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\begin{document}
\title{\vspace{-0.5in}Supplementary Material on\\
``Semiparametric Estimation of Random Coefficients in
Structural Economic Models"}
\author{
\begin{tabular}{ccccc}
Stefan Hoderlein\thanks{%
Department of Economics, Boston College, 140 Commonwealth Avenue, Chestnut
Hill, MA 02467, USA, email: stefan\_hoderlein@yahoo.com. } &  & Lars Nesheim
\thanks{%
Department of Economics, University College London, Gower Street, London,
WC1E 6BT, UK, email: l.nesheim@ucl.ac.uk} &  & Anna Simoni\thanks{%
CREST, 15 Boulevard Gabriel P\'{e}ri, 92240 Malakoff (France), email:
simoni.anna@gmail.com} \\
{\small Boston College } &  & {\small University College London } &  &
{\small CNRS - CREST} \\
&  &  &  &
\end{tabular}%
}
\date{\today\vspace{0.5cm} }
\maketitle
This document presents several appendices that contain supplementary results to be published online alongside the main paper, ``Semiparametric Estimation of Random Coefficients in
Structural Economic Models". All references to equation numbers and to section numbers are references to equations and sections in the main paper. The main paper contains Appendices \ref{s_Appendix}
 and \ref{section:figures}. This document begins with Appendix \ref{section:compactness}.

\appendix
\setcounter{section}{2}
\section{Compactness}\label{section:compactness}
The following proposition shows that under Assumptions \ref{AssDGP}-\ref{Assumption: Compact operator} the operator $T$ defined in (\ref{eq_th1_kernel_structural_model}) is compact with infinite dimensional range. As discussed in Section \ref{Section_Identification} in the paper, compactness of the operator is useful because then $T$ admits a SVD.%, and that this depends both on the structural model (characterized by the function $\Psi $ and the density $f_{\varepsilon|WZ\theta }$) and on the weights $\pi _{cz}$ and $\pi_{\theta}.$

\begin{proposition}
\label{prop_compactness} Let $T$ be the operator defined in (\ref{eq_th1_kernel_structural_model}) with domain $L_{\pi_{\theta }}^{2}$ and
let Assumptions \ref{AssDGP} - \ref{Assumption: Compact operator} be satisfied. If $f_{C|WZ\theta}/\pi_{\theta}$ is square integrable with respect to $\pi_{\theta}\times \pi_{cz}$ then $\mathcal{R}(T)\subset L_{\pi_{cz}}^2$ and $T: L_{\pi_{\theta}}^2\rightarrow L_{\pi_{cz}}^2$ is an a.s. bounded and compact operator. % with range included in $L_{\pi _{cz}}^{2}$.
%Moreover, if $f_{C|WZ\theta}\in\mathcal{F}_{C|WZ\theta}$ then $\mathcal{R}(T|_{\mathcal{F}_{\theta|W}})\subset L_{\pi_{cz}}^2$ and, if in addition $\sup_{\theta \in\Theta}\pi_{\theta}(\theta)>0$, then $\mathcal{R}(T)\subset L_{\pi_{cz}}^2$.
\end{proposition}
The proof is detailed in Appendix  \ref{section:proof compact} below.

\section{Identification and completeness}

In addition to the large class of functions that satisfy the sufficient conditions for identification given in Proposition \ref{point_identification_sufficient_condition}, we provide here further examples of
families $\mathcal{F}_{C|WZ\theta }$ for which the corresponding $\mathcal{F}_{\theta|CWZ}$ is $\mathcal{T}$-complete.\\ %These examples are confined to the single random coefficient case.\\

\textsc{Additively-closed one-parameter family of distributions}. Let 
$\Theta =\mathbb{R}_{+}$ and $\mathcal{F}_{C|WZ\theta }$ be \emph{additively
closed}. That is, $\forall f_{C|WZ\theta },h_{C|WZ\theta }\in
\mathcal{F}_{C|WZ\theta }$ and $\forall \theta _{1},\theta _{2}\in \Theta$,
\begin{equation}
f_{C|WZ\theta }(c,w,z,\theta _{1})\ast h_{C|WZ\theta }(c,w,z,\theta
_{2})=f_{C|WZ\theta }(c,w,z,\theta _{1}+\theta _{2}), \nonumber
\end{equation}
where $\ast $ denotes
the convolution operation. Then, $\mathcal{F}_{\theta|CWZ}$ is $\mathcal{T}-
$complete. Some distributions that belong to the additively-closed
one-parameter family, and that are relevant for our application, are the
following, see \cite{Teicher1961}.

\begin{itemize}
\item[-] Gamma distribution: $f_{C|WZ\theta }=\frac{g\left( z,w\right)
^{\theta }}{\Gamma (\theta )}c^{\theta -1}e^{-g\left( z,w\right) c}$, $c>0$,
$g\left( z,w\right) >0$, $\theta >0$ or $f_{C|WZ\theta }=\frac{\theta
^{g\left( z,w\right) }}{\Gamma (g\left( z,w\right) )}c^{g\left( z,w\right)
-1}e^{-\theta c}$, $c>0$, $g\left( z,w\right) >0$, $\theta >0$.

\item[-] Uniform distribution with support depending on $\theta$: $%
f_{C|WZ\theta} = \mathcal{U}[\theta - g(Z,W), \theta + g(Z,W)]$, where $%
g(\cdot,\cdot)$ is some positive and bounded function of $(Z,W)$. Therefore,
\begin{equation*}
f_{C|WZ\theta} = \frac{1}{2g(Z,W)}1\left\{ \theta - g(Z,W)<c<\theta + g(Z,W)\right\}. 
\end{equation*}
However, if $f_{C|WZ\theta}$ has a uniform distribution with support that does not depend on $\theta$ then, $f_{\theta|W}$ is not identified.
\end{itemize}

\textsc{Location-scale one-parameter family of distributions}. Let $\Theta =%
\mathbb{R}_{+}$ and $\mathcal{F}_{C|WZ\theta }$ be the one-parameter family
induced by $f_{C|WZ}$ via location or scale changes. That is, 
$\forall f_{C|WZ\theta }\in \mathcal{F}_{C|WZ\theta }$, $f_{C|WZ\theta }(c,w,z,\theta
)=f_{C|WZ}(c-\theta,w,z)$ or $f_{C|WZ\theta }(c,w,z,\theta
)=f_{C|WZ}(c\theta ,w,z)$. For the location (resp. scale) family, if the
conditional characteristic function of $C$ (resp. $\log C$), given $(W,Z)$,
does not vanish a.s. in some non-degenerate real interval, then the $%
f_{\theta \left\vert W\right. }$ is identified, see \cite{Teicher1961}.%

%=================================================================================
\subsection{Identification without nuisance unobservables\label{Section: no
epsilon}}

In this section we briefly describe the case where we do not have $%
\varepsilon $ so that $f_{C|WZ\theta }$ cannot be recovered as in Theorem %
\ref{th1}. This is relevant in models where all the unobservable variables
are of interest so $\varepsilon $ is included in $\theta $. In our setup,
this implies that the general structural model (\ref{eq_DGP_Ass_1}) reduces
to
\begin{equation}
\Psi (C,W,Z,\theta )=0\qquad a.s.
\end{equation}%
and Assumption \ref{AssDGP} is replaced by the following one.

\renewcommand{\theass}{1$'$}

\begin{ass}
\label{AssDGP_bis} The random element $(C,W,Z,\theta)$ satisfies a
structural economic model

\begin{equation}
\Psi (C,W,Z,\theta)=0\quad a.s.  \label{eq_DGP_Ass_1_bis}
\end{equation}%
where $\Psi $ is a \textbf{known} Borel measurable real-valued function. We
assume that (\ref{eq_DGP_Ass_1_bis}) has a unique global solution in terms
of $C$:
\begin{equation}
C=\varphi (W,Z,\theta),\quad a.s.  \label{eq_DGP_Ass_2_bis}
\end{equation}%
where $\varphi :\mathbb{R}^{k+l+d}\rightarrow \mathbb{R}$ is a
Borel-measurable function.
\end{ass}

Indeed, even in this setup where $\varphi $ is not strictly monotonic in $\theta $ and $\theta $ is multivariate, we can characterize the structural \textit{pdf} $f_{\theta |W}$ as a solution to a constrained functional equation. Let $F_{C|WZ}$ be the cumulative distribution function associated with $P_{C|WZ}$ and assumed to be in $L_{\pi _{cz}}^{2}$ for every $w\in\mathcal{W}$. Then, we have the following analog to Theorem \ref{th1}.
\begin{theorem}\label{th1_bis} 
Let Assumptions \ref{AssDGP_bis} and \ref{AssIndeta} be satisfied. If $P_{\theta|W}$ admits a \textit{pdf} $f_{\theta|W}$ with respect to the Lebesgue measure, then $f_{\theta|W}$ is a solution of:
\begin{equation}
 F_{C|WZ}(c,w,z) = Sf_{\theta |W}(\theta ,w)\quad \text{subject to }\quad f_{\theta|W}\in \mathcal{F}_{\theta |W},\qquad a.s.
\label{eq_estimating_without_noise}
\end{equation}
where $S$ is a linear operator defined as

\begin{equation}\label{eq_estimating_without_noise_2}
  Sh = \int_{\Theta }1_{\{\varphi (w,z,\theta )\leq c\}}(\theta) h(\theta ,w)d\theta ,\quad \forall h\in L_{\pi _{\theta }}^{2}.
\end{equation}
%\noindent Moreover, if $\int_{\Theta}\pi_{\theta}(\theta)d\theta<\infty$ and $\int_{\mathcal{C}\times \mathcal{Z}}\pi_{c,z}(c,z)dc dz<\infty$ then $S$ is bounded.
\end{theorem}
\begin{proof}
 Equations \eqref{eq_estimating_without_noise}-\eqref{eq_estimating_without_noise_2} follow from the fact that, 
under Assumption \ref{AssDGP_bis}, 
$F_{C|WZ}(c,w,z) = \mathbb{E}\left[\left. 1_{\{\varphi(w,z,\theta)\leq c\}}(\theta) \right|W,Z\right] = \int 1_{\{\varphi(w,z,\theta)\leq c\}}(\theta) dP_{\theta|W,Z}(\theta,w,z)$ 
and from Assumption \ref{AssIndeta}. %To show boundedness of the operator $S$ we write: $\forall h\in L_{\pi_{\theta}}^2$
%\begin{eqnarray*}
%  \|Sh\|^2 =
%\end{eqnarray*}
\end{proof}

The kernel of the operator $S$ is $\frac{1\{\varphi (w,z,\theta )\leq c\}}{\pi _{\theta }(\theta )}$ and the adjoint $S^{\ast }$ is given in the following proposition:

\begin{proposition}[Adjoint of $S$]\label{prop_adjoint_S} Let $S$ be the operator defined in (\ref{eq_estimating_without_noise_2}). Assume that $S:L_{\pi _{\theta }}^{2}\rightarrow L_{\pi_{cz}}^{2}$ is bounded. Then, the operator $S^{\ast }$ defined as: $\forall \psi \in L_{\pi _{cz}}^{2}$,
\begin{equation*}
S^{\ast }h=\int_{\mathcal{C}}\int_{\mathcal{Z}}1_{\{\varphi (w,z,\theta )\leq
c\}}(\theta) \frac{\pi _{cz}(c,z)}{\pi _{\theta }(\theta )}h(c,w,z)dcdz.
\end{equation*}
exists and is the adjoint of $S$. The operator $S^{\ast}:L_{\pi_{cz}}^{2}\rightarrow L_{\pi _{\theta }}^{2}$ is bounded and linear.
\end{proposition}

The proof is similar to the proof of Proposition \ref{prop_adjoint} and is omitted.
%
Note that when there are nuisance unobservables $\varepsilon$, the estimating equation (\ref{functional_equation_th_1}) can be trivially recovered from (\ref%
{eq_estimating_without_noise}) by differentiating with respect to $c$. If $\int_{\mathcal{C}\times\mathcal{Z}}\int_{\Theta}\frac{1}{\pi_{\theta}}d\theta\pi_{cz}dcdz < \infty$, then the bounded operator $S:L_{\pi _{\theta }}^{2}\rightarrow L_{\pi_{cz}}^{2}$ is compact.

Identification of $f_{\theta|W}$ depends on injectivity of $S|_{\mathfrak{D}}$ which, in turn, depends on the exogenous variation in $Z$. The estimation procedure for this case is the same as that one proposed in
Section \ref{s_Estimation} with the operator $T$ replaced by $S$. The rate
of the mean integrated squared error will improve since $F_{C|WZ}$ can be
estimated at a better rate than $f_{C|WZ}$. Moreover, the degree of
ill-posedness will not be as severe as in the case where the kernel of $T$
is exponential.




%=================================================================================
\section{Case with non-random parameters: Iterative two-step method}\label{App_C}

In this section we describe the two-step estimator in the case in which some components of $\theta$ are deterministic as described in Section \ref{rem_deterministic_theta}. This is an iterative algorithm similar to that proposed in \cite{HeckmanSinger1984}. The algorithm is as follows:

\begin{itemize}
\item[I.] For a given $\theta _{1}^{(j)}$ compute the \emph{indirect
Tikhonov regularized estimator} of $f_{\theta_{2}|W}$ using the two-step procedure described in Section \ref{ss_two_steps_approach}. That is, in the
\emph{first step} solve the minimization problem%
\begin{equation*}
\hat{f}_{\theta_{2}|W(j)}^{\alpha }=\arg \min_{h\in L_{\pi _{\theta_2}}^{2}}\left\{ ||T_{\theta_{1}^{(j)}}h-\hat{f}_{C|WZ}||^{2}+\alpha||h||^{2}\right\}
\end{equation*}%
and in the second step compute the metric projection of $\hat{f}_{\theta_{2}|W(j)}^{\alpha }$ onto the set $\mathcal{F}_{\theta |W}$ as%
\begin{equation}
\mathcal{P}_{c}\hat{f}_{\theta_{2}|W(j)}^{\alpha }=\max \left\{ 0,\hat{f}_{\theta _{2}|W(j)}^{\alpha }-\frac{c}{\pi_{\theta_2}}\right\}
\end{equation}%
\noindent where $c$ is such that $\int_{\Theta} \mathcal{P}_{c}\hat{f}_{\theta_{2}|W(j)}^{\alpha }d\theta = 1$. Fix $\hat{f}_{\theta_{2}|W}^{(j)}=\mathcal{P}_{c}\hat{f}_{\theta_{2}|W(j)}^{\alpha }$.

\item[II.] For a given $\hat{f}_{\theta_{2}|W}^{(j)}$ compute $\theta_{1}^{(j+1)}$ by solving the nonlinear least-squares problem:
\begin{equation*}
\theta_{1}^{(j+1)}=\arg \min_{\theta _{1}\in \Theta_1}\left(||T_{\theta_{1}}\hat{f}_{\theta _{2}|W}^{(j)}(\theta _{2},w)-\hat{f}_{C|WZ}||^{2}+\alpha ||\hat{f}_{\theta _{2}|W}^{(j)}||^{2}\right) .
\end{equation*}%
Then, iterate steps $I$ and $II$ until convergence. The algorithm should be
run using different starting values for $\theta_{1}$ to avoid convergence
to a local optimum.
\end{itemize}

%=================================================================================
\section{Proofs of minor results}
\subsection{{Proof of Proposition \protect\ref{prop_adjoint}}}

{\small By definition, the adjoint operator $T^{\ast }$ of the bounded linear operator $T$ satisfies: $\forall h\in L_{\pi _{\theta }}^{2}$, $\forall \psi \in L_{\pi _{cz}}^{2}$, $\langle Th,\psi \rangle =\langle
h,T^{\ast }\psi \rangle $. Thus,
\begin{eqnarray*}
\langle Th,\psi \rangle & = &\int_{\mathcal{C}}\int_{\mathcal{Z}}(Th)(c,w,z)\psi (c,z)\pi_{cz}(c,z)dcdz \\
& = & \int_{\mathcal{C}}\int_{\mathcal{Z}}\int_{\Theta }f_{C|WZ\theta
}(c,w,z,\theta)h(\theta)d\theta \psi (c,z)\pi_{cz}(c,z)dcdz \\
&=&\int_{\Theta }h(\theta)\pi_{\theta }(\theta)\int_{\mathcal{C}}\int_{%
\mathcal{Z}}f_{C|WZ\theta }(c,w,z,\theta )\psi(c,z)\frac{\pi_{cz}(c,z)}{%
\pi_{\theta }(\theta )}dcdzd\theta =\langle h,T^{\ast }\psi \rangle
\end{eqnarray*}%
where the third equality follows from the Fubini's theorem. Existence and
linearity follow from the Riesz representation theorem. Boundedness of $T^*$
follows from the boundedness of $T$ since $||T^*|| = ||T||$.}

%--------------------------------------------------------------------------------------------------------------------------------------------

\subsection{{Proof of Proposition \protect\ref{prop_compactness}}}\label{section:proof compact}

{\small We first show that $\mathcal{R}(T)\subset L_{\pi_{cz}}^2$. By the Cauchy-Schwarz inequality, $\forall w \in\mathcal{W}$ and $\forall h\in L_{\pi_{cz}}^2$:
\begin{eqnarray}
  ||Th||^{2} & = &\int_{\mathcal{C}}\int_{\mathcal{Z}}\Big\langle\frac{f_{C|WZ\theta }}{\pi _{\theta }},h\Big\rangle^{2}\pi _{cz}(c,z)dcdz \\
  &\leq&
   \int_{\mathcal{C}}\int_{\mathcal{Z}}\Big|\Big|\frac{f_{C|WZ\theta }}{\pi_{\theta }}\Big|\Big|^{2}||h||^{2}\pi _{cz}(c,z)dcdz  \notag \\
  & = &||h||^{2}\int_{\mathcal{C}}\int_{\mathcal{Z}}\int_{\Theta }\frac{f_{C|WZ\theta }^{2}}{\pi _{\theta }}\pi _{cz}d\theta dcdz. \nonumber
\end{eqnarray}
 The expression is finite if the multiple integral is bounded. This is shown below in the second part of the proof. Thus, after showing this we establish that $\mathcal{R}(T)\subset L_{\pi _{cz}}^{2}$.
 
Next, we show compactness of $T$. This can be shown by showing that $T$ is Hilbert-Schmidt. An integral operator from $L_{\pi _{\theta}}^{2}$ to $L_{\pi _{cz}}^{2}$ is Hilbert-Schmidt if its kernel is square
integrable with respect to $\pi _{\theta }\times \pi _{cz}$. An
Hilbert-Schmidt operator is bounded and compact. Under the conditions of the
proposition we compute $\int_{\mathcal{C}}\int_{\mathcal{Z}}\int_{\Theta }\frac{f_{C|WZ\theta }^{2}}{\pi _{\theta }^{2}}\pi _{\theta }\pi _{cz}$ and show that it is bounded:
\begin{eqnarray*}
&& \int_{\mathcal{C}}\int_{\mathcal{Z}}\int_{\Theta }\frac{f_{C|WZ\theta }^{2}}{\pi _{\theta }^{2}}\pi _{\theta }\pi _{cz} \\
&=&  \int_{\mathcal{C}}\int_{\mathcal{Z}}\int_{\Theta }\left[\sum_{i=1}^{s}f_{\varepsilon |WZ\theta }(\varphi _{i}^{-1}(w,z,\theta,c),w,z,\theta ) \left\vert \partial _{c}\varphi _{i}^{-1}(w,z,\theta,c)\right\vert 1_{\mathcal{C}_{i}}(c)\right] ^{2}\frac{\pi _{\theta }(\theta)\pi _{cz}(c,z)}{\pi _{\theta }^{2}}d\theta dcdz \\
&  \leq&  \int_{\Theta }\int_{\mathcal{Z}}2^{s-1}\sum_{i=1}^{s}\int_{\mathcal{C}}f_{\varepsilon |WZ\theta }^{2}(\varphi _{i}^{-1}(w,z,\theta ,c),w,z,\theta) \left\vert \partial _{c}\varphi _{i}^{-1}(w,z,\theta ,c)\right\vert^{2}1_{\mathcal{C}_{i}}(c)\frac{\pi _{cz}(c,z)}{\pi _{\theta }}dcdzd\theta \\
  &=& \int_{\Theta }\int_{\mathcal{Z}}2^{s-1}\sum_{i=1}^{s}\int_{\mathcal{E}_{i}}f_{\varepsilon |WZ\theta }^{2}(\varepsilon _{i},w,z,\theta ) \left\vert \partial _{\varepsilon _{i}}\varphi (w,z,\theta ,\varepsilon_{i})\right\vert ^{-1}\frac{\pi _{cz}(\varphi (w,z,\theta ,\varepsilon
_{i}),z)}{\pi _{\theta }(\theta )}d\varepsilon _{i}dzd\theta \\
&<& \infty
\end{eqnarray*}%
where the first inequality follows from the Fubini's theorem and the
Cauchy-Schwarz's inequality and the second equality follows from the change
of variable $\varphi _{i}^{-1}(w,z,\theta ,c)=\varepsilon _{i}$. The final
inequality follows from Assumption \ref{Assumption: Compact operator}. This result shows that $\mathcal{R}(T)\subset L_{\pi_{cz}}^2$, and that $T$ is Hilbert-Schmidt and then bounded and
compact. %To show that $\mathcal{R}(T|_{\mathcal{F}_{\theta|W}})\subset L_{\pi_{cz}}^2$, remark that for every $h\in\mathcal{F}_{\theta|W}$ and every $w\in\mathcal{W}$: $||Th||^{2} = \int_{\mathcal{C}}\int_{\mathcal{Z}}\left(\int_{\Theta}f_{C|WZ\theta}(c,w,z,\theta)h(\theta,w) d\theta\right)^2\pi_{cz}(c,z)dcdz$ and by the Jensen's inequality:
%
%\begin{eqnarray*}
%  ||Th||^{2} & \leq & \int_{\mathcal{C}}\int_{\mathcal{Z}}\int_{\Theta}f_{C|WZ\theta}(c,w,z,\theta)^2 h(\theta,w) d\theta\pi_{cz}(c,z)dcdz\\
%  & = & \int_{\Theta}\int_{\mathcal{C}}\int_{\mathcal{Z}}f_{C|WZ\theta}(c,w,z,\theta)^2 \pi_{cz}(c,z)dcdz h(\theta,w) d\theta
%\end{eqnarray*}
%\noindent which is finite since $f_{C|WZ\theta}\in\mathcal{F}_{C|WZ\theta   }$
%By using the inequality:
%\begin{eqnarray}
%||Th||^{2} &=&\int_{\mathcal{C}}\int_{\mathcal{Z}}\Big\langle\frac{%
%f_{C|WZ\theta }}{\pi _{\theta }},h\Big\rangle^{2}\pi _{cz}(c,z)dcdz\leq
%\int_{\mathcal{C}}\int_{\mathcal{Z}}\Big|\Big|\frac{f_{C|WZ\theta }}{\pi
%_{\theta }}\Big|\Big|^{2}||h||^{2}\pi _{cz}(c,z)dcdz  \notag \\
%&=&||h||^{2}\int_{\mathcal{C}}\int_{\mathcal{Z}}\int_{\Theta }\frac{%
%f_{C|WZ\theta }^{2}}{\pi _{\theta }}\pi _{cz}d\theta dcdz<\infty ,
%\end{eqnarray}%
%this result shows that $\mathcal{R}(T)\subset L_{\pi _{cz}}^{2}$.
}

\section{Technical lemmas}
%=====================================================================================
\begin{lemma}\label{Lem_intermed_results_Cor_1}
  Let the assumptions of Corollary \ref{Cor_Tikhonov_rate_kernel_1} be satisfied and $\hat{f}_{C|WZ}$ be as defined in \eqref{eq_kernel_estimator_f_CWZ}. Then, 
  \begin{enumerate}
  \item[(i)] $\left[ \mathbb{E}(T^{\ast }\hat{f}_{C|WZ}-T^{\ast }f_{C|WZ})\right]^2=\mathcal{O}\left( \max\{h_{n}^{4},h_{d}^{4}\}\right)$
  \item[(ii)] $Var(T^{\ast }\hat{f}_{C|WZ}) = \mathcal{O}\left[ n^{-1}\left(\min \{h_{n},h_{d}\}\right)^{-k}\right].$
  \end{enumerate}
\end{lemma}

%---------------- PROOF ----------------
\begin{proof}
Note that $\hat{f}_{C|WZ}-f_{C|WZ}=\frac{1}{f_{WZ}}\left(\hat{f}_{CWZ}-f_{C|WZ}\hat{f}_{WZ}\right)
\left[1-\left(\hat{f}_{WZ}-f_{WZ}\right) / \hat{f}_{WZ}\right]$. And, since $\left(\hat{f}_{WZ}-f_{WZ}\right)/\hat{f}_{WZ}=o_{p}(1)$ 
we can use the approximation $\hat{f}_{C|WZ}-f_{C|WZ}\simeq \frac{1}{f_{WZ}}\left(\hat{f}_{CWZ}-f_{C|WZ}\hat{f}_{WZ}\right)$.
We start by showing result \textit{(i)}.

Let $t$ be a $k$-dimensional vector and $v$ a $l$-dimensional vector. We use the notation $\overrightarrow{vt}=(v^{\prime },t^{\prime })$ and $\overrightarrow{uvt}=(u,v^{\prime },t^{\prime })$. Moreover, we let $p=k+l$ and let $D^{2}(h)$ be the Hessian matrix of a function $h$. We use a single integral symbol to denote the multiple integral either with respect to $dvdt $ or  $dudvdt$. We start by computing the bias term $b(w,\theta )=\mathbb{E} \left(T^{\ast }\hat{f}_{C|WZ}-T^{\ast }f_{C|WZ} \right)$. 

By standard Taylor series approximations we get: $b\left(w,\theta \right) \simeq T^{\ast}\frac{1}{f_{WZ}}
\left[ \mathbb{E}\left(\hat{f}_{CWZ}\right)-f_{C|WZ}\mathbb{E}\left(\hat{f}_{WZ}\right)\right]$ and then
\begin{small}
\begin{eqnarray*}
  b(w,\theta ) & \simeq&
   T^{\ast }\frac{1}{f_{WZ}}\left\{ \left[ \mathbb{E} \left( \hat{f}_{CWZ}\right)-f_{CWZ}\right]+f_{C|WZ}\left[ f_{WZ}-\mathbb{E}\left(\hat{f}_{WZ}\right)\right]\right); \\
  \mathbb{E}\left(\hat{f}_{CWZ}\right) -f_{CWZ} &=&
  \frac{h_{n}^{2}}{2}tr\left(D^{2}(f_{CWZ})\int \overrightarrow{uvt}^{\prime }\overrightarrow{uvt} K(u,c)K(v,z)K(t,w)dudvdt\right) +o(h_{n}^{2}); \\
  \mathbb{E}\left(\hat{f}_{WZ}\right) -f_{WZ} &= &
  \frac{h_{d}^{2}}{2}tr\left(D^{2}(f_{WZ})\int \overrightarrow{vt}^{\prime }\overrightarrow{vt}K(v,z)K(t,w)dvdt\right) +o(h_{d}^{2}); \\
  b\left(w,\theta \right) &\simeq& \int_{\mathcal{C}}\int_{\mathcal{Z}}\frac{f_{C|WZ\theta}}{f_{WZ}}\Big[h_{n}^{2}tr\left( D^{2}(f_{CWZ})(c,w,z)\int \overrightarrow{uvt}^{\prime }\overrightarrow{uvt}K(u,c)K(v,z)K(t,w)dudvdt\right) dcdz \\
  && -h_{d}^{2}tr\left( D^{2}(f_{WZ})(w,z)\int \overrightarrow{vt}^{\prime }\overrightarrow{vt}K(v,z)K(t,w)dvdt\right) \Big]\frac{\pi _{cz}(c,z)}{\pi_{\theta }(\theta )}dcdz+o\left( \max \{h_{n}^{2},h_{d}^{2}\}\right) \\
  & =& h_{n}^{2}b_{1}(w,\theta )-h_{d}^{2}b_{2}(w,\theta )+o\left( \max\{h_{n}^{2},h_{d}^{2}\}\right).
\end{eqnarray*}
\end{small}
Therefore, $b^{2}(w,\theta )=\mathcal{O}\left( \max\{h_{n}^{4},h_{d}^{4}\}\right) $ which proves \textit{(i)}. 

Now consider the variance term (part \textit{(ii)} of the Lemma).
\begin{eqnarray}
  Var \left(T^{\ast }\hat{f}_{C|WZ}\right)  &=&      Var\left[T^{\ast }\left(\hat{f}_{C|WZ}-f_{C|WZ}\right)\right] \nonumber \\ 
&  \simeq & Var\left[ T^{\ast }\frac{1}{f_{WZ}}\left(\hat{f}_{CWZ}-f_{C|WZ}\hat{f}_{WZ}\right)\right] \nonumber \\
  & =& Var\left( T^{\ast }\frac{\hat{f}_{CWZ}}{f_{WZ}}\right) +Var\left( T^{\ast }\frac{f_{C|WZ}\hat{f}_{WZ}}{f_{WZ}}\right) \nonumber \\
 && -2Cov\left( T^{\ast }\frac{\hat{f}_{CWZ}}{f_{WZ}},T^{\ast }\frac{f_{C|WZ}\hat{f}_{WZ}}{f_{WZ}}\right). \notag
\end{eqnarray}
In the following we use the notation: $K_{h,i}(z,w) = K_{h}(z_{i}-z,z)K_{h}(w_{i}-w,w)$. We start by analysing the first term:
\begin{small}
\begin{eqnarray}
  Var \left( T^{\ast }\frac{\hat{f}_{CWZ}}{f_{WZ}}\right) &=&
  Var \left[ \int_{\mathcal{Z}}\int_{\mathcal{C}}\frac{f_{C|WZ\theta }(c,w,z,\theta )}{f_{WZ}(w,z)nh_{n}^{p}}\sum_{i=1}^{n}\frac{K_{h}(c_{i}-c,c)}{h_{n}}K_{h,i}(z,w)\frac{\pi_{cz}(c,z)}{\pi _{\theta }}dcdz\right]  \label{proof_cor_Tikhonov_eq_1_Var_1} \\
  & =& Var \left[ \frac{1}{nh_{n}^{k}}\sum_{i=1}^{n}f_{C|WZ\theta }(c_{i},w,z_{i},\theta )\frac{\pi _{cz}(c_{i},z_{i})}{f_{WZ}(w,z_{i})}\frac{K_{h}(w_{i}-w,w)}{\pi _{\theta }}\right]
  +o\left( (nh_{n}^{k})^{-1}\right)\notag \\
  & =& \frac{1}{nh_{n}^{2k}}\int f_{C|WZ\theta }^{2}(c_{i},w,z_{i},\theta )\frac{\pi_{cz}^{2}(c_{i},z_{i})}{f_{WZ}^{2}(w,z_{i})}\frac{K_{h}^{2}(w_{i}-w,w)}{\pi_{\theta }^{2}}f_{CWZ}(c_{i},w_{i},z_{i})dc_{i}dw_{i}dz_{i} \nonumber \\
  && -\frac{1}{nh_{n}^{2k}}\left[ \int f_{C|WZ\theta }(c_{i},w,z_{i},\theta )\frac{\pi _{cz}(c_{i},z_{i})}{f_{WZ}(w,z_{i})}\frac{K_{h}(w_{i}-w,w)}
  {\pi_{\theta }}f_{CWZ}(c_{i},w_{i},z_{i})dc_{i}dw_{i}dz_{i}\right]^{2} \nonumber \\
  && +o\left( \frac{1}{nh_{n}^{k}}\right)  \notag \\
  & =& \frac{1}{nh_{n}^{k}}\int f_{C|WZ\theta }^{2}(c_{i},w,z_{i},\theta )\frac{\pi _{cz}^{2}(c_{i},z_{i})}{f_{WZ}(w,z_{i})}\frac{\int K^{2}(t,w)dt}{\pi_{\theta }^{2}}f_{C|WZ}(c_{i},w,z_{i})dc_{i}dz_{i} \nonumber \\
  &&+o\left((nh_{n}^{k})^{-1}\right). \nonumber 
\end{eqnarray}
\end{small}
Next,
\begin{eqnarray}
  Var \left(T^{\ast } \frac{f_{C|WZ}\hat{f}_{WZ}}{f_{WZ}}\right) &=&
  Var\left( \int_{\mathcal{Z}}\int_{\mathcal{C}}\frac{f_{C|WZ\theta }(c,w,z,\theta )}{f_{WZ}(w,z)nh_{d}^{p}}
   \sum_{i=1}^{n}f_{C|WZ}(c,w,z)K_{h,i}(z,w)\frac{\pi_{cz}(c,z)}{\pi _{\theta }}dcdz\right)  \label{proof_cor_Tikhonov_eq_1_Var_2} \\
  & =& Var\left( \frac{1}{nh_{d}^{k}}\sum_{i=1}^{n}\int_{\mathcal{C}}\frac{f_{C|WZ\theta }(c,w,z_{i},\theta )}{f_{WZ}(w,z_{i})}f_{C|WZ}(c,w,z_{i})K_{h}(w_{i}-w,w)\frac{\pi _{cz}(c,z_{i})}
  {\pi _{\theta }}dc\right) \nonumber \\
  &&+o\left( \frac{1}{nh_{d}^{k}}\right)  \notag \\
 & =& \frac{1}{nh_{d}^{k}}\int_{\mathcal{Z}}\left( \int_{\mathcal{C}}\frac{f_{C|WZ\theta }(c,w,z_{i},\theta )}{f_{WZ}(w,z_{i})}f_{C|WZ}(c,w,z_{i})\pi_{cz}(c,z_{i})dc\right) ^{2}\times  \notag \\
  && \qquad \frac{\int K^{2}(t,w)dt}{\pi _{\theta }^{2}}f_{WZ}(w,z_{i})dz_{i}+o\left( (nh_{d}^{k})^{-1}\right) ,  \nonumber
\end{eqnarray}
where the results are obtained by standard Taylor series approximations. 

Finally, we have to compute the covariance term:
\begin{small}
\begin{eqnarray}
  &&Cov \left(T^{\ast } \frac{\hat{f}_{CWZ}}{f_{WZ}}, T^{\ast}\frac{f_{C|WZ}\hat{f}_{WZ}}{f_{WZ}}\right) \label{proof_cor_Tikhonov_eq_1_Cov} \\
  &=&  
  \frac{1}{n^{2}h_{n}^{k}h_{d}^{k}}\sum_{i=1}^{n}
  Cov \left(
  \begin{array}{c}
    \frac{f_{C|WZ\theta }(c_{i},w,z_{i},\theta )}{f_{WZ}(w,z_{i})}K_{h}(w_{i}-w,w)
    \frac{\pi _{cz}(c_{i},z_{i})}{\pi _{\theta }},  \\
   \int_{\mathcal{C}} \frac{f_{C|WZ\theta }(c,w,z_{i},\theta )}{f_{WZ}(w,z_{i})}
         K_{h}(w_{i}-w,w)\frac{\pi _{cz}(c,z_{i})}{\pi _{\theta}}f_{C|WZ}(c,w,z_{i})dc
     \end{array} 
     \right) \nonumber \\
 &&+o\left\{ \left[n \left(\min \{h_{n},h_{d}\}\right)^{k}\right]^{-1} \right\} \notag \\
  & =&\frac{1}{nh_{d}^{k}}\int \left[ \int_{\mathcal{C}}f_{C|WZ\theta}(c,w,z_{i},\theta )f_{C|WZ}(c,w,z_{i})\frac{\pi _{cz}(c,z_{i})}{\pi_{\theta }}dc\right] ^{2}\frac{1}{f_{WZ}(w,z_{i})}dz_{i}  \notag \\
  && \qquad \quad\times \int K(t,w)K\left( \frac{th_{n}}{h_{d}},w\right) dt+o\left((n(\min \{h_{n},h_{d}\})^{k})^{-1}\right). \nonumber
\end{eqnarray}
By putting (\ref{proof_cor_Tikhonov_eq_1_Var_1}), (\ref{proof_cor_Tikhonov_eq_1_Var_2}) and (\ref{proof_cor_Tikhonov_eq_1_Cov}) together we obtain
\begin{eqnarray}
  &&Var(T^{\ast }\hat{f}_{C|WZ}) \label{V_1_V_2_V_3_def} \\
  &\simeq&
   \int_{\mathcal{Z}}\frac{1}{nh_{n}^{k}}\left[ \mathbb{E}\left( f_{C|WZ\theta }^{2}\pi _{cz}^{2}\left\vert w,z_{i}\right.\right) 
   +\frac{1}{nh_{d}^{k}}  \mathbb{E}\left( f_{C|WZ\theta}\pi _{cz}\left\vert w,z_{i}\right. \right) ^{2}\right]
   \frac{\int K^{2}(t,w)dt}{f_{WZ}(w,z_{i})\pi _{\theta }^{2}}dz_{i}  \notag \\
  & &-\frac{2}{nh_{d}^{k}}\int \frac{ \mathbb{E}\left(f_{C|WZ\theta }\pi _{cz}\left\vert w,z_{i}\right. \right)^{2}}
  {f_{WZ}(w,z_{i})}dz_{i}\frac{\int K(t,w)K\left( \frac{th_{n}}{h_{d}}\right) dt}{\pi _{\theta }^{2}}+o\left( \frac{1}{n(\min \{h_{n},h_{d}\})^{k}}\right)  \notag \\
  & =&\frac{1}{nh_{n}^{k}}V_{1}(w,\theta )+\frac{1}{nh_{d}^{k}}V_{2}(w,\theta )-2\frac{1}{nh_{d}^{k}}V_{3}(w,\theta )+o\left( \frac{1}{n(\min\{h_{n},h_{d}\})^{k}}\right) .  \nonumber
\end{eqnarray}
\end{small}
%Therefore
%
%\begin{eqnarray*}
%  \mathbb{E}||T^{\ast }(\hat{f}_{C|WZ}-f_{C|WZ})||^{2} &=&\int_{\Theta }\Big(\frac{1}{nh_{n}^{k}}V_{1}(w,\theta )+\frac{1}{nh_{d}^{k}}V_{2}(w,\theta )-2\frac{1}{nh_{d}^{k}}V_{3}(w,\theta )+h_{n}^{4}b_{1}^{2}(w,\theta)+h_{d}^{4}b_{2}^{2}(w,\theta ) \\
%  && \quad -2h_{n}^{2}h_{d}^{2}b_{1}(w,\theta )b_{2}(w,\theta )\Big)\pi_{\theta }d\theta +o\left( \frac{1}{n(\min \{h_{n},h_{d}\})^{k}}\right)+o\left( \max \{h_{n}^{4},h_{d}^{4}\}\right)
%\end{eqnarray*}
%
%\noindent and the rate of the MISE is:
%
%\begin{equation*}
%  \mathbb{E}||\mathcal{P}_{c}\hat{f}_{\theta |W}^{\alpha }-f_{\theta|W}^{\dagger c}||^{2}=\mathcal{O}\left( \phi ^{2}(\alpha )+\frac{1}{\alpha^{2}}\left( \max \{h_{n}^{4},h_{d}^{4}\}+\frac{1}{n(\min \{h_{n},h_{d}\})^{k}}\right) \right) .
%\end{equation*}
\end{proof}


%=====================================================================================
\begin{lemma}\label{Lem_intermed_results_Lemma_2}
  Let the assumptions of Theorem \ref{lem_pointwise_asymptotic_normality} be satisfied, $\hat{f}_{C|WZ}$ be as defined in \eqref{eq_kernel_estimator_f_CWZ} and $Z_{ni}$ be as defined in the proof of Theorem \ref{lem_pointwise_asymptotic_normality}. Then, $$\sum_{i=1}^{n}\mathbb{E}\left\vert Z_{ni}/\sqrt{nVar(Z_{ni})}\right\vert^{3}\rightarrow 0\qquad \textrm{ as } n\rightarrow \infty $$
  if $\alpha ^{3}/(nh_{n}^{k})\rightarrow 0$.
\end{lemma}

%---------------- PROOF ----------------
\begin{proof}
Note that
\begin{align}
  \sum_{i=1}^{n}\mathbb{E}\left\vert Z_{ni}/\sqrt{nVar(Z_{ni})}\right\vert^{3}& =n(nVar(Z_{n1}))^{-\frac{3}{2}}\mathbb{E}|Z_{n1}|^{3}  \notag \\
  & =n^{-\frac{1}{2}}(Var(Z_{n1}))^{-\frac{3}{2}}\mathbb{E}|Z_{n1}|^{3}\label{Liapounov_eq}
\end{align}
and $\mathbb{E}|Z_{n1}|^{3}=\mathcal{O}\left( \alpha^{-3/2}h_{n}^{-2k}\right)$ by Assumption \ref{Ass_technical_asymptotic_Normality} \textit{(i)}. Moreover, by Assumption \ref{Ass_technical_asymptotic_Normality} \textit{(ii)} there exists a constant $\kappa>0$ such that $Var(Z_{n1})>\kappa \alpha^{-2} h_n^{-k}$. Therefore,
\begin{equation*}
  \sum_{i=1}^{n}\mathbb{E}\left\vert Z_{ni}/\sqrt{nVar(Z_{ni})}\right\vert^{3}=\mathcal{O}\left( \sqrt{\frac{1}{n\alpha ^{3}h_{n}^{4k}(Var(Z_{n1}))^{3}}}\right) = \mathcal{O}\left( \sqrt{\frac{\alpha^6 h_n^{3k}}{n\alpha ^{3}h_{n}^{4k}}}\right)
\end{equation*}
which converges to $0$ if $\alpha ^{3}/(nh_{n}^{k})\rightarrow 0$.

%Hence, we have to analyze $Var(Z_{n1})$:
%
%\begin{eqnarray}
%  Var(Z_{n1}) &=&Var\left( \sum_{j=1}^{\infty }\frac{\lambda _{j}}{(\alpha+\lambda _{j}^{2})}\left\langle \left( K_{h}(c_{i}-c,c)-\mathbb{E}_{WZ}(K_{h}(c_{i}-c,c))\right) \frac{K_{h,i}(z,w)}{h_{n}^{k+l+1}\hat{f}_{WZ}},\psi _{j}\right\rangle \varphi _{j}(\theta ,w)\right)  \notag \\
%  &=&\sum_{j=1}^{\infty }\frac{\lambda _{j}^{2}}{(\alpha +\lambda _{j}^{2})^{2}}Var\left( \left\langle \left( K_{h}(c_{i}-c,c)-\mathbb{E}_{WZ}(K_{h}(c_{i}-c,c))\right) \frac{K_{h,i}(z,w)}{h_{n}^{k+l+1}\hat{f}_{WZ}},\psi _{j}\right\rangle \varphi _{j}(\theta ,w)\right)  \notag \\
%  &&+2\sum_{j>j^{\prime }}\frac{\lambda _{j}\lambda _{j^{\prime }}}{(\alpha + \lambda _{j}^{2})(\alpha +\lambda _{j^{\prime }}^{2})}Cov\Big(\left\langle\left( K_{h}(c_{i}-c,c)-\mathbb{E}_{WZ}(K_{h}(c_{i}-c,c))\right) \frac{K_{h,i}(z,w)}{h_{n}^{k+l+1}\hat{f}_{WZ}},\psi_{j}\right\rangle ,  \notag \\
%  &&\qquad \left\langle \left( K_{h}(c_{i}-c,c)-\mathbb{E}_{WZ}(K_{h}(c_{i}-c,c))\right) \frac{K_{h,i}(z,w)}{h_{n}^{k+l+1}\hat{f}_{WZ}},\psi _{j^{\prime }}\right\rangle \Big)\varphi _{j}(\theta ,w)\varphi_{j^{\prime }}(\theta ,w)  \label{proof_asymptotic_normality_var}
%\end{eqnarray}
%
%\noindent which is strictly positive by Assumption \ref{Ass_technical_asymptotic_Normality} \textit{(ii)}. By a Taylor expansion of $\frac{\lambda _{j}^{2}}{(\alpha +\lambda _{j}^{2})^{2}}$ (resp. of $\frac{\lambda _{j}}{(\alpha +\lambda _{j}^{2})}$) around $\lambda _{j}=0$ it results that $\frac{\lambda _{j}^{2}}{(\alpha +\lambda _{j}^{2})^{2}}=\frac{1}{\alpha ^{2}}\lambda _{j}^{2}+o(\lambda _{j})$ (resp. $\frac{\lambda_{j}\lambda_{j^{\prime }}}{(\alpha +\lambda _{j}^{2})(\alpha +\lambda_{j^{\prime }}^{2})}=\frac{1}{\alpha ^{2}}\lambda _{j}\lambda _{j^{\prime}}+o(\lambda _{j}\lambda _{j^{\prime}})$). Moreover, from standard arguments in kernel estimation it is possible to show that the variance and covariance terms in (\ref{proof_asymptotic_normality_var}) are at most of the order $h_{n}^{-k}$. It follows that $\alpha ^{2}h_{n}^{k}Var(Z_{n1})$ is bounded. Therefore from (\ref{Liapounov_eq}) and the previous results we get
%
%\begin{equation*}
%  \sum_{i=1}^{n}\mathbb{E}\left\vert Z_{ni}/\sqrt{nVar(Z_{ni})}\right\vert^{3}=\mathcal{O}\left( \sqrt{\frac{\alpha ^{3}}{nh_{n}^{k}(\alpha^{2}h_{n}^{k}Var(Z_{n1}))^{3}}}\right)
%\end{equation*}
%
%\noindent which converges to $0$ if $\alpha ^{3}/(nh_{n}^{k})\rightarrow 0$.

\end{proof}

%=====================================================================================
\begin{lemma}\label{Lem_intermed_results_Theorem_4}
  Let the assumptions of Theorem \ref{thm_convergence_constrained_Tikhonov_rate_2} be satisfied. Then,
  $$(\alpha \tilde{P}_{f}+\tilde{P}_{f}T^{\ast }T\tilde{P}_{f})f_{\theta |W}^{\alpha ,L_{f}} = \tilde{P}_{f}T^{\ast }T\tilde{P}_{f}f_{\theta |W}^{\dagger c}$$
where $f_{\theta |W}^{\alpha ,L_{f}}$ is the solution of (\ref{eq_contrained_tikhonov_min}) with $\widehat{f}_{\left.C \right|WZ}$ replaced
by $f_{\left.C \right|WZ}$ and $\mathcal{F}_{\left. \Theta \right| W}$
replaced by $L_f = \left\{h \in L_{\pi_{\theta}}^2 : \left< f_{\left. \theta \right| W}^{\dagger c} - f,h-f_{\left. \theta \right| W}^{\dagger c} \right> = 0 \right\}$.
\end{lemma}

\begin{proof}
From \eqref{eq_proof_theorem_constrained_Tikh_rate_3} in the proof of Theorem \ref{thm_convergence_constrained_Tikhonov_rate_2} and because $f_{\theta |W}^{\alpha ,L_{f}}\in L_{f}$ implies $(f_{\theta |W}^{\alpha ,L_{f}}-f_{\theta |W}^{\dagger c})\in \tilde{L}_{f}$, which in turn implies that $\tilde{P}_{f}(f_{\theta |W}^{\alpha,L_{f}}-f_{\theta |W}^{\dagger c})=(f_{\theta |W}^{\alpha ,L_{f}}-f_{\theta |W}^{\dagger c})$, we get: $\tilde{P}_{f}(T^{\ast }T+\alpha I)f_{\theta |W}^{\alpha ,L_{f}} = \tilde{P}_{f}T^{\ast}f_{C|WZ}$ by using \eqref{eq_proof_theorem_constrained_Tikh_rate_3}. Using these results, the following equivalences hold:
\begin{eqnarray*}
    \tilde{P}_{f}T^{\ast }T(f_{\theta |W}^{\alpha,L_{f}}-f_{\theta |W}^{\dagger c})+\alpha \tilde{P}_{f}f_{\theta |W}^{\alpha,L_{f}} &=&\tilde{P}_{f}T^{\ast }f_{C|WZ}-\tilde{P}_{f}T^{\ast }Tf_{\theta|W}^{\dagger c} \\
    \Leftrightarrow \quad \tilde{P}_{f}T^{\ast }T\tilde{P}_{f}(f_{\theta|W}^{\alpha ,L_{f}}-f_{\theta |W}^{\dagger c})+\alpha \tilde{P}_{f}f_{\theta|W}^{\alpha ,L_{f}} &=&\tilde{P}_{f}T^{\ast }f_{C|WZ}-\tilde{P}_{f}T^{\ast}Tf_{\theta |W}^{\dagger c} \\
    \Leftrightarrow \quad (\alpha \tilde{P}_{f}+\tilde{P}_{f}T^{\ast }T\tilde{P}_{f})f_{\theta |W}^{\alpha ,L_{f}} &=&\tilde{P}_{f}T^{\ast }f_{C|WZ}+\tilde{P}_{f}T^{\ast }T(\tilde{P}_{f}-I)f_{\theta |W}^{\dagger c} \\
    &=&\tilde{P}_{f}T^{\ast }\left( Tf_{\theta |W}^{\dagger c}+T(\tilde{P}_{f}-I)f_{\theta |W}^{\dagger c}\right) \\
    &=&\tilde{P}_{f}T^{\ast }T\tilde{P}_{f}f_{\theta |W}^{\dagger c}.
\end{eqnarray*}
\end{proof}

%====================================================================================================================================================================
%====================================================================================================================================================================

%Denote $\mathcal{G} = \Theta_1 \times \mathcal{F}_{\theta|W}$, $\widehat{Q}_n(\theta_1, h) = ||T_{\theta_1}h-\hat{f}_{C|WZ}||^{2}$ and $Q(\theta_1, h) = ||T_{\theta_1}h - f_{C|WZ}||^{2}$. Moreover, denote by $g = (\theta_1,h)$ a generic element of $\mathcal{G}$, with $\theta_1\in\Theta_1$ and $h\in \mathcal{F}_{\theta|W}$, and by $g^{\dagger} = (\theta_1^{\dagger},h^{\dagger})$ the true value of $g$. The estimator computed by solving \eqref{eq_contrained_tikhonov_min_deterministic} will be denoted by $\hat{g}=(\hat{\theta}_1,\hat{f}_{\theta|W})$ and belongs to $\mathcal{G}$.
%Assumptions:
%\begin{ass}\label{Ass_deterministic}
%  \textit{(i).} The subset $\Theta_1\subset \mathbb{R}^{d_1}$ is compact.\\
%  \textit{(ii).} The family of functions $\{\widehat{Q}_n(\cdot,h) + \alpha \|h\|^2\}$ is equidifferentiable at every $\theta_1\in\Theta_1$.\\
%  \textit{(iii).} Let $\widehat{Q}_{n,1}(\theta_1,h)$ denote the first derivative of $\widehat{Q}_{n,1}(\theta_1,h)$ with respect to $\theta_1$ evaluated at $\theta_1$. We assume $\sup_{h\in\mathcal{F}_{\theta_2|W}}|\widehat{Q}_{n,1}(\theta_1,h)|<\infty$ for every $\theta_1\in\Theta_1$.\\
%  \textit{(iv).} $Q(g^{\dagger}) = 0$ and any $(\theta_1,h)\in\mathcal{H}$ that satisfies $Q(\theta_1,h) = 0$ also satisfies $\theta_1 = \theta_1^{\dagger}$ and $h = h^{\dagger}$ almost everywhere.\\
%  \textit{(v).} The function $f_{C|WZ\theta}(c,w,z,\theta_1,\theta_2)$ is continuous in $\theta_1$.
%\end{ass}
%
%Proof of Theorem \ref{thm_convergence_deterministic_case}
%
%
%Let $\mathcal{U}_w(g^{\dagger})$ denote an open neighborhood in $\mathcal{G}$ in the weak topology around $g^{\dagger}$ and $\mathcal{U}_w(\theta_1^{\dagger})$ denote its projection onto $\Theta_1$, that is,
%
%$$\mathcal{U}_w(\theta_1^{\dagger}) = \left\{\theta_1\in\Theta_1; \exists h\in\mathcal{F}_{\theta|W} \textrm{ such that } (\theta_1,h)\in \mathcal{U}_w(g^{\dagger})\right\}.$$
%
%Define $\|\hat{g} - g^{\dagger}\|=\|\hat{\theta}_1\|_{E} + \|\hat{f}_{\theta|W} - f_{\theta|W}^{\dagger}\|$, where $\|\cdot\|_{E}$ denotes the Euclidean norm induced by the scalar product $\langle \cdot,\cdot\rangle_{E}$. Hence, because $\Theta_1\subset \mathbb{R}^d$ and because the weak and norm topologies coincides on finite dimensional spaces, then $\|\hat{\theta}_1\|_{E}\rightarrow 0$ in probability if and only if $P(\hat{\theta}_1 \in \mathcal{U}_w(\theta_1^{\dagger}))\rightarrow 1$. This follows from Lemma \ref{lem_Prob_weakly_open_set} and the inequality $P(\hat{\theta}_1 \in \mathcal{U}_w(\theta_1^{\dagger}))\geq P(\hat{g} \notin \mathcal{U}_w(g^{\dagger}))$.\\
%Next, we show $\|\hat{f}_{\theta|W} - f_{\theta|W}^{\dagger}\|\rightarrow 0$ in probability. Lemma \ref{lem_Prob_weakly_open_set} shows consistency under the weak topology which implies $\langle \tilde{g},\hat{f}_{\theta|W} - f_{\theta|W}^{\dagger} \rangle$ for every $\tilde{g}\in \Theta_1\times \mathcal{F}_{\theta|W}$. From Lemma \ref{lem_strongly_convex}
%
%\begin{eqnarray*}
%  \|\hat{f}_{\theta|W}\|^2 - \|f_{\theta|W}^{\dagger}\|^2 & \geq & \langle f_{\theta|W}^{\dagger}, \hat{f}_{\theta|W} - f_{\theta|W}^{\dagger}\rangle  + c \|\hat{f}_{\theta|W} - f_{\theta|W}^{\dagger}\|^{2}\\
%  & = & \langle g^{\dagger}, \hat{g} - g^{\dagger}\rangle  - \langle \theta_1^{\dagger}, \hat{\theta}_1 - \theta_{1}^{\dagger}\rangle + c \|\hat{f}_{\theta|W} - f_{\theta|W}^{\dagger}\|^{2}.
%\end{eqnarray*}
%
%\noindent The first two terms of the right hand side converge to zero in probability by Lemma \ref{lem_Prob_weakly_open_set} and the left hand side converges to zero in probability by Lemma ???. Hence, $\|\hat{f}_{\theta|W} - f_{\theta|W}^{\dagger}\|^{2}\rightarrow 0$ in probability.

\begin{lemma}\label{lemma_existence_deterministic}
  Suppose that assumptions \ref{Ass_deterministic} \textit{(i)}-\textit{(iii)} and \textit{(v)} hold. Then, a solution to the minimization problem \eqref{eq_contrained_tikhonov_min_deterministic} exists.
\end{lemma}

\begin{proof}
Problem \eqref{eq_contrained_tikhonov_min_deterministic} is numerically equivalent to the following procedure computed in two steps, where in the first step one computes, for each $\theta_1\in\Theta_1$
\begin{equation}
  m(\theta_1) = \min_{h\in \mathcal{F}_{\theta_2 \left\vert W\right.}}\left\{||T_{\theta_1}h - \hat{f}_{C|WZ}||^{2}+\alpha ||h||^{2}\right\}\label{eq_contrained_tikhonov_min_deterministic_profiled}
\end{equation}
and in the second step one computes:
\begin{equation}\label{eq_contrained_tikhonov_min_deterministic_profiled_theta_1}
  \hat{\theta}_1 = \min_{\theta_1\in\Theta_1} m(\theta_1).
\end{equation}
A solution to \eqref{eq_contrained_tikhonov_min_deterministic_profiled} exists for every $\theta_1\in\Theta_1$ since it is a convex problem. Moreover, under Assumptions \ref{Ass_deterministic} \textit{(ii)}-\textit{(iii)} and \textit{(v)}, by Theorem 3 of \cite{MilgromSegal2002} the value function $m(\cdot)$ is continuous. This together with compactness of $\Theta_1$ implies the existence of a solution to \eqref{eq_contrained_tikhonov_min_deterministic_profiled_theta_1}.
\end{proof}

%=======================================================================================================================
\begin{lemma}\label{lem_strongly_convex}
  The functional $\xi(h)=\|h\|^2$ defined on $L_{\pi_{\theta}}^2$ satisfies:
  \begin{displaymath}
    \|h_1\|^2 - \|h_2\|^2 - \langle h_2,(h_1 - h_2)\rangle \geq c \|h_1 - h_2\|^2 \quad \textrm{ for any } 0< c \leq 1.
  \end{displaymath}
\end{lemma}

\begin{proof}
Note that the G\^{a}teaux derivative of $\|\cdot \|^2$ at $h_0\in L_{\pi_{\theta}}^2$, denoted by $D(h_0)$ is equal to the linear functional $D(h_0) = \langle \cdot, h_{0}\rangle$ on $L_{\pi_{\theta}}^2$. Hence, for every $h_1,h_2\in L_{\pi_{\theta}}^2$
\begin{equation}\label{eq_1_proof_lem_strongly_convex}
D(h_0)(h_1 - h_2) =  \langle D(h_{1}) - D(h_2),(h_1 - h_2)\rangle \geq c\|h_1 - h_2\|^2, \qquad \textrm{ for any } 0< c \leq 1.
\end{equation}
Define $\varphi(t) = \|h_t\|^2$ where $h_t = th_1 + (1 - t) h_2$,  for $h_1,h_2\in L_{\pi_{\theta}}^2$ and for every $t\in[0,1]$. Note that $h_t \in \mathcal{F}_{\theta|W}$ if $h_1,h_2\in \mathcal{F}_{\theta|W}$. Moreover, $d\varphi(t)/dt = D(h_t)(h_1 - h_2)$. Then, for $0\leq t' < t \leq 1$
\begin{eqnarray*}
  \frac{d\varphi(t)}{dt} - \frac{d\varphi(t')}{dt} & = & \langle D(h_t) - D(h_{t'}),h_1 - h_2\rangle\\
  & = & \left\langle D(h_t) - D(h_{t'}),\frac{h_t - h_{t'}}{t - t'}\right\rangle\\
  & \geq & c\frac{\|h_t - h_{t'}\|^2}{t - t'}
\end{eqnarray*}
where the second equality is due to the equality $h_t - h_{t'} = (t - t')(h_1 - h_2)$ and the last inequality is due to \eqref{eq_1_proof_lem_strongly_convex}. Now, by setting $t' = 0$ we get $\frac{d\varphi(t)}{dt} - \frac{d\varphi(0)}{dt} \geq c t \|h_1 - h_2\|^2$. Therefore, by $\varphi(1) - \varphi(0) - \frac{d\varphi(0)}{dt} = \int_0^1 [\frac{d\varphi(t)}{dt} - \frac{d\varphi(0)}{dt}] dt \geq c \|h_1 - h_2\|^2$. By replacing $\varphi(1)$ with $\|h_1\|^2$ and $\varphi(0)$ with $\|h_2\|^2$ we get the result.
 \end{proof}

%=======================================================================================================================
\begin{lemma}\label{lem_bounded}
Under Assumption \ref{Ass_deterministic} \textit{(iv)} and \textit{(vi)} we have: \textit{(i)} $\|\check{f}_{\theta_2|W}^{\alpha,c}\|^2  - \|f_{\theta_2|W}^{0}\|^2 = O_p(\alpha^{-1}\delta_n)$ for $\delta_n = o(1)$; \textit{(ii)} if $\delta_n = O(\alpha)$, then there exists an $M_0$ such that:
  \begin{displaymath}
    P\left(\|\check{f}_{\theta_2|W}^{\alpha,c}\|^2> M_0\right) \rightarrow 0 \qquad \textrm{ as } n\rightarrow \infty.
  \end{displaymath}
\end{lemma}

\begin{proof}
 By definition of $\hat{g}$ and since $\widehat{Q}_n(\hat{g}) \geq 0$: $\alpha \|\check{f}_{\theta_2|W}^{\alpha,c}\|^2 \leq \widehat{Q}_n(\hat{g}) + \alpha \|\check{f}_{\theta_2|W}^{\alpha,c}\|^2$. This implies that
\begin{eqnarray*}
  \alpha \left(\|\check{f}_{\theta_2|W}^{\alpha,c}\|^2  - \|f_{\theta_2|W}^{0}\|^2\right) & \leq & \widehat{Q}_n(\hat{g}) + \alpha \left(\|\check{f}_{\theta_2|W}^{\alpha,c}\|^2 - \|f_{\theta_2|W}^{0}\|^2\right)\\
  & \leq & \widehat{Q}_n(g^{0}) + \alpha \left(\|f_{\theta_2|W}^{0}\|^2 - \|f_{\theta_2|W}^{0}\|^2\right)\\
  & = & Q(g^{0}) + |\widehat{Q}_n(g^{0}) - Q(g^{0})| + \alpha \left(\|f_{\theta_2|W}^{0}\|^2 - \|f_{\theta_2|W}^{0}\|^2\right)\\
  & = & O_p(\delta_n)
\end{eqnarray*}
by Assumption \ref{Ass_deterministic} \textit{(iv)} and \textit{(vi)}, where $\delta_n = o(1)$ and the second inequality follows from the fact that $\check{f}_{\theta_2|W}^{\alpha,c}$ is the minimizer of the criterion (and hence, $\widehat{Q}_n(\hat{g}) + \alpha \|\check{f}_{\theta_2|W}^{\alpha, c}\|^2 \leq \widehat{Q}_n(g^{0}) + \alpha \|f_{\theta_2|W}^{0}\|^2$). This shows \textit{(i)}. To show \textit{(ii)} we use result \textit{(i)} and observe that
\begin{eqnarray*}
  P\left(\|\check{f}_{\theta_2|W}^{\alpha,c}\|^2  - \|f_{\theta_2|W}^{0}\|^2> M\right) & = & P\left(\alpha(\|\check{f}_{\theta_2|W}^{\alpha,c}\|^2  - \|f_{\theta_2|W}^{0}\|^2) > \alpha M\right)\\
  & = & P\left(O_p(\delta_n) > \alpha M\right)
\end{eqnarray*}
which converges to zero for every $M>0$ if $\delta_n = O(\alpha)$. Finally, because $\|f_{\theta_2|W}^{0}\|^2$ is bounded, we can choose a finite $M_0>0$ sufficiently large so that $P\left(\|\check{f}_{\theta_2|W}^{\alpha,c}\|^2> M_0\right) \rightarrow 0$ as $n\uparrow 0$.
\end{proof}

%=======================================================================================================================
\begin{lemma}\label{lem_Prob_weakly_open_set}
Let $\mathcal{U}_w(g^{0})$ denote an open neighborhood in $\mathcal{G}$ in the weak topology around $g^{0}$. Under Assumptions 7 and 10 (i),(iv)-(vi), and if $\delta_n = O(\alpha)$, where $\delta_n$ is as in Assumption 10 (vi), then:
  \begin{displaymath}
    P(\hat{g} \notin \mathcal{U}_w(g^{0})) \rightarrow 0 \qquad \textrm{ as } n\rightarrow \infty.
  \end{displaymath}
\end{lemma}

\begin{proof}  
Since $P(A)\leq P(A\cap B) + P(B^c)$ for any measurable sets $A$ and $B$ and by recalling the notation $\hat{g} = (\hat{\theta}_1,\check{f}_{\theta_2|W}^{\alpha,c})$ then
\begin{equation}
  P(\hat{g} \notin \mathcal{U}_w(g^{0})) \leq P(\hat{g} \notin \mathcal{U}_w(g^{0}), \|\check{f}_{\theta_2|W}^{\alpha,c}\|^2 \leq M_0) + P(\|\check{f}_{\theta_2|W}^{\alpha,c}\|^2 > M_0)
\end{equation}
for some $M_0>0$ large. Lemma \ref{lem_bounded} shows that for any $\varepsilon > 0$ there exists an $M_0 = M_0(\varepsilon) > 0$ such that $P(\|\check{f}_{\theta_2|W}^{\alpha,c}\|^2 > M_0) < \varepsilon$. So, we focus on the first probability in the right hand side.
\begin{eqnarray}
  &&P(\hat{g} \notin \mathcal{U}_w(g^{0}), \|\check{f}_{\theta_2|W}^{\alpha,c}\|^2 \leq M_0) \label{eq_1_lem_Prob_weakly_open_set} \\
  & \leq & P\left(\inf_{g\notin \mathcal{U}_w(g^{0}); \|f\|^2\leq M_0}[\widehat{Q}_n(g) + \alpha \|f\|^2] \leq \widehat{Q}_n(g^{0}) + \alpha \|f_{\theta_2|W}^{0}\|^2\right)\nonumber\\
  & \leq & P\Big(\inf_{g\notin \mathcal{U}_w(g^{0}); \|f\|^2\leq M_0}[Q(g) + \alpha \|f\|^2] - \sup_{g\notin \mathcal{U}_w(g^{0}); \|f\|^2\leq M_0}|\frac{1}{2}\widehat{Q}_n(g) - Q(g)|\nonumber\\
  & & \qquad \leq Q(g^{0}) + \alpha \|f_{\theta_2|W}^{0}\|^2 + |\widehat{Q}_n(g^{0}) - Q(g^{0})|\Big). \nonumber
\end{eqnarray}

Note that $|\frac{1}{2}\widehat{Q}_n(g) - Q(g)| = \frac{1}{2}\|T_{\theta_1}h-\hat{f}_{C|WZ}\|^{2} - ||T_{\theta_1}h-f_{C|WZ}||^{2} \leq \|\hat{f}_{C|WZ} - f_{C|WZ}\|^2$ by using the inequality $\frac{a^2}{2} - b^2 \leq (a - b)^2$. So, by Assumption \ref{Ass_iid_data}, $\sup_{g\notin \mathcal{U}_w(g^{0}); \|f\|^2\leq M_0}|\frac{1}{2}\widehat{Q}_n(g) - Q(g)| = \mathcal{O}_p(\eta_n)$ for some $\eta_n = o(1)$. Then, from \eqref{eq_1_lem_Prob_weakly_open_set} and because $|\widehat{Q}_n(g^{0}) - Q(g^{0})| = O_p(\delta_n)$, it follows:
\begin{eqnarray}
&&  P(\hat{g} \notin \mathcal{U}_w(g^{0}), \|\check{f}_{\theta_2|W}^{\alpha,c}\|^2 \leq M_0) \nonumber \\
   &\leq& P\left(\inf_{g\notin \mathcal{U}_w(g^{0}); \|f\|^2\leq M_0}[Q(g) + \alpha \|f\|^2] \leq \alpha \|f_{\theta_2|W}^{0}\|^2 + O_p(\eta_n) + O_p(\delta_n)\right) \nonumber \\
 & \leq & P\left(\inf_{g\notin \mathcal{U}_w(g^{0}); \|f\|^2\leq M_0}Q(g) + \inf_{\|f\|^2\leq M_0}\alpha \|f\|^2 \leq \alpha \|f_{\theta_2|W}^{0}\|^2 + O_p(\eta_n) + O_p(\delta_n)\right)
 \nonumber \\
 & \leq& P\left(\inf_{g\notin \mathcal{U}_w(g^{0}); \|f\|^2\leq M_0}Q(g) \leq O_p(\max\{\alpha ,\eta_n,\delta_n\})\right). \nonumber
\end{eqnarray}
%Observe that $\inf_{\|f\|^2\leq M_0}\alpha \|f\|^2$ cannot be zero since $f$ is a density function.

Note that the set $\mathcal{U}_w^c(g^{0})$ is closed and $\Theta_1 \times \{f\in\mathcal{F}_{\theta_2|W}; \|f\|^2\leq M_0\}$ is closed and bounded. Thus, under assumption \ref{Ass_deterministic} \textit{(v)}, $Q$ is continuous which implies that there exists a $g^*\in\{g\in \mathcal{U}_w(g^{0})^c; \|f\|^2\leq M_0\}$ such that $\inf_{g\notin \mathcal{U}_w(g^{0}); \|f\|^2\leq M_0}Q(g) = Q(g^*)$. Moreover, by Assumption \ref{Ass_deterministic} \textit{(iv)} it must be $Q(g^*)>0$. If this was not the case, then we would have $g^* = g^{0}$, but this is a contradiction of the fact that $g^*\in \mathcal{U}_w^c(g^{0})$.\\
Because $O_p(\max\{\alpha ,\eta_n,\delta_n\})\rightarrow 0$ as $n\rightarrow \infty$ and $Q(g^*)>0$, we conclude that 
\begin{equation*}
P(\hat{g} \notin \mathcal{U}_w(g^{0}), \|\check{f}_{\theta_2|W}^{\alpha,c}\|^2 \leq M_0) \rightarrow 0
\end{equation*}
 which in turn implies that $P(\hat{g} \notin \mathcal{U}_w(g^{0}))\rightarrow 0$ as $n\rightarrow \infty$.
\end{proof}

%=================================================================================
%\subsection{Identification without nuisance unobservables\label{Section: no
%epsilon}}
%
%In this section we briefly describe the case where we do not have $%
%\varepsilon $ so that $f_{C|WZ\theta }$ cannot be recovered as in theorem %
%\ref{th1}. This is relevant in models where all the unobservable variables
%are of interest so $\varepsilon $ is included in $\theta $. In our setup,
%this implies that the general structural model (\ref{eq_DGP_Ass_1}) reduces
%to
%\begin{equation}
%\Psi (C,W,Z,\theta )=0\qquad a.s.
%\end{equation}%
%and assumption \ref{AssDGP} is replaced by the following one.
%
%\renewcommand{\theass}{1'}
%
%\begin{ass}
%\label{AssDGP_bis} The random element $(C,W,Z,\theta)$ satisfies a
%structural economic model
%
%\begin{equation}
%\Psi (C,W,Z,\theta)=0\quad a.s.  \label{eq_DGP_Ass_1_bis}
%\end{equation}%
%where $\Psi $ is a \textbf{known} Borel measurable real-valued function. We
%assume that (\ref{eq_DGP_Ass_1_bis}) has a unique global solution in terms
%of $C$:
%\begin{equation}
%C=\varphi (W,Z,\theta),\quad a.s.  \label{eq_DGP_Ass_2_bis}
%\end{equation}%
%where $\varphi :\mathbb{R}^{k+l+d}\rightarrow \mathbb{R}$ is a
%Borel-measurable function.
%\end{ass}
%
%Indeed, even in this setup where $\varphi $ is not strictly monotonic in $%
%\theta $ and $\theta $ is multivariate, we can characterize the structural
%\textit{pdf} $f_{\theta |W}$ as a solution to a constrained functional
%equation. Let $F_{C|WZ}$ be the cumulative distribution function of $C$
%conditioned on $(W,Z)$ and assumed to be in $L_{\pi _{cz}}^{2}$. Then, under
%assumptions \ref{AssDGP_bis}, \ref{Assmu'} and \ref{AssIndeta}
%\begin{equation}
%F_{C|WZ}(c,w,z)=Sf_{\theta |W}(\theta ,w)\quad \text{and}\quad f_{\theta
%|W}\in \mathcal{F}_{\theta |W},\qquad a.s.
%\label{eq_estimating_without_noise}
%\end{equation}%
%where $S:L_{\pi _{\theta }}^{2}\rightarrow L_{\pi _{cz}}^{2}$ is a bounded
%linear operator defined as
%\begin{equation*}
%Sh=\int_{\{\theta :\varphi (w,z,\theta )\leq c\}}h(\theta ,w)d\theta \equiv
%\int_{\Theta }1\{\varphi (w,z,\theta )\leq c\}h(\theta ,w)d\theta ,\quad
%\forall h\in L_{\pi _{\theta }}^{2}.
%\end{equation*}%
%The kernel of the new operator is $\frac{1\{\varphi (w,z,\theta )\leq c\}}{%
%\pi _{\theta }(\theta )}$ and the adjoint $S^{\ast }$ is:%
%\begin{equation*}
%S^{\ast }h=\int_{\mathcal{C}}\int_{\mathcal{Z}}1\{\varphi (w,z,\theta )\leq
%c\}\frac{\pi _{cz}(c,z)}{\pi _{\theta }(\theta )}h(c,w,z)dcdz,\quad \forall
%h\in L_{\pi _{cz}}^{2}.
%\end{equation*}%
%Note that when there are nuisance unobservables, the estimating equation (%
%\ref{functional_equation_th_1}) can be trivially recovered from (\ref%
%{eq_estimating_without_noise}) by differentiating with respect to $c$.
%
%The estimation procedure for this case is the same as that one proposed in
%Section \ref{s_Estimation} with the operator $T$ replaced by $S$. The rate
%of the mean integrated squares error will improve since $F_{C|WZ}$ can be
%estimated at a better rate than $f_{C|WZ}$. Moreover, the degree of
%ill-posedness will not be as severe as in the case where the kernel of $T$
%is exponential.

%\bibliographystyle{alphanat}
\bibliography{HNSBib}

\end{document} 