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%%%%%%%%%%%%%%% MY DEFINITIONS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Put your own definitions
%
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\def\eeqrs{\end{eqnarray*}}             %\eeqrs

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

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\title[Stat 609 Lecture 16]{Stat 609: Mathematical Statistics \\ Lecture 16}
\author[UW-Madison]{Jun Shao}
\institute[Statistics]{%
%  Department of Statistics\\
%  University of Wisconsin\\
%  Madison, WI 53706, USA\\
%~~~\\
  }
\date[2014]{}

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\AtBeginSection[]{\frame{\frametitle{Outline}\tableofcontents[current]}}

\begin{document}

%\frame{\titlepage}

\frame {
\frametitle{
Lecture 16: Inequalities}\vspace{-2mm}
\begin{alertblock}{}
Inequalities are useful for statistical theory.
\end{alertblock}\vspace{-2mm}
\begin{block}{Theorem 3.6.1 (Chebychev's inequality)}<2->
Let $X$ be a random variable and let $g(x)$ be  a nonnegative function. For any $r >0$, \vspace{-2mm}
\[ P(g(X) \geq r ) \leq \frac{E[g(X)] }{r} \]
\end{block}\vspace{-2mm}
\begin{block}{Proof.}<2->
Assuming that $X$ has a pdf (if $X$ has a pmf, we replace integral by summation), we have  \vspace{-2mm}
\beqrs
 P(g(X) \geq r ) & = & \int_{-\infty}^\infty g(x)f_X(x) dx \geq
 \int_{-\infty}^\infty I( \{ g(x) \geq r\} )  g(x)f_X(x) dx \\
 & \geq & r \int_{-\infty}^\infty I( \{ g(x) \geq r\} )  f_X(x) dx
= r \int_{ \{ g(x) \geq r\}} f_X(x) dx \\
 & = & r P(g(X) \geq r )
 \eeqrs
 where $I(A) $ is the indicator function of the set $A$.
\end{block}
}
\frame{
\vspace{-2mm}
\begin{alertblock}{Different forms of  Chebychev's inequality}
\begin{itemize}
\item
If $g$ is nondecreasing, then another form of  Chebychev's inequality is, for $\epsilon >0$, \vspace{-2mm}
\[ P(|X| \geq \epsilon ) \leq \frac{E[g(X)] }{g(\epsilon )} \vspace{-2mm} \]
\item
Suppose that $X$ has expectation $\mu$ and variance $\sigma^2$.

For $g(x) = (x- \mu)^2/\sigma^2$, we have \vspace{-2mm}
\[ P(|X-\mu| \geq t\sigma ) = P\left( \frac{(X-\mu)^2}{\sigma^2} \geq t^2 \right) \leq \frac{1}{t^2} E\frac{(X-\mu)^2}{\sigma^2} = \frac{1}{t^2}\vspace{-2mm}  \]
\item
If $X$ has a finite $k$th moment with an integer $k$, then, for $t>0$,
 \vspace{-2mm}
\[ P(|X-\mu| \geq t )  \leq \frac{E|X-\mu|^k}{t^k} \vspace{-2mm} \]
\item
If $X$ has a finite mgf $M_X(t)$ for $t \in (-h,h)$, then, for $r>0$ and $t>0$,  \vspace{-2mm}
\[ P(X \geq r )  \leq \frac{E(e^{tX})}{e^{tr}} = \frac{M_X(t)}{e^{tr}}, \quad \quad
 P(X \leq -r )  \leq \frac{E(e^{-tX})}{e^{tr}} = \frac{M_X(-t)}{e^{tr}}\vspace{-2mm}  \]
\[ P(|X| \geq r) \leq \frac{M_X(t)+M_X(-t)}{e^{tr}} \]
\end{itemize}
\end{alertblock}
}
\frame{ \vspace{-1mm}
\begin{block}{}
Chebychev's inequality is useful, but sometimes it is too loose because it does not require much from the distribution of $X$ except some moment conditions.
\vspace{1mm}

More useful probability inequality can be derived when we know something about the distribution of $X$.
\end{block}\vspace{-2mm}
\begin{block}{Example 3.6.3 (A normal probability inequality}
If $Z \sim N(0,1)$, then \vspace{-2mm}
\[ P(|Z| \geq t) \leq \frac{\sqrt{2}e^{-t^2/2}}{\sqrt{\pi} t} , \quad \quad t>0 \vspace{-2mm} \]
because \vspace{-2mm}
\beqrs
P(|Z|\geq t) & = & 2P(Z\geq t) = \frac{2}{\sqrt{2\pi}} \int_t^\infty e^{-z^2/2} dz \\
& \leq & \frac{\sqrt{2}}{\sqrt{\pi}} \int_t^\infty \frac{z}{t} e^{-z^2/2} dz =
\frac{\sqrt{2}}{\sqrt{\pi} t}e^{-z^2/2}\bigg|_\infty^t \\
& = & \frac{\sqrt{2}e^{-t^2/2}}{\sqrt{\pi} t}
\eeqrs
\end{block}
}
\frame{
\begin{block}{}
Comparing this inequality with Chebychev's inequality obtained previously, \vspace{-2mm}
\[ P(|Z| \geq t) \leq \frac{M_X(t)+M_X(-t)}{e^{t^2}} = \frac{2 e^{t^2/2} }{e^{t^2}} = 2 e^{-t^2/2} \vspace{-2mm} \]
 we find that this inequality is more accurate by a factor of $\sqrt{2\pi} t$, which is large when $t$ is large.
\end{block}\vspace{-2mm}
\begin{alertblock}{Cauchy-Schwartz's inequality}<2->
This is another simple but very useful inequality.
\vspace{1mm}

If $X$ and $Y$ are random variables with $a^2 = E(X^2) < \infty$ and $b^2 = E(Y^2) < \infty$, then
the following Cauchy-Schwartz's inequality holds: \vspace{-1mm}
\[ [E(XY)]^2 \leq E(X^2)E(Y^2) \vspace{-1mm} \]
with equality holds iff $P(X = cY ) =1$ for a constant $c$.
\vspace{1mm}

In fact, we also have \vspace{-1mm}
\[ [E|XY| ]^2 \leq E(X^2)E(Y^2)  \]
\end{alertblock}
}
\frame{\vspace{-2mm}
\begin{alertblock}{Proof of Cauchy-Schwartz's inequality}

\end{alertblock}\vspace{-2mm}
\begin{block}{Natural exponential family}<2->
If $\eta_i = w_i(\theta )$, $i=1,...,k$, and $\eta = (\eta_1,...,\eta_k)$, the form of $f_\theta$ in the exponential family becomes \vspace{-3mm}
\[ f^*_\eta (x) = h(x)c^*(\eta ) \exp \left( \sum_{i=1}^k \eta_i t_i(x) \right) \vspace{-3mm} \]
$\eta$ is called the natural parameter.

The set of $\eta$'s for which $f^*_\eta (x)$ is a well-defined pdf is called the natural parameter space.
\end{block}\vspace{-2mm}
\begin{alertblock}{Full or curved exponential families}<3->
In an exponential family, if the dimension of $\theta$ is $k$ (there is an open set $\subset \Theta$, then
the family is a full exponential family. Otherwise the family is a curved exponential family.
\end{alertblock}}
\frame{ \vspace{-2mm}
\begin{alertblock}{}
An example of a full exponential family is $N(\mu , \sigma^2)$, $\mu \in {\cal R}$, $\sigma >0$.

An example of a curved exponential family is $N(\mu, \mu^2)$, $\mu \in {\cal R}$.
\end{alertblock}\vspace{-2mm}
\begin{block}{How to show a family is not an exponential family}<2->
It may be difficult to show that a family is not an exponential family.
\vspace{1mm}

If $f_\theta$, $\theta \in \Theta$ is an exponential family, then \vspace{-2mm}
\[ \{ x: f_\theta (x) > 0 \} = \{ x: h(x) > 0 \} \vspace{-2mm} \]
which does not depend on $\theta$ values.
\vspace{1mm}

This fact can be used to show a family is non-exponential, i.e.,
if $\{ x: f_\theta (x)>0\}$ depends on $\theta$, then $f_\theta $, $\theta \in \Theta$, is not an exponential family.
\vspace{1mm}

Consider the family of two parameters exponential distributions with pdf's\vspace{-2mm}
\[ f_\theta (x) = \left\{ \begin{array}{ll}
\lambda^{-1} e^{-(x-\mu)/\lambda} & \quad x > \mu \vspace{1mm} \\
0 & \quad x \leq \mu \end{array} \right. \quad \mu \in {\cal R}, \  \lambda > 0 \vspace{-2mm} \]
It is not an exponential family because \vspace{-2mm}
\[ \{ x: f_\theta (x) > 0 \} = \{ x: x > \mu \}  \]
\end{block}
}
\frame{\vspace{-2mm}
\begin{block}{Definition 3.5.2 (location family)}
Let $f(x)$ be a given pdf. The family of pdf's, $f(x-\mu)$, $\mu \in {\cal R}$, is called a location family with location parameter $\mu$.
\end{block}\vspace{-2mm}
\begin{alertblock}{}
\begin{itemize}
\item
Examples of location families are normal,
double exponential, Cauchy, logistic, and two-parameter exponential distributions with location parameter $\mu \in {\cal R}$ and the other parameter $\sigma$ or $\lambda$ fixed.
\item
The pdf $f(x-\mu)$ is obtained by shifting the entire curve $f(x)$ by an amount $\mu$ (see the figure) without changing the structure of $f(x)$.
\item
It can be shown that $X \sim f(x- \mu)$ iff $X=Z+ \mu $ with $Z \sim f(x)$.
\end{itemize}
\end{alertblock}\vspace{-2mm}

}
\frame{\vspace{-3mm}
\begin{alertblock}{Definition 3.5.4 (scale family)}
Let $f(x)$ be a given pdf. The family of pdf's, $\sigma^{-1}f(x/\sigma )$, $\sigma >0$, is called a scale family with scale parameter $\sigma$.
\end{alertblock}\vspace{-3mm}
\begin{alertblock}{}
\begin{itemize}
\item
Examples of scale families are normal,
double exponential, Cauchy, and logistic distributions with scale parameter $\sigma >0$ and $\mu$ fixed,
$gamma (\alpha , \beta )$ with $\beta >0$ and $\alpha$ fixed, and
 two-parameter exponential distributions with $\lambda >0$ and $\mu$ fixed.
\item
The pdf $\sigma^{-1} f(x/\sigma )$ is obtained by stretching ($\sigma >1$) or contracting ($\sigma <1$) the curve $f(x)$ while still maintaining the same shape.
\item
It can be shown that $X \sim \sigma^{-1} f(x/\sigma )$ iff $X=\sigma Z $ with $Z \sim f(x)$.
\end{itemize}
\end{alertblock}\vspace{-3mm}
}
\frame{
\begin{block}{Definition 3.5.5 (location-scale family)}
Let $f(x)$ be a given pdf. The family of pdf's, $\sigma^{-1}f((x-\mu)/\sigma )$, $\mu \in {\cal R}$, $\sigma >0$, is called a location-scale family with location parameter $\mu$ and scale parameter $\sigma$.
\end{block}\vspace{-2mm}
\begin{alertblock}{}
\begin{itemize}
\item
A location-scale family is a combination of a location family and a scale family:
it contains a sub-family that is a location family with any fixed $\sigma$, and a sub-family that is a scale
family with any fixed $\mu$.
\end{itemize}
\end{alertblock}\vspace{-2mm}
}
\frame{
\begin{alertblock}{}
\begin{itemize}
\item
Examples of location-scale families are normal,
double exponential, Cauchy, logistic, and  two-parameter exponential  distributions with location parameter $\mu \in {\cal R}$ and
scale parameter $\sigma >0$. Except for the two-parameter exponential distribution, all others are symmetric about $\mu$.
\item
If $f(x)$ is symmetric about 0, then $\sigma^{-1}  f((x-\mu)/\sigma)$ is symmetric about $\mu$ and $\mu$ is the median of
$X \sim \sigma^{-1}  f((x-\mu)/\sigma)$; furthermore, if the expectation of $f(x)$ exists, then $\mu$ is the expectation of
$\sigma^{-1}  f((x-\mu)/\sigma)$.
\item
It can be shown that $X \sim \sigma^{-1} f((x- \mu)/\sigma) $ iff $X=\sigma Z + \mu$ with $Z \sim f(x)$; furthermore, if $E(Z^2)< \infty$, then
$E(X) = \sigma E(Z) + \mu$ and Var$(X ) = \sigma^2$Var$(Z)$.
\item
The pdf $f(x)$ in a location-scale family is standard iff the expectation $\int_{-\infty}^\infty xf(x)dx =0$ and the variance
$\int_{-\infty}^\infty x^2f(x) dx = 1$.
\item
Typically, we choose a standard $f(x)$ to generate a location-scale family, in which case $\mu$ and $\sigma^2$ are the expectation and variance
of  $\sigma^{-1}  f((x-\mu)/\sigma)$, respectively.
\end{itemize}
\end{alertblock}
}
\end{document}
