $\rhat_c$ and the marginal integration
estimator $\rtilde_c$ as in Section 2, now using only the complete observations.
We mention that minimax properties of such complete-case estimators
in nonparametric regression are obtained by Efromovich (2011).
Define residuals $\hve_{c,j} = Y_j - \hvt_c^\top U_j - \rhat_c(X_j)$, 
and $\tve_{c,j} = Y_j - \hvt_c^\top U_j - \rtilde_c(X_j)$.
The complete-case versions of the empirical distribution functions 
$\emp$, $\hemp$ and $\temp$ are
\[
\emp_c(t) = \frac{1}{N}\sj\delta_j\1[\ve_j\leq t], \quad
\hemp_c(t) = \frac{1}{N}\sj\delta_j\1[\hve_{c,j}\leq t], \quad
\temp_c(t) = \frac{1}{N}\sj\delta_j\1[\tve_{c,j}\leq t].
\]
Using Koul et al.\ (2012) again, from the uniform stochastic expansions 
for $\hemp$ and $\temp$ in Theorems \ref{thm.1} and \ref{thm.2}
we obtain uniform stochastic expansions for the complete-case versions,
\be*
& \displaystyle \sup_{t\in \R} 
\Big| \hemp_c(t) - \emp_c(t) 
- f(t) \avj\frac{\delta_j}{E[\delta]}\ve_j \Big| 
= \opn, & \\
& \displaystyle \sup_{t\in \R} \Big|
\temp_c(t) - \emp_c(t)
- f(t) \avj\frac{\delta_j}{E[\delta]} \ve_j\Big(1+h_c(X_j) 
- \nu_c^{\top} W_c^{-1} (U_j-\mu_c(X_j))\Big) \Big| 
= \opn. &
\no*
It follows from these two expansions that under the hypothesis of additivity
the complete-case test statistic 
\[
T_c = \sup_{t\in \R} N^{1/2}|\hemp_c(t)-\temp_c(t)|
\]
satisfies the stochastic expansion
\[
T_c = \norm{f} (E[\delta])^{-1/2}
\Big| n^{-1/2}\sum_{j=1}^n\delta_j\ve_j 
\Big(h(X_j)-\nu_c^{\top}W_c^{-1} (U_j-\mu_c(X_j))\Big) 
\Big| + \op.
\] 
The test is now 
$$
\1[T_c > \norm{\hat f_c} \hat \sigma_c \hat \gamma_c z_{\alpha/2}],
$$
where $\hat f_c$, $\hat \sigma_c$ and $\hat \gamma_c$ 
are complete-case versions of the estimators $\hat f$, $\hat \sigma$,
and $\hat \gamma$ from the previous section.
For example, if $\hat f$ is the kernel estimator 
$$
\hat f(t)= \dfrac{1}{nb_n} \sj k\Big(\frac{\hve_j-t}{b_n}\Big), \quad t\in \R, 
$$
based on the kernel $k$ and bandwidth $b_n$, 
then its complete-case version is of the form
$$
\hat f_c(t)= \dfrac{1}{Nb_N} \sj \delta_j 
k\Big(\frac{\hve_{c,j}-t}{b_N}\Big), \quad t\in \R.
$$

Since the pairs $(U,X)$ are always abserved, 
one might be tempted to use $\hat \gamma$ 
instead of $\hat \gamma_c$. However, for missing data 
we need to estimate $\gamma_c$, which is a functional 
of the conditional distribution of $(U,X)$ given $\delta=1$, 
and not $\gamma$, which is the same functional 
of the unconditional distribution of $(U,X)$.  
Thus $\gamma_c$ is typically different from $\gamma$, 
and it is imperative to use $\hat \gamma_c$ in the present case. 
Note also that the use of $\hat \gamma$ instead of 
$\hat \gamma_c$ would not result in a complete-case estimator.


\section{Simulations}
\label{secsim}

We ran simulations to study the performance of the proposed test. 
Since the test requires estimation of quantiles
(which demands special care and attention), we focus on the
case $q=2$ and $\vt=0$, i.e.\ $X$ is two-dimensional
and the regression function does not have a linear part. 
We generated covariates $X = (X_1,X_2)$ from a quasi-uniform 
distribution on $\cube=[0,1]^2$ specified by the density
$$
g(x) = g(x_1,x_2) = 1+ 0.5 \, \sign(x_1- 0.5)\sign(x_2- 0.5), 
\quad x = (x_1,x_2) \in \cube.
$$
Note that $g$ alternately takes the values $0.5$ and $1.5$ on the four
quarters of $\cube$ and that 
$g_{1}=g_{2}= g_{-1}=g_{-2}=1$ on $[0,1]$. The conditional probability
of $\delta =1$ given the covariates is $\pi(X) = \pi(X_1) = \cos(X_1)$
so that the data contain on average about $84\%$ complete cases. 
The errors are generated from a normal distribution with mean zero and 
standard deviation \nolinebreak $\sigma$. The test is
$$
\1\Big[T_c> \norm{\hat f_c} 
\hat\sigma_c \Big(\avj \hat h_c^2(X_j)\Big)^{1/2} z_{\alpha/2}\Big].
$$
In order to implement the test statistic $T_c$ we used 
a locally linear smoother provided by the R routine
``loess''. We looked at several choices of the smoothing parameter ``span''
(between zero and one). Under the null hypothesis of additivity
we chose $\vr(X) = X_1 + X_2$. Since our smoother is locally 
{\em linear}, it is not too surprising that the 
test performs best for large values of ``span'' (see Table \ref{table1};
note that the default value in R is $0.75$).

\input{table1}

For estimation of the quantiles we used the estimator ``density''
implemented in R to estimate $f_c$, and a particularly small bandwidth 
(the range of the residuals $\hve_j$ divided by 20) to avoid oversmoothing
and to obtain a good estimate of the maximum. Our estimator
of $\sigma^2_c$ is the residual-based empirical estimator.
Finally we hand coded $\hat h_c$ (which involves estimators
of the bivariate covariate density and of the marginal densities) 
using a uniform kernel function and the bandwidth $1/3$.

In Table \ref{table1} we consider the situation when the 
regression function is additive and the null hypothesis
should be rejected. The entries are the proportions of tests 
that reject the hypothesis of additivity in 1000 trials ($n=100$)
and in 500 trials ($n=500$).
We study both the scenario with missing responses and the scenario where
all data are completely observed ($\pi(X) =1$). In order to keep the 
significance level of the test ($\alpha = 0.05$ in all simulations), 
we work with a large ``span''. 
For completely observed data and our choice of $g$, we have $Eh^2(X) = 4/3$. 
The `true' quantile for the simulation scenario thus computes to $0.903$.
The rejection rates of the tests that use this quantile are given in 
Table \ref{table1} in parentheses. The results are apparently similar
to the case when the quantile is estimated. Also we observe that
there is no big difference between the two cases missing data / no
missing data, which can perhaps be explained by the fact that only a
relatively small percentage of responses (about 16\%) are missing.

\input{table2-neu}

In Table \ref{table2} we work with the regression function 
$\vr(X)+cX_1X_2$, $c \in \{0.1,0.5,1\}$, which violates the null 
hypothesis since it contains an additional multiplicative part.
Note that for $c=0.1$ the additional part
can be regarded a local alternative since $cX_1X_2 = n^{-1/2}X_1X_2$.
We observe that the rejection rates for this particular case are quite
low and not much affected by the error variance $\sigma^2$. 
The situation is different for $c=0.5$ and $c=1$, which
denotes alternatives that are easier to detect. 
As expected, the rejection rates are high for small $\sigma$ and large $c$,
and decrease as $\sigma$ becomes larger. Consider, 
for example, the rejection rates for span 1 (which yielded 
reasonable results under the null hypothesis) and $c=1$:
for $\sigma=0.1$ about $98\%$ of the simulated tests reject 
the (false) null hypothesis, but only about $12\%$ of the tests 
reject it when $\sigma=0.75$.

\input{table3}

The third study (Table \ref{table3}) continues the discussion of the 
asymptotic power under local alternatives $\vr + n^{-1/2} \Delta$ from the 
last part of Section \ref{sec3}. In order to keep the focus on the behavior
under those alternatives, we restrict our attention to the case of fully 
observed data. (The case with missing responses can be treated analogously.)
For our illustrations we now consider a different alternative,
namely $\vr + n^{-1/2} \Delta$, with $\vr$ as before and with
$$
\Delta(x) = \Delta(x_1,x_2)=\sign(x_1- 0.5)\sign(x_2- 0.5) - 0.5
\quad x \in \cube,
$$ 
which takes alternately the values $-1.5$ and $0.5$ on the four
quarters of $\cube=[0,1]^2$.
It is easy to see that $\Delta$ satisfies the requirements given
in Section \ref{sec3}: it belongs to $L_2(G)$ and it is orthogonal 
to the space of additive functions. 

In Section \ref{sec3} we have shown that under local alternatives
the limiting distribution of the test statistic $T$ is shifted by
the value $\int h \Delta \, dG$, which must be non-zero in order 
to detect these alternatives. This is indeed satisfied for our choice of
$g$ since $h(x) = 2[1-g(x)]/g(x)$ takes alternately values $2$ and $-2/3$
on the four quarters of $\cube$. This combined with the identities
$$
\int_0^1 \Delta(x_1,x_2)\,dx_1= -0.5= \int_0^1 \Delta(x_1,x_2)\,dx_2
$$
makes it possible to calculate the shift explicitly: 
$$
\int h \Delta \, dG
=
E[h(X_1,X_2) \Delta(X_1,X_2)]
=
2\iint \Delta(x_1,x_2)(1-g(x_1,x_2)) \,dx_1\,dx_2
=-1. 
$$
In Table \ref{table3} (a) we study the power of the test 
for sample sizes $n=100$ and $n=500$ for which 
$\vr + n^{-1/2} \Delta = \vr + 0.1 \Delta$ and 
$\vr + n^{-1/2} \Delta \approx \vr + 0.045 \Delta$, respectively.
We also list the rejection rates for the larger alternative 
$\vr + \Delta$. Our test performs very well unless the standard deviation
is large: for $\sigma=0.5$ and a properly chosen span ($1$ and $0.95$ for $n=100$
and $n=500$, respectively; cf.\ Table \ref{table1}) only roughly a fourth 
of the local alternatives can be detected. 

In panel (b) we list the rejection rates when the covariates are 
generated from a uniform distribution on $\cube = [0,1]^2$. 
As already explained in Section \ref{sec3}, the test has 
no local asymptotic power in this case since the shift
is zero. In particular, since $h=0$, we have $T=\opn$ and hence no
(theoretical) quantile that can be used for comparisons.
The rejection rates of the test using estimated quantiles 
are zero or almost zero, in line with the theoretical findings.
We therefore expect that our test will perform poorly if the
distribution of the covariates is uniform or close to uniform.


\section{Proof of \thmref{2}}
\label{sec6}


The marginal integration estimator $\rtilde$ is based on the local
polynomial smoother $\rhat$. We begin by recalling results on $\rhat$
from M\"uller et al.\ (2012). Order the multi-indices $i\in I(m)$ 
lexicographically. Let $\psi$ be the vector with components $\psi_i$, 
$i\in I(m)$.
By definition, $\rhat$ is the component 
$\hb_0$ of $\hb$, where $\hb$ is the solution of the normal equation
$$
R(x)=W(x)\hb(x)
$$
with
$$
R(x)= \wav (Y_j-U_j^{\top}\hvt)\psi\Big(\dfrac{X_j-x}{c_n}\Big) 
$$
and
$$
W(x)= \wav \psi\Big(\dfrac{X_j-x}{c_n}\Big) 
\psi^{\top}\Big(\dfrac{X_j-x}{c_n}\Big).
$$

On the H\"older space $\H_q(m, \gamma)$ we introduce the norm
$$
\norm{h}^{(q)}_{m,\gamma} = \max_{i\in I(m)} \sup_{x\in \cube} | D^i h(x)| + 
\max_{i\in J(m)} \sup_{x,y\in \cube, x\neq y} 
\dfrac{|D^i h(y)- D^ih(x)|}{\|x-y\|^{\gamma}}
$$
with 
$$
D^i h(x)  =  
\frac{\partial^{i_1+\dots+i_q}}{\partial x_1^{i_1} \cdots \partial x_q^{i_q}}
h(x), \quad x=(x_1,\dots,x_q)\in \cube.
$$
Let $\B_q(m,\gamma)$ denote the unit ball of $\H_q(m,\gamma)$ for this norm.
In the general partially linear regression model,
M\"uller et al.\ (2012) have obtained the following uniform
stochastic expansion of the regression function estimator,
\bel{use}
\sup_{u \in \R^p, x \in \cube} 
\big| \hvt^\top u + \rhat(x) - \vt^\top u - \vr(x) -
(\hvt - \vt)^\top (u - \mu(x)) - \hat c(x) 
\big| = \opn
\ee
with 
$$
\hat c(x)= e^{\top} (E[W(x)])^{-1} \wav 
\ve_j\psi\Big(\dfrac{X_j-x}{c_n}\Big)
$$
satisfying, for some $\alpha >0$ and $\xi>q/(2s-q)$,
\bel{a1}
P\big(\hat c \in \B_q(q,\alpha)\big) \to 1,
\ee
\bel{a2}
\int |\hat c(x)|^{1+\xi} g(x) \, dx = \opn.
\ee

In the \emph{additive} partially linear regression model, let 
us set 
%$r(u,x)=\vt^{\top}u + \vr(x)$, 
%$\tilde r(u,x)= \hvt^{\top}u + \tilde \vr(x)$ and 
$$
\tilde a(u,x)= (\hvt-\vt)^\top(u-\mu_*(x))+ \tilde c(x),
$$
where 
\bea
\mu_*(x) &= (1-q)E[U] + \sum_{l=1}^qE[\mu(X_{1,-l}(x_l))],
\\
\tilde c(x) &= (1-q)\avj \ve_j + \sum_{l=1}^q \avj \hat c(X_{j,-l}(x_l)).
\ea

For the H\"older space $\H_1(q, \alpha)$ the above norm simplifies to 
$$
\| a\|^{(1)}_{q,\alpha} = \sum_{i=1}^q \sup_{0\le t\le 1} |a^{(i)}(t)| + 
\sup_{0\le s \le t \le 1} \dfrac{|a^{(q)}(s)-a^{(q)}(t)|}{|s-t|^{\alpha}}.
$$
Let $\B(q,\alpha)$ denote the unit ball of $\H_1(q,\alpha)$ for this norm.
We write $\mathcal D$ for the set of functions of the form 
$$
a(u,x)= b^{\top}(u-\mu_*(x)) + a_1(x_1) + \dots + a_q(x_q)
$$
with $|b|\le 1$ and $a_l \in \B(q,\alpha)$ for $l=1,\dots,q$.
Here $\alpha$ is so small that \Ref{a1} holds.
Let $Q$ denote the joint distribution of $U$ and $X$.
Recall that $f$ is bounded. By Theorem (2.2) in M\"uller et al.\ (2007),
the desired result then follows from the following statements:
\ben
\label{t1} 
&
P(\tilde a \in \D) \to 1;
&
\\
\label{t2}
&
\displaystyle
\int |\tilde a|^{1+\xi} \,dQ = \opn,  \quad \xi > q/(2s-q);
&
\\
\label{t3}
&
\displaystyle
\int \tilde a\,dQ = \avj \ve_j
+ \avj \ve_j h(X_j) - (\hvt -\vt)^{\top} \nu
=\opn;
&
\\
\label{t4}
&
\displaystyle
\sup_{u\in \R^k, x\in \cube} \big|
\hvt^\top u + \rtilde(x) - \vt^\top u - \vr(x) - \tilde a(u,x)\big| 
= \opn.
&
\non
Note that requirement (2.1) of M\"uller et al.\ (2007) on the
bracketing numbers of the class $\D$ is verified as in that paper, 
but now using the bound (1.5) in M\"uller et al.\ (2009) in place of
the bound (3.1) in M\"uller et al.\ (2007). 
Statements \Ref{t1} and \Ref{t2} above are simple consequences 
of the $\sqrt{n}$-consistency of $\hvt$ and the properties 
\Ref{a1} and \Ref{a2} of $\hat c$.
Statement \Ref{t3} follows from the identities 
$$
\int \tilde a\,dQ = (\hvt-\vt)^\top\big(E[U]-E[\mu_*(X)]\big) 
+ (1-q)  \avj \ve_j
+ \sum_{l=1}^q \avj \int \hat c(X_{j,-l}(t))g_l(t)\,dt
$$
and
$$
E[U]-E[\mu_*(X)] = qE[U] - \sum_{l=1}^q \int \mu(x)g_{(l)}(x)\,dx
= - \nu
%\int \mu(x) h(x) g(x) \,dx
$$
and from the expansions 
$$
\avj \int \hat c(X_{j,-l}(t)) g_l(t) \,dt =  
\int \hat c(x) g_{(l)}(x)\,dx  + \opn
$$
and
$$
\sum_{l=1}^q \int \hat c(x) g_{(l)}(x) \,dx = 
\int \hat c(x) (q+h(x))g(x)\,dx = \avj \ve_j (q+h(X_j)) + \opn.
$$
These expansions are proved as in M\"uller et al.\ (2009).
We omit the details.

We now verify \Ref{t4}.  Note that \Ref{use} implies the expansion 
$$
\sup_{x\in \cube} 
\big|\hat \vr(x) - \vr(x) 
+(\hvt-\vt)^{\top} \mu(x) - \hat c(x)\big| = \opn.
$$
{From} this we can conclude that 
$$
\begin{aligned}
&\sup_{x\in \cube}
\Big|\sum_{l=1}^q \avj \Big(\hat \vr(X_{j,-l}(x_i))- \vr(X_{j,-l}(x_l))
+(\hvt-\vt)^{\top}\mu(X_{j,-l}(x_l))-\hat c(X_{j,-l}(x_l))\Big)\Big| \\
&\qquad = \opn.
\end{aligned}
$$
In view of the additivity of $\vr$, we have the identity 
$$
\sum_{l=1}^q \avj \vr(X_{j,-l}(x_l)) = \vr(x) + (q-1)\avj \vr(X_j).
$$
Using the representation \Ref{ybar} for $\bar Y_*$, we have 
$$
\sup_{x\in \cube}
%|\tilde \vr(x) - \vr(x)+(q-1)(\avj \ve_j -(\hvt-\vt)^{\top}\bar U) 
%+(\hvt-\vt)^{\top}\tilde \mu(x) -\tilde c(x)| 
\big|\tilde \vr(x) - \vr(x) 
+(\hvt-\vt)^{\top}\tilde \mu(x) -\tilde c(x)\big| 
= \opn,
$$
where 
$$
\tilde \mu(x)= (1-q)\avj U_j+\sum_{l=1}^q \avj \mu(X_{j,-l}(x_l)).
$$
Since $\mu$ is continuous, we obtain 
$$
\sup_{x\in \cube} |\tilde \mu(x)-\mu_*(x)| =\op.
$$
Combining the above, we obtain \Ref{t4}.


%\section*{Acknowledgments}
%Ursula U. M\"uller was supported by NSF Grant DMS 0907014.
%Anton Schick was supported by NSF Grant DMS 0906551.
%The authors thank two referees for suggestions which 
%improved the manuscript. For example, they led us to add two 
%sections on testing. 


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%\newpage 

\end{document}