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\def\coursename{Statistics 330b/600b, Math 330b spring 2013}
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\HW{9}{Thursday 11 April}
{\it Please attempt at least the starred problems.}
\starproblem[Neyman.fact]
(Neyman factorization theorem cf.\ UGMTP Example~5.31)
Suppose $\PP$ and $\PP_\th$, for $\th\in\Th$, are probability measures defined on a sigma-field~$\ff$, for some index set~$\Theta$.
Suppose also that~$\gg$ is a sub-sigma-field of~$\ff$ and that there exist versions of densities
\beqN
\frac{d\PP_\th}{d\PP} = g_\th(\om) h(\om)
\qt{with $g_\th\in\mm^+(\Om,\gg)$ for each~$\th$}
\eeqN
for a fixed $h\in\mm^+(\Om,\ff)$ that doesn't depend on~$\theta$.
Define $H$ to be a version of~$\PP_\gg h$. [That is, choose one $H$ from the $\PP$-equivalence class of possibilities.]
\ppart
Show that
$
\PP_\th\{H\in B\}= \PP g_\th(\om)\{H\in B\}H$ for each~$\th$ and each~$B\in\bb(\RRbar)$.
\ppart
Deduce that
$
\PP_\th\{H=0\}=0=\PP_\th\{H=\infty\}$ for each~$\th$.
\ppart
For each $X$ in~$\mm^+(\Om,\ff)$ and some fixed choice of~$\PP_\gg(Xh)$ define
\beqN
Y(\om) = \dfrac{\PP_\gg(Xh)}{H}\{00$
with
$\a_1+\a_2=1$. Let
$X$ denote the coordinate map from $\xx\times \yy$ onto the~$\xx$ space. Show that the conditional
probability distribution $P_x=P(\cdot\mid X=x)$ concentrates on the points $(x,g_1(x))$
and~$(x,g_2(x))$. Find the conditional probabilities assigned to each point.
Hint: Consider the density of~$\mu_i$ \wrt/~$\a_1\mu_1+\a_2\mu_2$.\index{Bayes theory}
\problem[gg.measurable]
Suppose $X\in\ll^1(\Omega,\ff,\PP)$ and $\gg$ is a sub-$\sig$-field of $\ff$ containing all
$\PP$-negligible sets. Show that $X$ is $\gg$-measurable if and only if $\PP(XW)=0$ for
every bounded random variable~$W$ with $\PP_\gg W=0$ almost
surely by the following steps.
\Remark
Compare with the corresponding statement for random variables that
are square integrable: $Z\in\ll^2(\gg)$ if and only if it is orthogonal to every
square integrable~$W$ that is orthogonal to~$\ll^2(\gg)$.
\endRemark
\ppart
Suppose~$t$ is a real number~$t$ for which $\PP\{X=t\}=0$. Define $Z_t=
\PP_\gg\{X>t\}$ and $W_t=\{X>t\}-Z_t$.
Show that $(X-t)W_t\ge0$ almost surely.
\ppart
Explain why
$\PP\left((X-t)W_t\right)=0$. Deduce that $(X-t)W_t=0$ almost surely.
\ppart
Deduce that $\{X>t\}\in\gg$ for every~$t$ with $\PP\{X=t\}=0$. Conclude
that $X$ is $\gg$-measurable.
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