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\begin_body

\begin_layout Title
Problem Set 7
\end_layout

\begin_layout Author
Daryl Larsen
\end_layout

\begin_layout Section
Lecture note
\end_layout

\begin_layout Standard
\begin_inset Formula $(\implies)$
\end_inset

 Suppose not, so 
\begin_inset Formula $Z=g(X_{i},\theta_{0})-g(X_{i},\theta)=0$
\end_inset

.
 Then 
\begin_inset Formula $Z^{2}=0$
\end_inset

 and 
\begin_inset Formula $EZ^{2}=0$
\end_inset

, which is a contradiction.
\end_layout

\begin_layout Standard
\begin_inset Formula $(\impliedby)$
\end_inset

 Suppose that 
\begin_inset Formula $P(Z\neq0)=P(Z^{2}>0)$
\end_inset

.
 Then 
\begin_inset Formula $\exists n\in\mathbb{N}\text{ s.t. }P(Z^{2}\geq n^{-1})>0$
\end_inset

 so that 
\begin_inset Formula 
\[
EZ^{2}\geq EZ^{2}\mathds{1}\{Z^{2}\geq n^{-1}\}\geq n^{-1}P(Z^{2}\geq n^{-1})>0
\]

\end_inset


\end_layout

\begin_layout Section
Lemma 11.1
\end_layout

\begin_layout Standard
Lemma 11.1 is that 
\begin_inset Formula $\Theta$
\end_inset

 compact, 
\begin_inset Formula $Q(\theta)$
\end_inset

 continuous, and 
\begin_inset Formula $\theta_{0}$
\end_inset

 uniquely minimizes 
\begin_inset Formula $Q(\theta)$
\end_inset

 over 
\begin_inset Formula $\theta\in\Theta$
\end_inset

 implies 
\begin_inset Formula $\forall\epsilon<0,\inf_{\theta\notin B(\theta_{0},\epsilon)}Q(\theta)>Q(\theta_{0})$
\end_inset

.
 Suppose not, so 
\begin_inset Formula $\exists\epsilon>0\text{ s.t. }\inf_{\theta\notin B(\theta_{0},\epsilon)}Q(\theta)\leq Q(\theta_{0})$
\end_inset

.
 Since 
\begin_inset Formula $\Theta$
\end_inset

 is compact, 
\begin_inset Formula $\Theta\backslash$
\end_inset


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