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\title[Large Sample Theory]{Large Sample Theory: Lecture XIV}
\author{Charles B. Moss}
\date{\today}

\begin{document}

\frame{\titlepage}

\section[Outline]{}
\frame{\tableofcontents}

\section{Basic Sample Theory}
\frame
{
   \frametitle{Basic Sample Theory}
   \begin{itemize}
   \item The problems set up is that we want to discuss sample theory.
      \begin{itemize}
      \item First assume that we want to make an inference, either estimation or some test, based on a sample.
      \item We are interested in how well parameters or statistics based on that sample represent the parameters or statistics of the whole population.
      \end{itemize}
   \item The complete statistical term is known as convergence.
      \begin{itemize}
      \item Specifically, we are interested in whether or not the statistics calculated on the sample converge toward the population estimates.
      \item Let $\left\{\ X_n \right\}\ $  be a sequence of samples.  We want to demonstrate that statistics based on $\left\{\ X_n \right\}\ $ converge toward the population statistics for $X$ .
      \end{itemize}
   \end{itemize}
}
\frame
{
   \begin{itemize}
   \item Taking a slightly different tack:  The classical assumptions for ordinary least squares (OLS) as presented in White, Halbert Asymptotic Theory for Econometricians.
      \begin{itemize}
      \item {\bf Theorem 1.1}: The following are the assumptions of the classical linear model
         \begin{itemize}
         \item The model is known to be $y=X\beta + \epsilon$, $\beta < \infty$.
         \item $X$ is a nonstochastic and finite $n \times k$ matrix.
         \item $X'X$ is nonsingular for all $n \le k$.
         \item ${\rm E}\left(\epsilon\right) = 0$.
         \item $\epsilon \sim N\left( 0 , \sigma_0^2 I \right)$, $\sigma_0^2 < \infty$.
         \end{itemize}
      \end{itemize}
   \end{itemize}
}
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{
   \begin{itemize}
   \item Continued
      \begin{itemize}
      \item Given these assumptions, we can conclude that
         \begin{itemize}
         \item {\it Existence} given (i) - (iii) $\beta_n$ exists for all $n \ge k$ and is unique.
         \item {\it Unibiasedness} given (i) - (v) ${\rm E}\left[ \beta_n\right] = \beta_0$.
         \item {\it Normality} given (i) - (v) $\beta_n \sim N\left(\beta_0, \sigma^2 \left(X'X\right)^{-1} \right)$.
         \item {\it Efficiency} given (i) - (v) $\beta_n$ is the maximum likelihood estimator and the best unbiased estimator in the sense that any other unbiased estimator exceeds that of $\beta_n$  by a positive semi-definite matrix regardless of the value of $\beta_0$.
         \end{itemize}
      \end{itemize}
   \end{itemize}
}
\frame
{
   \begin{itemize}
   \item Continued
      \begin{itemize}
      \item Existence, unbiasedness normality and efficiency are small sample analogs of asymptotic theory.
         \begin{itemize}
         \item Unbiased implies that the distribution of $\beta_n$ is centered around $\beta_0$.
         \item Normality allows us to construct $t$-distribution or $F$-distribution tests for restrictions.
         \item Efficiency guarantees that the ordinary least squares estimates have the greatest possible precision.
         \end{itemize}
      \item Asymptotic theory involves the behavior of the estimator under the failure of certain assumptions.  Specifically, assumptions (ii) or (v).
      \item Finally, within the classical linear model the normality of the error term is required to strictly apply $t$-distributions or $F$-distributions.  However, the central limit theorem can be used if $n$ is large to guarantee that $\beta_n$ is approximately normal.
      \end{itemize}
   \end{itemize}
}
\section{Modes of Convergence}
\frame
{
   \frametitle{Modes of Convergence}
   \begin{itemize}
   \item {\bf Definition 6.1.1} A sequence of real numbers $ \left\{\ \alpha_n \right\}\ $ , $n = 1,2, \cdots $ is said to converge to a real number $\alpha$ if for any $\epsilon > 0$  there exists an integer $N$ such that for all $n > N$  we have

\begin{equation}
\left| \alpha_n - \alpha \right| < \epsilon
\label{eqn:lect14-01}
\end{equation}

      \begin{itemize}
      \item This convergence is expressed as $\alpha_n \rightarrow \alpha$  as $n \rightarrow \infty$ or $\lim_{n \rightarrow \infty} \alpha_n = \alpha$.
      \item This definition must be changed for random variables because we cannot require a random variable to approach a specific value.  Instead, we require the probability of the variable to approach a given value.  Specifically, we want the probability of the event to equal 1 or zero as $n$ goes to infinity.
      \end{itemize}
   \end{itemize}
}
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{
   \begin{itemize}
   \item {\bf Definition 6.1.2} (convergence in probability) A sequence of random variables $\left\{\ X_n \right\}\ $ , $n=1,2,\cdots $ is said to converge to a random variable $X$ in probability if for any $\epsilon > 0$  and $\delta > 0$  there exists an integer $N$   such that for all $n > N$  we have $P\left( \left| X_n - X \right| < \epsilon \right) > 1 - \delta $ .  We write

\begin{equation}
X_n \stackrel{P}{\rightarrow} X
\label{eqn:lect14-02}
\end{equation}

   \noindent $n \rightarrow \infty $  or ${\rm plim}_{n \rightarrow \infty} X_n = X$ .  The last equality reads “the probability limit of $X_n$is $X$.” (Alternatively, the if clause may be paraphrased as follows: if $\lim P\left( \left| X_n - X \right| < \epsilon \right) = 1 $  for any $\epsilon > 0$.
      \begin{itemize}
      \item Since we are dealing with convergence in the probability, we need to introduce two concepts: convergence in mean square and convergence in distribution.
      \end{itemize}
   \end{itemize}
}
\frame
{
   \begin{itemize}
   \item {\bf Definition 6.1.3.} ({\it Convergence in Mean Square}) A sequence $\left\{\ X_n \right\}\ $ is said to {\it converge} to $X$ in {\it mean square} if $\lim_{n \rightarrow \infty} {\rm E}\left( X_n - X \right)^2 = 0$ . We write

\begin{equation}
X_n \stackrel{M}{\rightarrow} X
\label{eqn:lect14-03}
\end{equation}

   \end{itemize}
}
\frame
{
   \begin{itemize}
   \item {\bf Definition 6.1.4.} ({\it Convergence in Distribution}) A sequence $\left\{\ X_n \right\}\ $  is said to {\it converge} to $X$ {\it in distribution} if the distribution function $F_n$ of $X_n$ converges to the distribution function $F$ of $X$ at every continuity point of $F$.  We write

\begin{equation}
X_n \stackrel{d}{\rightarrow} X
\label{eqn:lect14-04}
\end{equation}

   \noindent and call $F$ the limit distribution of $\left\{\ X_n \right\}\ $ .  If $\left\{\ X_n \right\}\ $  and $\left\{\ Y_n \right\}\ $  have the same limit distribution, we write
   
\begin{equation}
X_n \stackrel{LD}{=} Y_n
\label{eqn:lect14-05}
\end{equation}

   \end{itemize}
}
\frame
{
   \begin{itemize}
   \item {\bf Theorem 6.1.1} ({\it Chebyshev})

\begin{equation}
X_n \stackrel{M}{\rightarrow} X \Rightarrow X_n \stackrel{P}{\rightarrow} X
\label{eqn:lect14-06}
\end{equation}

   \item {\bf Theorem 6.1.2}
   
\begin{equation}
X_n \stackrel{P}{\rightarrow} X \Rightarrow X_n \stackrel{d}{\rightarrow} X
\label{eqn:lect14-07}
\end{equation}

   \end{itemize}
}
\frame
{
   \begin{itemize}
   \item Chebyshev's Inequality

\begin{equation}
P\left[ g\left(X_n\right) \ge \epsilon^2 \right] \le \frac{\displaystyle {\rm E}\left[ g\left(X_n\right) \right]}{\displaystyle \epsilon^2}
\label{eqn:lect14-08}
\end{equation}

   \item {\bf Theorem 6.1.3} Let $X_n$ be a vector of random variables with a fixed finite number of elements.  Let $g$ be a function continuous at a constant vector point $\alpha$.  Then

\begin{equation}
X_n \stackrel{P}{\rightarrow} \alpha \Rightarrow g\left(X_n\right) \stackrel{P}{\rightarrow} g\left(\alpha\right)
\label{eqn:lect14-09}
\end{equation}

   \end{itemize}
}
\frame
{
   \begin{itemize}
   \item {\bf Theorem 6.1.4} ({\it Slutsky}) If $X_n \stackrel{d}{\rightarrow} X$ and $Y_n \stackrel{d}{\rightarrow} \alpha $, then

\begin{equation}
\begin{array}{c}
\displaystyle X_n + Y_n \stackrel{d}{\longrightarrow} X + \alpha \\ \\
\displaystyle X_n Y_n \stackrel{d}{\longrightarrow} \alpha X \\ \\
\displaystyle \left( \frac{\displaystyle X_n}{\displaystyle Y_n} \right) \stackrel{d}{\longrightarrow} \frac{\displaystyle X}{\displaystyle \alpha} \,\,{\rm if}\,\, \alpha \neq 0
\end{array}
\label{lect14-10}
\end{equation}

   \end{itemize}
}

\end{document}