臺灣大學數學系

九十學年度第二學期碩博士班資格考試試題

 統計與機率

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1.
(20 points) The random variables $X_1,\ldots,X_n$ are independent and $X_i$ is normally distributed with mean $\theta_i$ and variance $1$.
(11) Show that the most-powerful size $0.05$ test of $H_0: \theta_i =0,\;1\leq i \leq n$ versus $H_1: \theta_i =1/2,\;1\leq i \leq r$ and $\theta_i =-1/2,\;r+1\leq i \leq n$ has critical region
\begin{displaymath}\left\{(x_1,\ldots,x_n): \sum_{i=1}^r x_i - \sum_{i=r+1}^n x_i > 1.645\sqrt{n}\right\}.\end{displaymath}

(12) How large must $n$ be to ensure that the power of this test is at least $0.9$?
2.
(20 points) Let $X_1,\ldots,X_n$ be a random sample from the pdf
\begin{displaymath}f(x\vert\theta) = \theta x^{-2},\;\;\;0 < \theta \leq x < \infty.\end{displaymath}

(21) (8 points) Find the maximum likelihood estimate (MLE) of θ.
(22) (12 points) Derive the asymptotic distribution the MLE of θ.
3.
(20 points) Importance sampling is a useful technique for calculating features of a distribution. Suppose we want to find $Eh(X)$ where the probability density function (pdf) of $Y_1,Y_2,\ldots,Y_m$ is $f$ and $h$ is a function. But the pdf $f$ is difficult to simulate from. Instead, generate $Y_1,Y_2,\ldots,Y_m$ from a pdf $g$ where the supports of $f$ and $g$ are the same and $Varh(X)< \infty$.
(31) Show that $E\left(m^{-1} \sum_{i=1}^m \frac{f(Y_i)}{g(Y_i)} h(Y_i)\right) = Eh(X)$.
(32) Show that $m^{-1} \sum_{i=1}^m \frac{f(Y_i)}{g(Y_i)}h(Y_i)$ converges to $Eh(X)$ in probability.
(33) Although the estimator of (31) has the correct expectation, in practice the estimator
\begin{displaymath}\sum_{i=1}^m \left(\frac{f(Y_i)/g(Y_i)}{\sum_{j=1}^m f(Y_j)/g(Y_j)}\right)h(Y_i)\end{displaymath}

is preferred. Show that this estimator converges in probability to $Eh(X)$. Moreover, show that this estimator is superior to the one in (31).
4.
(20 points) Let $Y_1,Y_2,\ldots,Y_m$ and $Y$ be independent exponential random variables, with $f(x\vert\lambda) = \lambda^{-1} \exp(-x/\lambda)$, $x>0$, and $f(y\vert\mu) = \mu^{-1} \exp(-y/\mu)$, $y>0$. Due to data collection problem, we can observe $(Z_i,W_i)$ only, $1 \leq i \leq n$,with $Z_i = \min(X_i,Y_i)$. Here $X_i$ and $Y_i$ are assumed to be independent and identically distributed and
\begin{displaymath}W_i = \left\{\begin{array}{ll} 1 & \mbox{if $Z_i =X_i$} \\ 0 & \mbox{if $Z_i=Y_i$}\\ \end{array} \right. .\end{displaymath}

Find the maximum likelihood estimate of λ.
5.
(20 points) Find the asymptotic distribution of $\hat{p}_n(1- \hat{p}_n)$ where $\hat{p}_n$ is the proportion of success of a binomial distribution with $n$ trials and the probability of success $p$. (Hint: You may consider $p=1/2$ and $p\not = 1/2$ separately.)


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