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%	\vspace*{20pt}
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\title[Invariance principle and its applications]{Invariance principle for
multiparameter summation processes and its applications}
\author{Vaidotas Zemlys}
\institute[Vilnius University] {
    Vilnius University\\
    Universite des Sciences et Technologies de Lille
}


\date{1$^{\text{st}}$ of April, 2008}

\begin{document}
\begin{frame}
    \titlepage
\end{frame}
\begin{frame}
    \frametitle{Outline} 
    \begin{enumerate}
	\item Introduction
	    \begin{enumerate}
		\item Definition of the invariance principle.
		\item Applications in the change point problem.
		\item Panel data.
	    \end{enumerate}
	\item Theoretical results
	    \begin{enumerate}
		\item Invariance principle for \iid{} variables in space
		    $\Hao([0,1]^d, \bm{H})$.
		\item FCLT for series schema in $\Hao([0,1]^d)$.
	    \end{enumerate}
	\item Applications 
	    \begin{enumerate}
		\item CUSUM type statistics (FCLT for panel regresion
		    residuals).
		\item Generalisation of epidemic alternative for two-dimensional
		    case.
	    \end{enumerate}
    \end{enumerate}
\end{frame}

\section{Motivation}
\begin{frame}
    \frametitle{What is the invariance principle?} 
    \begin{itemize}
	\item Convergence of partial sums process in some space of functions. 
	    Generalization of the Central Limit Theorem.
	\item Example of partial sums process:
	    \begin{align*}
		\xi_{n}(t)=S_{[nt]}+(nt-[nt])X_{[nt]+1}, 
		\quad S_k=X_1+\dots+X_k
	    \end{align*}
	%\item $X_i$, $i=1,\dots,n$ - \iid{} $S_k=X_1+\dots+X_k$.
	\item Invariance principle:
	    \begin{align*}
		n^{-1/2}\xi_{n}\xrightarrow{C[0,1]} W \Leftrightarrow  EX_1^2<\infty
	    \end{align*}
	\item $W$ - Wiener process, $EW(t)W(s)=\min(t,s)$.	
    \end{itemize}
\end{frame}
\begin{frame}
    \frametitle{The change point problem} 
    \begin{itemize}
	\item Having a sample we want to test whether there is a point where the
	    mean changes. 
	\item Null and alternative hypotheses:
	    \begin{align*}
		H_0&: EX_i=\mu_0
		\intertext{against}
		H_A&:\exists k^* \text{ such that } EX_i=\mu_0+(\mu_1-\mu_0)\indf{k^*< i\le n} 
	    \end{align*}
    \end{itemize}
\end{frame}
\begin{frame}
    \frametitle{Change point is known} 
    \begin{itemize}
 	\item Simple idea: compare the means.
	    \begin{align*}
		\frac{1}{k^*}S_{k^*}-\frac{1}{n-k^*}\left(S_{n}-S_{k^*}\right)=
		\frac{n}{k^*(n-k^*)}\left(S_{k^*}-\frac{k^*}{n}S_{n}\right)
	    \end{align*}
	\item Denote
	    \begin{align*}
		R=(S_{k^*}-\frac{k^*}{n}S_n)%=n^{-1/2}\left(1-\frac{k}{n}\right)S_k-\frac{k}{n}(S_n-S_k)
	    \end{align*}
	\item Assume $k^*/n\to c$. Then under null due to CLT we have
	    \begin{align*}
		n^{-1/2}R\to N(0,\sigma^2c(1-c))
	    \end{align*}
	\item Under alternative we have
	    \begin{align*}
		n^{-1/2}R=n^{1/2}\frac{k^*}{n}\left(1-\frac{k^*}{n}\right)(\mu_1-\mu_0)+O_P(1)\to\infty
	    \end{align*}
    \end{itemize}
\end{frame}
\begin{frame}
    \frametitle{Change point is unknown} 
    \begin{itemize}
	\item If the change point is unknown, ``look'' for it:
	    \begin{align*}
		Q=\max_{1<k<n}|S_{k}-\frac{k}{n}S_n|
	    \end{align*}
	\item Remember $\xi_n=S_{[nt]}+(nt-[nt])X_{[nt]+1}:$
	    \begin{align*}
		Q=\max_{1<k<n}|\xi_{n}(k/n)-k/n\xi_n(1)|
	    \end{align*}
	\item We have that $Q$ is actually a functional of $\xi_n$. 
	    \begin{align*}
		Q=f(\xi_{n}), \text{ where } f(x)=\sup_{0<t<1}|x(t)-x(1)|
	    \end{align*}
	\item Functional $f:C[0,1]\to R$ is continuous in $C([0,1])$.
	\item Invariance principle and continuous mapping gives us: 
	    \begin{align*}
		n^{-1/2}Q\xrightarrow{D} \sup_{0<t<1}|W(t)-tW(1)|
	    \end{align*}
	    \end{itemize}
\end{frame}
\begin{frame}
    \frametitle{Epidemic alternatives}
    \begin{itemize}
	\item We want to test epidemic: the mean changes then reverts to
	    previous value.
	
	\item Alternative hypothesis:  $EX_i=\mu_0+(\mu_1-\mu_0)\indf{k^*< i\le
	    m^*}$ 
	\item The same argumentation gives test statistic:
	    \begin{align*}
		UI(n,\alpha)=\max_{1\le i<
		j\le n}\frac{|S_{j}-S_{i}-(j-i)/nS_n|}{[(j-i)/n]^\alpha}
	    \end{align*}
	\item Division by $[(j-i)/n]^\alpha$ let us detect shorter epidemics.	
   \end{itemize}
\end{frame}
\begin{frame}
    \frametitle{Epidemic alternatives II}
    \begin{itemize}
	\item Shortness of detected epidemic depends on $\alpha$.	
	\item Under alternative 
	    \begin{align*}
		n^{-1/2}UI(n,\alpha)\ge 
		\frac{l^{*(1-\alpha)}}{n^{1/2-\alpha}}\left(1-\frac{l^*}{n}\right)|\mu_1-\mu_0|+O_P(1)
	    \end{align*}
	\item If $l^*=n^\gamma$ then 
	    \begin{align*}
		n^{-1/2}UI(n,\alpha)\to \infty \text{ for }
		\gamma>\frac{1/2-\alpha}{1-\alpha}
	    \end{align*}
%	\item Note $0<\alpha<1/2$, since $W$ ``lives only'' in $\Hao([0,1])$ for
%	    $\alpha<1/2$.
    \end{itemize}
\end{frame}
\begin{frame}
    \frametitle{H\" older space} 
    \begin{itemize}
	\item Problem:
	    \begin{align*}
		f(x)=\sup_{0<s<t<1}\frac{|x(t)-x(s)-(t-s)x(1)|}{|t-s|^\alpha}
	    \end{align*}
	 $f$ is not continous in $C[0,1]$!

	\item Functional $f(x)$ is continuous in H\"older space:
	    \begin{align*}
		\Hao([0,1])=\{x\in C[0,1]: |x(t+h)-x(t)|=o(h^\alpha)\}.
	    \end{align*}
       \item If $n^{-1/2}\xi_n\to W$ in H\" older space then
	    \begin{align*}
		n^{-1/2}UI(n,\alpha)\to\sup_{0<s<t<1}\frac{|W(t)-W(s)-(t-s)W(1)|}{[t-s]^\alpha}
	    \end{align*}
	\item Note $0<\alpha<1/2$, since $W$ ``lives only'' in $\Hao([0,1])$ for
	    $\alpha<1/2$.

    \end{itemize}
\end{frame}
\begin{frame}
    \frametitle{Short conclusion} 
    \begin{itemize}
	\item Invariance principle gives us easy way to get limiting
	    distributions for tests for the change point problems.

	\item For statistical applications it makes sense to investigate
	    invariance principle in different spaces of functions.
    \end{itemize}
\end{frame}
\begin{frame}
    \frametitle{Panel data} 
    \begin{itemize}
	\item Usual situation: we observe some random variable at given time
	    points (price of stock), or we observe group of variables at certain
	    given time (Inflation rate of EU countries in 2007). In both cases
	    our observation have one index.

	\item Panel data - observations of group of random variables at certain
	    time points. Yearly inflation rate history of EU countries.      
	    Observed variables have two indexes: $X_{it}$, $i$ - individuals,
	    $t$ - time.
	%\item Least square estimate
	%    \begin{align*}
	%	\wh{\bb}=\bb+\left(\sum_{i=1}^n\sum_{t=1}^T\bx_{it}\bx_{it}'\right)^{-1}\sum_{i=1}^n\sum_{t=1}^T\bx_{it}u_{it}
	%    \end{align*}
	\item We have double index, hence the need to use  invariance
	    principle for multi-indexed processes. 
    \end{itemize}
\end{frame}
\begin{frame}
    \frametitle{Thesis goals} 
    \begin{itemize}
	\item Investigate invariance principle for multiparameter processes
	\item Investigate change point problems in panel data.
    \end{itemize}
\end{frame}
\section{Invariance principle}
\subsection{Summation process}
%\begin{frame}
%    \frametitle{Other types of invariance principles}
%   
%    \begin{itemize}
%	\item Other trajectory spaces:	
%   	\item Space $D[0,1]$. Convergence of $\zeta_n(t)=S_{[nt]}$.
%	\item Space $\Hao[0,1]=\{x\in C[0,1]: |x(t+h)-x(t)|=o(h^\alpha)\}$.
%	    Process remains the same.
%	
%    \end{itemize}
%\end{frame}
\begin{frame}
    \frametitle{General summation process} 
    \begin{itemize}
	\item Double indexed sum:
	    \begin{align*}
		S_{nm}=\sum_{i=1}^n\sum_{j=1}^mX_{ij}
	    \end{align*}
	\item In one-dimensional setting we have
	    \begin{align*}
		\xi_n(i/n)=S_i
	    \end{align*}    
	\item In two-dimensional setting then put
	    \begin{align*}
		    \xi_{nm}(i/n,j/m)=S_{ij}		
	    \end{align*}

    \end{itemize}
\end{frame}
\begin{frame}
    \frametitle{General summation process II}
    \begin{itemize}
	\item For $\bj,\bn\in \mathbb{N}^d$ write $\bj\le\bn$ iff $j_i\le n_i$,
	    $i=1,..,d$.
	\item Continuous version, $\bt\in [0,1]^d$
	    \begin{align*}
		\xi_{\bm{n}}(\bt)=\sum_{\bm{j}\le \bm{n}}|R_{\bm{n},\bm{j}}|^{-1}|R_{\bm{n},\bm{j}}\cap
		[0,\bt]|X_{\bm{j}},
	    \end{align*}
	    where
	    \begin{align*}
	    R_{\bm{n},\bm{j}} := [(j_1-1)/n_1, j_1/n_1]\times\cdots\times
[(j_d-1)/n_d, j_d/n_d].
	    \end{align*}
	    and
	    \begin{align*}
		[0,\bm{t}]=[0,t_1]\times\dots \times[0,t_d]
	    \end{align*}
	\item Note $\xi_{\bn}(\bj/\bn)=S_{\bj}$	
    \end{itemize}
\end{frame}
\begin{frame}
    \frametitle{Invariance principle} 
    \begin{itemize}
	\item Limiting process Wiener sheet:
	    $EW(\bm{t})W(\bm{s})=\min(t_1,s_1)\dots\min(t_d,s_d)$
	\item When
	    \begin{align*}
		(n_1\dots n_d)^{-1/2}\xi_{\bn}\to W?
	    \end{align*}
    \end{itemize}
\end{frame}
\subsection{Results}
\begin{frame}
    \frametitle{Known results} 
    \begin{itemize}
	\item Bass (1985), Alexander, Pyke (1986), space of continuous
	     functions. Necessary and sufficient condition $EX_1^2<\infty$. Not
	     multiindex.

	\item Erickson (1981), H\" older function space.  The FCLT
	    holds in $H_\alpha([0,1]^d)$, if $E|X_{\bm{1}}|^q<\infty$, for
	    $q>d/(1/2-\alpha)$.
	\item Lots of results for $\bn=(n,\dots,n)$ and different dependence assumptions
    \end{itemize}
\end{frame}
\begin{frame}
    \frametitle{I. i. d. random variables}
    \begin{itemize}
	\item Ra\v ckauskas, Suquet, Zemlys (2007)
     \begin{align*}
	(n_1\dots n_d)^{-1/2}\xi_{\bm{n}} \xrightarrow[m(\bm{n})\to\infty]{\mathcal{D}} W \text{
	in the space } \ab{H}^o\bigl([0,1]^d\bigr),
    \end{align*}
    
    \vspace*{15pt}
    
    if and only if
    
    \vspace*{5pt}
   
    \begin{align*}
	    n_1\dots n_d\mathbf{P}\bigl(|X_{\bm{1}}|>n_1^{1/p}n_2^{1/2}\dots
	    n_d^{1/2}\bigr)\to 0,
	    \text{ when } m(\bm{n}) \to \infty,
	   \end{align*}
	   where $p=1/(1/2-\alpha)$.
    \end{itemize}
\end{frame}
\begin{frame}
    \frametitle{Improvements} 
    \begin{itemize}
	\item Also proved for case when $X_{\bj}$ are random elements from
	   Hilbert space $\mathbb{H}$.
       \item Improves on Erickson's result, weakens moment assumption.
       \item For $d=1$ moment condition is $\lim_{t\to0}t^pP(|X_1|>t)=0$. Proven
	   by Račkauskas and Suquet.
       \item For $d>1$ in general case $\sup_{t>0}t^pP(|X_1|>t)<\infty$, but if
	   we assume $\bn=(n,\dots,n):$ 
	   \begin{align*}
	       \lim_{t\to\infty}t^{\frac{2d}{d-2\alpha}}P(|X_{\bm{1}}|> t)=0
    	   \end{align*}
	
    \end{itemize}
\end{frame}

\begin{frame}
    \frametitle{Series schema: definitions} 
    \begin{itemize}
	\item $(X_{\bm{n},\bm{k}},\; \bm{1}\le \bm{j}\le \bm{k}_{\bm{n}}),\; \bm{n}\in
    \mathbb{N}^d$
    \item 
	\begin{align*}
	S_\bn(\bm{k})=\sum_{\bm{j}\le \bm{k}}X_\nj,\quad b_\bn(\bm{k})=\sum_{\bm{j}\le
	\bm{k}}\snk
	\end{align*}

	\item $
	    b_i(k)=b_\bn(k^1_\bn,\dots,k^{i-1}_\bn,k,k^{i+1}_\bn,\dots,k^d_\bn)$
	\item ``Adaptive'' grid: 
	    \begin{align*}
		Q_{\nj} :=\bigg[b_1(j_1-1),\; b_1(j_1)\bigg)\times \dots\times
    \bigg[b_d(j_d-1),\;b_d(j_d)\bigg)
\end{align*}
\item Summation process	
   \begin{align*}
		\xi_{\bm{n}}(\bm{t})=\sum_{\bm{1}\le \bm{j}\le
		\bm{n}}|Q_{\bm{n},\bm{j}}|^{-1}|Q_{\bm{n},\bm{j}}\cap
		[0,\bm{t}]|X_{\bm{j}}	
    \end{align*}
    
    \end{itemize}

\end{frame}
\begin{frame}
    \frametitle{Series schema results:} 
    \begin{itemize}
	\item If	
	    \begin{align*}
		\mu_\bn(\bm{t})=\sum_{\bm{1}\le\bm{k}\le\kn}\indf{\B(\bm{k})\in
		[0,\bm{t}]} \snk\to \mu (\bm{t})
	    \end{align*}
	\item Then (with moment conditions),  Zemlys (2007):
	    \begin{align*}
		\xi_{\bn}\xrightarrow{D} G, \text{ in } \Hao([0,1]^d)
	    \end{align*}
	    where $\{G(\bm{t}), \bt\in[0,1]^d\}$ - Gaussian with
	    \begin{align*}
		EG(\bm{t})G(\bm{s})=\mu(\min(t_1,s_1),\dots,\min(t_d,s_d))
	    \end{align*}     
    \end{itemize}
\end{frame}
\section{Applications}
\subsection{CUSUM statistics}
\begin{frame}
    \frametitle{Time series regression} 
    \begin{itemize}
	\item Model:
	    \begin{align}\label{treg} 
		y_t=\bx_t\bb+u_t,\quad t=1,\dots,T
	    \end{align}
	\item Regression residuals
	    \begin{align*}
		\wh{u}_t=y_t-\bx_t\wh{\bb}=u_t-\bx_t(\wh{\bb}-\bb)
	    \end{align*}
	    where $\wh{\bb}$ is least squares solution of \eqref{treg}. 
	\item Summation process $\xi_T(z)=T^{-1/2}\sum_{t=1}^{[Tz]}\wh{u}_t$ 
	\item Ploberger, Kramer (1992):
	\begin{align*}
	    T^{-1/2}\xi_T\to W(z)-zW(1)
	\end{align*}
    \end{itemize}
\end{frame}
\begin{frame}
    \frametitle{CUSUM test} 
    \begin{itemize}
	\item Null hypothesis: $\bb_t=\bb_0$.
	\item Alternative hypothesis: $\bb_t$ depends on $t$.
	\item Under null
	    \begin{align*}
		\sup_{0\le z\le 1}|\xi_T(z)|\to \sup_{0\le z\le
		1}|W(z)-zW(1)|. 
	    \end{align*}
    \end{itemize}
\end{frame}
\begin{frame}
    \frametitle{Panel regression} 
    \begin{itemize}
	\item Ordinary least squares
	    \begin{align*}
		y_{it}=\bx_{it}\bb+u_{it},\quad i=1,\dots,n; t=1,\dots,T
	    \end{align*}
	\item Fixed effects
	    \begin{align*}
		y_{it}=\mu_i+\bx_{it}\bb+u_{it},\quad i=1,\dots,n; t=1,\dots,T
	    \end{align*}
    \end{itemize}
\end{frame}
\begin{frame}
    \frametitle{Invariance principle} 
    If invariance principle holds for $u_{it}$:
	    \begin{align*}
		(nT)^{-1/2}\xi_{nT}\xrightarrow{D} W
	    \end{align*}
	    then

    \begin{itemize}
	\item Ordinary least squares
	    \begin{align*}
		(nT)^{-1/2}\xi_{nT}^{(OLS)}(u,v) \to W(u,v)-uvW(1,1)
	    \end{align*}
	\item Fixed effects
	    \begin{align*}
		(nT)^{-1/2}\xi_{nT}^{(FE)}(u,v) \to W(u,v)-vW(u,1)
	    \end{align*}

    \end{itemize}
\end{frame}
\begin{frame}
    \frametitle{Local alternatives} 
Suppose 
	    \begin{align*}
			\bb_{ij}=\bb+\frac{1}{\sqrt{nm}}\bm{g}\left(\frac{i}{n},\frac{j}{m}\right)
	    \end{align*}

    \begin{itemize}
	\item Ordinary least squares
	    \begin{align*}
		(nT)^{-1/2}\xi_{nT}^{(OLS)}(t,s)\xrightarrow{D}
		W(t,s)-tsW(1,1)\\+
	\int_0^t\int_0^s\bm{c}'\bm{g}(u,v)dudv-ts\bm{c}'\int_0^1\int_0^1\bm{g}(u,v)dudv
	    \end{align*}
	\item Fixed effects
	    \begin{align*}
	(nT)^{-1/2}\xi_{nT}^{(FE)}(t,s)\xrightarrow{D} W(t,s)-sW(t,1)\\+
	\int_0^t\int_0^s\bm{c}'\bm{g}(u,v)dudv-s\int_0^t\int_0^1\bm{c}'\bm{g}(u,v)dudv,
	    \end{align*}
    \end{itemize}
\end{frame}
\subsection{Epidemic alternatives}
\begin{frame}
\frametitle{Definitions} 
    \begin{itemize}
	\item $(H_0):$ {$X_{ij}$ have all the same mean $\mu_{0}$.}
	\item 	$(H_A):$ {There are integers $1<a^*\le b^*<n$, 
	    $1<c^*\le d^*<m$ and a constand $\mu_1\neq \mu_0$ such that}
\begin{align*}
    EX_{ij}=\mu_0+\mu_1\indf{(i,j)\in[a^*,b^*]\times[c^*,d^*]\cap \mathbb{N}^2}
\end{align*}
	\item Test statistic:
	    \begin{align*}
		DUI(nm,\alpha)=\max_{\substack{1\le a<b\le n\\  1\le c< d\le m}} 
    \frac{|\Delta^1_{b-a}\Delta^2_{d-c}S_{bd}-(s_{b}-s_{a})
    (t_{d}-t_{c})S_{nm}|}{\max\{s_{b}-s_{a},t_{d}-t_{c}\}^\alpha}
	    \end{align*}
	where $s_i=i/n$, $t_j=j/m$, for $i=1,\dots,n$, $j=1,\dots,m$ and
	\begin{align*}
	    \Delta^1_{b-a}\Delta^2_{d-c}S_{bd}
	    =S_{bd}-S_{ad}-S_{bc}+S_{ac}
	\end{align*}
    \end{itemize}
\end{frame}
\begin{frame}
    \frametitle{Null hypothesis} 
  %  \begin{center}
%	\item Functional
	    \begin{center}
$		T_{\alpha}(x)=\sup_{\bm{0}\le \bm{s}<\bm{t}\le
		\bm{1}}\frac{|x(\bm{t})-x(s_1,t_2)-x(t_1,s_2)+x(\bm{s})-(t_1-s_1)(t_2-s_2)x(1,1)|}{|\bm{t}-\bm{s}|^\alpha}$
	    \end{center}

is continuous in $\Hao([0,1]^2)$ hence
	    \begin{align*}
		(nm)^{-1/2}DUI(nm,\alpha)\to T_{\alpha}(W)
	    \end{align*}
 %   \end{center}
\end{frame}
\begin{frame}
    \frametitle{Alternative hypothesis} 
   Assume under $(H_A)$ that the $X_{ij}$ are independent and
    $\sigma_0^2=\sup_{\bm{n}}\var(X_{\bn})$ is finite. If
    \begin{align*}%\label{c:hndm}
	\lim_{\bn\to\infty}
	(nm)^{1/2}\frac{h_{nm}}{d_{nm}^\alpha}|\mu_1-\mu_0|\to\infty, 
    \end{align*}
where
    \begin{align*}
	h_{nm}=\frac{k^*l^*}{nm}\left(1-\frac{k^*l^*}{nm}\right) 
	\text{ and }
	d_{nm}=\max\left\{\frac{k^*}{n},\frac{l^*}{m}\right\},
    \end{align*}
then
\begin{align*}%\label{l:duih1}
    (nm)^{-1/2}DUI(nm,\alpha)\to\infty
\end{align*}

\end{frame}
\begin{frame}
    \frametitle{Open problems} 
    \begin{itemize}
	\item Extend invariance principle for dependent random variables
	\item More general alternative hypothesis for epidemic alternative
	\item Improve rates for alternative hypothesis
    \end{itemize}
\end{frame}
\end{document}

