\input{../courseDefs}

\begin{center}
 Second Set of Notes
\end{center}
\section{More on Testing and Confidence Sets}
See Lehmann, TSH, Ch. 3, 4, 5.
\begin{Definition}
$S(X)$ is a confidence level $1-\alpha$ confidence set for the parameter
$\tau(\Prob_\theta)$ if
\beq
\Prob_\nu \{ S(X) \ni \tau(\nu) \} \ge 1-\alpha \;\; \forall \nu \in \Theta .
\eeq
\end{Definition}

\begin{Definition}Accuracy of Confidence Sets.
The accuracy at $\tau(\gamma)$ 
of the confidence set $S(X)$ for the parameter $\tau( \cdot )$ is
\beq \label{eq:Saccuracy}
\Prob_\nu \{S(X) \ni \tau(\Prob_\gamma)\}, \tau(\nu) \ne \tau(\gamma).
\eeq
That is, it is the probability that the confidence set contains
$\tau(\gamma)$, when that is not the true value of $\tau$.
\end{Definition}

The accuracy is the same as the ``false coverage'' probability that appeared on the
right hand side of the Ghosh-Pratt identity.
Typically, when there are nuisance parameters, there is no confidence set
$S(X)$ that minimizes \ref{eq:Saccuracy} for all $\gamma$ such that $\tau(\nu)
\ne \tau(\gamma)$ (there is no {\em uniformly most accurate} confidence set).
However, if one restricts the class of confidence sets in ways that are sometimes
reasonable, there are then optimal sets in the restricted class.

\begin{Definition}
A $1-\alpha$ confidence set $S(X)$ for the parameter 
$\tau(\theta)$ is {\em unbiased} if for all $\nu$,
\beq
\Prob_\nu \{ S(X) \ni \tau(\gamma) \} \le 1-\alpha \;\; \forall \gamma \mbox{ s.t. }
\tau(\nu) \ne \tau(\gamma).
\eeq
\end{Definition}

That is, a confidence set is unbiased if
the probability of covering a false value of $\tau$ is smaller than the
confidence level $1-\alpha$.
In some sense, a biased confidence set treats some parameter values $\tau$ specially.
For example, a biased 95\% confidence set for the mean $\theta$ of a unit variance normal is
$\{0\} \cup [X - 1.96, X+1.96 ]$. The coverage probability is 95\% for all $\theta \in \bfR$
except $\theta = 0$, which is in the set with probability one.

The analogous property of a test is that there is no alternative value of $\tau$
for which the probability of rejection is less than the level $\alpha$ of the test:
 
\begin{Definition}
A level $\alpha$ test $\delta$ of $H$ against the alternative $K$ is
{\em unbiased} if  
\begin{eqnarray}
\beta_\delta (\Prob_\nu) \le \alpha \;\;&& \forall \nu \in H \\
\mbox{and }
\beta_\delta (\Prob\nu ) \ge \alpha \;\; &&\forall \nu \in K .
\end{eqnarray}
If the second inequality is strict ($>$), the test is {\em strictly
unbiased}.
\end{Definition}

There exist UMP unbiased tests in many problems for which there is no UMP test,
in particular, when $\tau(\nu) \ne \nu$ (when there are nuisance parameters on which
the distribution of $X$ depends, but which are irrelevant to the truth of the hypothesis).

\begin{Lemma} (Lehmann, TSH, Lemma 4.1.1)
Suppose $\Theta \subset \bfR$, and let $\Omega$ be the common boundary of $\{ \nu : \Prob_\nu
\in H\}$ and $\{ \nu : \Prob_\nu \in K \}$.
If $\Prob_\Theta$ is such that
the power function $\beta_\delta(\Prob_\gamma)$ is a continuous function of
$\gamma$ for every $\delta$, and if $\delta_0$ is UMP among all level $\alpha$ tests
such that 
\beq
\beta_{\delta}(\gamma) = \alpha \;\; \forall \gamma \in \Omega ,
\eeq
then 
$\delta_0$ is UMP unbiased.
\end{Lemma}
This lemma reduces questions about UMP unbiased tests to their behavior on the
boundary between the null and alternative, provided the level is a continuous function
of the parameter.

In one-parameter exponential families, we saw that there exist UMP tests of one-sided
hypotheses about the parameter.
For two-sided hypotheses $H: \gamma_1 \le \theta \le \gamma_2$ versus 
$K: \theta < \gamma_1$ or $\theta > \gamma_2$, there exist
UMP unbiased tests; the form of their decision functions is (Lehmann, TSH, 4.2)
\beq \label{eq:oneParamExpUnbiased}
\phi(x) = \left \{ \begin{array}{ll}
1, & T(x) < c_1, \; T(x) > c_2 \\
a_i & T(x) = c_i, \; i = 1, 2 \\
0 & c_1 < T(x) < c_2),
\end{array}
\right .
\eeq
with $c_1$ and $c_2$ chosen s.t. 
\beq \label{eq:oneParamExpConstr}
E_{\Prob_{\gamma_1}} \phi(X) = E_{\Prob_{\gamma_1}} \phi(X) = \alpha .
\eeq


\noindent
{\bf Example.} (Lehmann, TSH, \S 4.2)
Suppose $\Prob_\Theta$, $\Theta = [0, 1]$ is distributions of the number of
successes $X$ in a sequence of a fixed number $n$ of independent trials,
each with probability $\theta$ of success. ({\em I.e.}, $X \sim \mbox{Bin}(n, \theta)$.)
This is a one-parameter exponential family.
Consider the null hypothesis $H: \Prob_\theta = \mbox{Bin}(n, \gamma)$, for fixed
$\gamma \in [0, 1]$. The null hypothesis is of the form \ref{eq:oneParamExpUnbiased}
with $\gamma_1 = \gamma_2 = \gamma$, and $T(x) = x$,
so the constraint \ref{eq:oneParamExpConstr}
reduces to
\beq
E_{\Prob_\gamma} \phi(X) = 
= 1 - \sum_{x = c_1 + 1}^{c_2 - 1} {}_nC_x \gamma^x (1-\gamma)^{n-x} -
a_1 {}_nC_{c_1} \gamma^{c_1} (1-\gamma)^{n-c_1} - 
a_2 {}_nC_{c_2} \gamma^{c_2} (1-\gamma)^{n-c_2}
= 1- \alpha 
\eeq
Note again that in practice, typically one would choose $\alpha$ so that $a_1 = a_2 = 0$,
rather than use a randomized test.

\begin{Definition}
A family of confidence level
$1-\alpha$ confidence sets $S(X)$ for $\tau(\theta)$ 
is {\em uniformly most accurate unbiased}
if for all $\nu \in \Theta$, it minimizes the probabilities
\beq
\Prob_\nu \{ S(X) \ni \tau(\gamma) \} \;\; \forall \gamma \mbox{ s.t. } 
\tau(\gamma) \ne \tau(\nu).
\eeq
\end{Definition}

Confidence sets derived by inverting uniformly most powerful unbiased tests are
uniformly most accurate unbiased.

\section{Equivariant and Invariant Procedures}
\begin{Definition}
A {\em group} is a set $\calG$ and an operation $\circ : \calG \times \calG \rightarrow \calG$
that satisfies for every $g$, $h$, $k$ in $\calG$
\begin{enumerate}
\item 
$g\circ(h\circ k) = (g\circ h)\circ k$ (associativity)
\item
There exists a unique $e \in \calG$ such that $e\circ g = g \circ e = g$ for 
every $g \in \calG$
(existence of an identity element)
\item
For each $g \in \calG$, there exists $g^{-1} \in \calG$ s.t. 
$g\circ g^{-1} = g^{-1}\circ g = e$.
\end{enumerate}
\end{Definition}

Typically, the dot in the notation will be omitted, so we shall write $gh$ (multiplication)
in place of
$g \circ h$.
Another symbol commonly used for the group operation is $+$.
Also, while formally a group is the pair $(\calG, \circ)$,
$\calG$ is commonly referred to as the group, with the group operation $\circ$ understood
from context.

\begin{Definition}
A {\em group of transformations on the set $\calX$} is a collection $\calG$ of transformations
$g: \calX \rightarrow \calX$ and an operation $\circ: \calG\times\calG \rightarrow \calG$
such that $(\calG, \circ)$ is a group.
\end{Definition}

For example, let $\calX = \bfRn$, and for each $\gamma \in \bfRn$, let 
\begin{eqnarray}
g_\gamma : &&\calX \rightarrow \calX \nonumber \\
&& \nu \rightarrow \nu + \gamma \mbox{ (vector addition in $\bfRn$)}
\end{eqnarray}
Let $+$ denote the group operation, and define $g_\gamma + g_\eta = g_{\gamma+\eta}$.
With these definitions, $(\calG = \{g_\gamma: \gamma \in \bfRn\}, +)$ is a group of transformations on
$\calX$.
The identity element of the group is $g_0$, the inverse of $g_\gamma$ is $g_{-\gamma}$,
and associativity of the group operation $+$ follows from the associativity of
vector addition on $\bfRn$. This group is called the {\em translation group
on $\bfRn$}.

\begin{Definition}
Invariance of decision procedures.
A statistical decision problem (a set $\Prob_\Theta$ of distributions on an outcome
space $\calX$, a loss function $L$, and a set of possible decisions $\calD$)
is {\em invariant} under the group
$\calG$ of transformations of the outcome space $\calX$ if 
\begin{enumerate}
\item
The family $\Prob_\Theta$ is closed under $\calG$, in the sense that
for any $\gamma \in \Theta$ and any $g \in \calG$, there exists $\nu \in \Theta$
such that if $X \sim \Prob_\gamma$, $gX \sim \Prob_\nu$, with $\nu \in \Theta$, 
and the mapping $\bar{g}: \Theta \rightarrow \Theta$, $\gamma \mapsto \nu$ 
is one-to-one and onto ($\bar{g} \Theta = \Theta$).
\item
For each $g \in \calG$, there is a transformation $h(g) : \calD \rightarrow \calD$
such that $h(g_1 g_2) = h(g_1)h(g_2)$, and 
$L(\bar{g}\gamma, h(g)d) = L(\gamma, d)$ for all $\gamma$, $g$, and $d$.
\end{enumerate}
\end{Definition}

\noindent
{\bf Example.}
Suppose $\Theta = \bfRm$, $\Prob_\Theta = \{N(\theta, I): \theta \in \Theta\}$ is the set
multivariate normal distributions with independent, unit variance components, 
$\tau(\theta) = \theta$, $\calD = \bfRm$,
and that $L(\theta, \gamma) = \|\theta - \gamma \|^2$.
Let $\calG$ be the translation group on $\bfRm$.
Clearly, the set $\Prob_\Theta$ is closed under $\calG$:
if $X \sim \Prob_\theta$, then $g_\gamma X \sim \Prob_{\theta + \gamma}$.
The set of transformations on $\Prob_\Theta$ induced by the action of $\calG$ on
$\calX$ is $\calGbar$, whose elements $\bar{g}(g_\gamma) = \bar{g}_\gamma$ map
$\Prob_\nu$ to $\Prob_{\nu+\gamma}$. ($\bar{\calG}$ is also a group, with the group
operation defined by $\bar{g}_\gamma + \bar{g}_\nu = \bar{g}(g_{\gamma+\nu})$).
If we define $h(g_\nu): \calD \rightarrow \calD$, $\gamma \mapsto \gamma+\nu$,
then $h(g_\nu g_\eta)(\gamma) = h(g_\nu)h(g_\eta)\gamma$, 
and $L(\bar{g}_\nu \theta, h(g_\nu)\gamma) = L(\theta, \gamma)$,
as required by the conditions of equivariance.


When the decision problem is invariant under $\calG$, it is reasonable to
consider only {\em invariant} decision rules $\delta : \calX \rightarrow \calD$, for which 
$\delta(gx) = h(g)\delta(x)$.
Lehmann (TSH, 1.5) draws a distinction between {\em invariant} and {\em equivariant} decision
rules: for the former, $h(g)d = d$ for all $d$, while for the latter, $\delta(gx)$ varies
with $g$. 
In the invariant case, the decision problem is unchanged under $X \mapsto gX$.

\begin{Definition}
Suppose we are testing $H$ against $K$. Given a transformation $g$ on $\calX$, if
$\{\Prob_{\bar{g}\nu} : \Prob_\nu \in H\} = H$ and 
$\{\Prob_{\bar{g}\nu}: \Prob_\nu \in K\} = K$,
we say {\em the problem of testing $H$ against $K$
is invariant under the transformation $g$}. 
\end{Definition}

The set of transformations under which a testing problem is invariant is always a group,
with the group operation defined in the natural way; the induced set $\calGbar$ of
transformations on $\Prob_\Theta$ also form a group ($\calGbar$ is a homomorphism of
$\calG$).

\begin{Definition}
A function $M: \calX \rightarrow \calY$ is
maximal invariant with respect to the $\calG$ of transformations
on a set $\calX$ if
\begin{enumerate}
\item
$M(x) = M(gx)$ for all $g \in \calG$ ($M$ is invariant under $\calG$)
and
\item
$M(x) = M(y) \;\Rightarrow\; x = gy$ for some $g \in \calG$.
\end{enumerate}
\end{Definition}

A test is invariant if and only if it depends on $x$ only through a maximal invariant
$M(x)$.
For example, suppose the observation $X$ is an iid sample of size $n$ from some 
common distribution.
The distribution of $X$ is clearly invariant under permutations of its components,
so we could
work instead with the set of order statistics without losing any information
about $\theta$.
The order statistics are in fact a maximal invariant of the permutation group, so 
invariant tests 
regarding $\theta$ need depend on $X$ only through its order statistics.
Any test that treated the components of $X$ differently would have an {\em ad hoc}
flavor.
{\em Ceteris paribus}, it makes sense to base tests on a maximal invariant function of
the data $X$ (with respect to $\calG$) if the testing problem is invariant under $\calG$.

Subject to some measurability considerations, the set
of all invariant tests is characterized by the set of all decision functions
$\phi(x) = h(M(x))$ where
$M(x)$ is a maximal invariant.
Basing tests on sufficient statistics reduces the outcome space $\calX$.
Invariant tests reduce not only the outcome space $\calX$, but also the parameter
space $\Theta$:

\begin{Theorem}
(See Lehmann, TSH, 6.3 Th.3.)
If $M(x)$ is maximal invariant with respect to $\calG$ and if $\upsilon(\theta)$ is
maximal invariant with respect to the induced group $\calGbar$, then the
distribution of $M(X)$ depends on $\theta$ only through $\upsilon(\theta)$.
\end{Theorem}

The utility of this theorem results from the fact 
that $M(x)$ and $\upsilon(\theta)$ can turn out
to be real-valued, even when the dimensions of $\calX$ and $\Theta$ are large, 
with the distribution of $M(X)$ having
monotone likelihood ratio in $\upsilon(\theta)$.
That makes it possible to find UMP invariant tests using the one-dimensional
optimality theory we saw earlier.

\begin{Definition}
A test with decision function $\phi$ is {\em almost invariant
with respect to $\calG$} if for all $g \in \calG$,
$\phi(gx) = \phi(x)$ except on a $\Prob_\Theta$-null
set $\calN_g$ that can depend on $g$.
\end{Definition}

\noindent
{\bf Remark.}
The power function of a test that is almost invariant under $\calG$ is invariant under
the induced group $\calGbar$.
The converse is not true in general.

\noindent
{\bf Remark.}
Unbiasedness and invariance are not equivalent in general, in that UMP unbiased
tests can exist when UMP almost-invariant ones do not, and {\em vice versa.} However,
Lehmann (6.6, Th. 7) shows that if in a given testing problem,
there exists a UMP unbiased test with decision function $\phi^*$ that is
unique up to sets of measure zero, and there also exists
a UMP almost-invariant test w.r.t. some group $\calG$, then the UMP almost-invariant
test is also unique up to sets of measure zero, and the two tests are the same
a.e.



\end{document}
