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\begin{center}
 {\Large Midterm 2}\\
28 March  2007, 2:15
\end{center}


\medskip
\medskip

\begin{enumerate}

\medskip
\item
Find conditions on
$f(\bxx,\theta )$ such that
\begin{displaymath}
I (\theta) = - E \left ( \frac{\partial^2}{\partial
\theta^2} \log f (\bx, \theta ) \right )
\end{displaymath}
holds.\\


\item
Let $f$ and $g
$ be two densities. Show that
\begin{displaymath}
\int_{-\infty
}^{\infty
}f(x)\log f(x)dx \geq \int_{-\infty
}^{\infty} f(x)\log g(x)dx
,
\end{displaymath}
assuming that both integrals exist. When do we have equality?\\


\item
Let $X_1, X_2, \ldots , X_n$ be i.i.d.r.v.'s, uniform on
$[0, \theta]\; (\theta > 0).$\\
(a) Find the maximum likelihood estimator for
     $\theta$.\\
(b) Get the asymptotic distribution of the maximum
     likelihood estimator.\\


\item
Let $X_1, X_2, \ldots, X_n$ be independent identically distributed random variables  with $P\{X_1 =
1\} = p$ and $P\{X_1 = 0\} = 1-p$.  We wish to test
$H_0: \; p=p_0$ against $H_A:\; p \neq p_0$.  Use the likelihood
ratio test and get its asymptotic distribution directly.\\



\item
Let $X_1, X_2, \ldots, X_n$ be independent identically distributed random variables  uniform on $[0,\theta]$.  We wish to test
$H_0: \; \theta=\theta_0$ against $H_A:\; \theta \neq \theta_0$.  Use the likelihood
ratio test and get its asymptotic distribution.\\


\item
$X_1, X_2, \ldots, X_n$ be independent identically distributed $N(\mu,\sigma^{2
})
$ random variables. We wish to test $H_{0
}:\;\mu=\mu_{0
}
$ against the alternative that $H_{A
}:\;\mu\neq\mu_{0
}
$. Show that the t--test and the likelihood ratio test are equivalent.\\


\item
$X_1, X_2, \ldots, X_n$ be independent identically distributed $N(\mu,\sigma^{2
})
$ random variables. We wish to test $H_{0 }:\mu\leq \mu_{0 }
$
against the alternative $H_{A}:\mu> \mu_{0 }
$. Find a test using the likelihood ratio.\\


\item
$X_1, X_2, \ldots, X_n$ be independent random variables. We assume that $X_{i
}
$
is an exponential$(\theta_{i
}
$ random variable. We wish to test $\theta_{1
}=\theta_{2
}=\ldots =\theta_{n
}
$ against the alternative that the null is not true.\\
(a) Find a test using the likelihood ratio.\\
(b)  Find the asymptotic distribution of the likelihood ratio test.\\

\item
Let
$X_1, X_2, \ldots, X_n$ be independent identically distributed $N(\mu_{1
},\sigma^{2
})
$ random variables. Let $Y_1, Y_2, \ldots, Y_n$ be independent identically distributed $N(\mu_{2
},\sigma^{2
})
$. We wish to test $H_{0
}:\;\;\mu_{1
}=\mu_{2
}
$ against the alternative  $H_{A
}:\;\;\mu_{1
}\neq\mu_{2
}
$. Show that the likelihood ratio test is equivalent with the two sample t--test.\\


\item
Let
$X_1, X_2, \ldots, X_n$ be independent identically distributed $N(\mu_{1
},\sigma^{2
}_{1
})
$ random variables. Let $Y_1, Y_2, \ldots, Y_n$ be independent identically distributed $N(\mu_{2
},\sigma^{2
}_{2
})
$. We wish to test $H_{0
}:\;\mu_{1
}=\mu_{2
}
$ against the alternative  $H_{A
}:\;\;\mu_{1
}\neq\mu_{2
}
$. Find a test using the generalized likelihood.\\

\item
We wish to test the null hypothesis that $\theta=\theta_0$ against the alternative that the null is not true . We reject the null if
$$
\frac{|\ell'_n(\theta_0)|}{\sqrt{-\ell''_n(\theta_0)}}\;\; \mbox{is large}.
$$
Determine the asymptotic rejection region.\\


\item
Let  $X_{1 }, X_{2 }, \dots $ be independent identically distributed normal $N(\mu,\sigma^{2 })
$ random variables. We wish to have an $1-\alpha
$ confidence interval for $\mu
$ of length not greater than $d
$. Use sequential method to achieve this goal.
\end{enumerate}
\end{document}
