\documentclass[utf8,hyperref={unicode}]{beamer}
%\documentclass[utf8]{beamer}

\mode<presentation>
{
  \usetheme{Warsaw}
  % or ...
  %\usecolortheme{dove} 

  % or whatever (possibly just delete it)
   \setbeamertemplate{footline}
   {%
     \leavevmode%
    \hbox{\begin{beamercolorbox}[wd=.5\paperwidth,ht=2.5ex,dp=1.125ex,leftskip=.3cm 	plus1fill,rightskip=.3cm]{author in head/foot}%
    \usebeamerfont{author in head/foot} \insertshortauthor
    \end{beamercolorbox}%
    \begin{beamercolorbox}[wd=.5\paperwidth,ht=2.5ex,dp=1.125ex,leftskip=.3cm,rightskip=.3cm plus1fil]{title in head/foot}%
    \usebeamerfont{title in head/foot}\insertshorttitle \hskip 2ex p.
	\insertpagenumber\enspace of \insertdocumentendpage\enspace
    \end{beamercolorbox}}%
  \vskip0pt%
}
%\hfill\insertshorttitle\enspace -- p. \insertpagenumber\enspace iš
%\insertdocumentendpage\enspace%
%   }
    
}
 \setbeamertemplate{navigation symbols}{}



\usepackage[english]{babel}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage{amsmath}
\usepackage{amssymb}
%\usepackage{theorem}
\usepackage{bm}
\usepackage{graphicx}

\newcommand{\ab}[1]{#1_{\alpha}}
\newcommand{\wab}[2][\delta]{w_{\alpha}(#2,#1)}
\newcommand{\normab}[1]{\lVert#1\rVert_{\alpha}}
\newcommand{\eps}{\varepsilon}
\newcommand{\sprod}[1]{\langle #1 \rangle}
\newcommand{\indf}[1]{\mathbf{1}\left( #1 \right)}
\newcommand{\bx}{\mathbf{x}}
\newcommand{\bb}{\bm{\beta}}
\newcommand{\bj}{\bm{j}}
\newcommand{\bn}{{\bm{n}}}
\newcommand{\bt}{\bm{t}}

\newcommand{\nj}{{\bm{n},\bm{j}}}
\newcommand{\snk}{\sigma^2_{\bm{n},\bm{k}}}
\newcommand{\snj}{\sigma^2_{\bm{n},\bm{j}}}
\newcommand{\kn}{\bm{k}_{\bm{n}}}
\newcommand{\B}{\bm{B}}


\DeclareMathOperator{\var}{var}

\newcommand{\wt}[1]{\widetilde{#1}}
\newcommand{\wh}[1]{\widehat{#1}}
\newcommand{\HH}{\mathrm{H}} % a set of notations for Holder spaces

\newcommand{\Hao}{\HH_{\alpha}^o}

\newcommand{\iid}{\mbox{i.i.d.\frenchspacing}}

%\renewcommand{\theenumi}{\roman{enumi}}
%\renewcommand{\labelenumi}{\theenumi)}
\theoremstyle{change}\newtheorem{teorema}{Proposition}
%\theoremstyle{change}\newtheorem{definit}{Definition}
%	\vspace*{20pt}
%	\vspace*{20pt}

\title[Change point problems for multi-indexed r.v.]{Change point problems for
multi-indexed random variables}
\author{Vaidotas Zemlys}
\institute[Vilnius University] {
    Vilnius University\\
    Universite des Sciences et Technologies de Lille
}


\date{17$^{\text{th}}$ of June, 2008}
\usecolortheme{crane}
\begin{document}
\begin{frame}
    \titlepage
\end{frame}
%\begin{frame}
%    \frametitle{Outline} 
%    \begin{enumerate}
%	\item Classical change point problem.
%	    \begin{enumerate}
%		\item Statistics when change point is known and unknown.
%		\item Epidemic alternatives and H\" older space.
%	    \end{enumerate}
%	\item Multidimensional index.
%	    \begin{enumerate}
%		\item General setting.
%		\item Rectangle alternatives.
%	    \end{enumerate}
%	\item Applications to panel data regression.
%    \end{enumerate}
%\end{frame}

%\section{Motivation}
\begin{frame}
    \frametitle{The change point problem} 
    \begin{itemize}
	\item Having a sample (one dimensional index) 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 (Cs\"orgo and Horvath (1997)): 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 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 the 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 and its limit:
	    \begin{align*}
		Q_E&=\max_{1\le i<j\le\ n}|S_{j}-S_{i}-(j-i)/nS_n|,\\
		n^{-1/2}Q_E&\to\sup_{0<|t-s|<1}|W(t)-W(s)-(t-s)W(1)|
	    \end{align*}
   \end{itemize}
\end{frame}
\begin{frame}
    \frametitle{Short epidemics}
    \begin{itemize}
	\item Division by the length of epidemic leads to detection of shorter
	    epidemics:
	    \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 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 Under null hypothesis the FCLT in H\" older space  (Ra\v ckauskas
	    and Suquet (2004)) gives
	   \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}
%\section{Multi-dimensional index data}

\begin{frame}
    \frametitle{Examples of multi-dimensional index 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*}
        \end{itemize}
\end{frame}

\begin{frame}
\frametitle{General setting} 
    \begin{itemize}
	\item $(H_0):$ {$X_{ij}$ have all the same mean $\mu_{0}$.}
	\item 	$(H_A):$ {There is a set $D^*\subset [0,1]^2$ such that}
\begin{align*}
    EX_{ij}=\mu_0+\mu_1\indf{\left(\frac{i}{n},\frac{j}{m}\right)\in D^*}
\end{align*}
\item The analog of $R$ statistic, when the change set is known.
\begin{align*}
    R=\sum_{i=1}^n\sum_{j=1}^mX_{ij}
    \indf{\left(\frac{i}{n},\frac{j}{m}\right)\in D^*}-
    \frac{N_{D^*}}{nm}\sum_{i=1}^n\sum_{j=1}^mX_{ij},
\end{align*}
where
\begin{align*}
    N_{D^*}=\sum_{i=1}^n\sum_{j=1}^m \indf{\left(\frac{i}{n},\frac{j}{m}\right)\in D^*}
\end{align*}
\end{itemize}
\end{frame}

\begin{frame}
    \frametitle{The limiting distribution} 
    \begin{itemize}
	\item If  $N_{D^*}/(nm)\to |D^*|>0$, then  
	    \begin{align*}
		(nm)^{-1/2}R\to N(0,\sigma^2|D^*|(1-|D^*|))
		\end{align*}
	\item What to do when $D^*$ is unknown? Restrict the class of possible
	    sets:
	    \begin{align*}
		(nm)^{-1/2}\sup_{D\in
		\mathcal{D}}|S_{nm}(D)-\frac{N_D}{nm}S_{nm}(T)|\to
		\sup_{D\in\mathcal{D}}|W(D)-|D|W(T)| 
    	    \end{align*}
	    due to resuls by Bass (1985), Alexander, Pyke
	    (1986). Here $T=[0,1]^2$,
	$\mathcal{D}$ is suitable class of subsets.
    \end{itemize}
\end{frame}

\begin{frame}
    \frametitle{Rectangles}
   
    \begin{itemize}
	\item Let us analyze the class of the rectangles
	    $\mathcal{D}=\{[\bm{s},\bm{t}]\subset [0,1]^2\}$. If the rectangle
	    is known:
	    \begin{align*}
    D^*=\left[\frac{a^*}{n},\frac{b^*}{n}\right]\times 
	\left[\frac{c^*}{m},\frac{d^*}{m}\right].
\end{align*}


	    \item Under alternative
    \begin{align*}
        (nm)^{-1/2}R=(nm)^{1/2}\frac{k^*l^*}{nm}\left(1-\frac{k^*l^*}{nm}
    \right) (\mu_1-\mu_0) +O_P(1),
    \end{align*}
    where $k^*=b^*-a^*$, $l=d^*-c^*$.

     \item  The divergence occurs when $k^*=O(n^{\gamma})$ and $l^*=O(m^\delta)$
       with $\gamma,\delta>1/2$.
       \end{itemize}
\end{frame}

\begin{frame}
    \frametitle{Epidemic rectangle}
    \begin{itemize}
	\item As in one dimensional case it is possible to improve the rates.  		
	\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{Alternative hypothesis} 
    \begin{itemize}
	\item Condition for divergence under alternative hypothesis
	    \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*}
\item Taking $k^*=O(n^\gamma)$ and $l^*=O(m^\delta)$
    we get
    \begin{align*}
    \frac{n^{\gamma-1/2}m^{\delta-1/2}}{[n^{\gamma-1}\vee
    m^{\delta-1}]^\alpha}\to \infty
	\end{align*}
If $n^{\gamma-1}>m^{\delta-1}$ then we have
\begin{align*}
    n^{\gamma(1-\alpha)+\alpha-1/2} m^{\delta-1/2}\to\infty
\end{align*}

  \end{itemize}
\end{frame}
\begin{frame}
    \frametitle{Null hypothesis} 
  
    \begin{itemize}
	\item Given the FCLT by  Ra\v ckauskas, Suquet, Zemlys (2007)
	 \begin{align*}
		(nm)^{-1/2}DUI(nm,\alpha)\to T_{\alpha}(W),
	    \end{align*}
where
	    \begin{align*}
		T_{\alpha}(x)=\sup_{\bm{0}\le \bm{s}<\bm{t}\le
\bm{1}}\frac{|\Delta_{[\bm{s},\bm{t}]}W(\bm{t})-(t_1-s_1)(t_2-s_2)W(1,1)|}{|\bm{t}-\bm{s}|^\alpha}
	    \end{align*}
	\item Division by volume is open question. Hard to find a suitable
	    functional space.   
	\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{CUSUM type statistics} 
    \begin{itemize}
	\item Generalisation of Ploberger, Kramer (1992) results:
	\item ``Plug-in'' residuals of panel regression to statistic
	    $DUI(nm,\alpha)$.
	\item For the ordinary least squares regression the limiting statistic
	    is the same.
	\item For the fixed effects regression:
	    \begin{align*}
		DUI^{FE}(\alpha)=\sup_{\bm{0}\le \bm{s}<\bm{t}\le\bm{1}}\frac{|\Delta_{[\bm{s},\bm{t}]}W-(t_2-s_2)[W(t_1,1)-W(s_1,1)]|}{|\bm{t}-\bm{s}|^\alpha}.	    \end{align*}
	   \item The results hold as $n\wedge m\to\infty$!	

    \end{itemize}
\end{frame}
\begin{frame}
    \frametitle{Remaining problems} 
    \begin{itemize}
	\item Calculation of critical values.
	\item Power calculations. 
	\item Small sample properties.
    \end{itemize}
\end{frame}
%\begin{frame}
    %\frametitle{Bibliography} 
%    \setbeamerfont{bibliography}{size=\tiny}
%\bibliographystyle{alpha}
%\bibliography{AlfCh,vzthesis}

%\end{frame}
\end{document}

