臺灣大學數學系

九十學年度第一學期碩博士班資格考試試題

 統計與機率

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1

A random sample, $X_1,\ldots,X_n$, is drawn from a Pareto population with pdf

$$f(x\vert\theta,\tau) = \frac{\theta \tau^{\theta}}{x^{\theta+1}} I_{ [\tau,\infty ]}(x), \;\;\;\theta > 0,\;\;\;\tau>0.$$

Here $I_{ [\tau,\infty ]}(x)$ is equal to $1$ if $x \geq \tau$; $0$, otherwise.

(11) Find the MLE of θ and τ.
Hint: (12) may help you to solve this problem.

(12) Show that the likelihood ratio test of

$$H_0: \theta = 1, \tau \;\;\mbox{unknown versus } H_1: \theta \not = 1,\tau \;\;\mbox{unknown},$$

has critical region of the form $\{(x_1,\ldots,x_n): T((x_1,\ldots,x_n)) \leq c_1\;\;\mbox{or}\;\; T((x_1,\ldots,x_n)) \geq c_2\}$, where $0 < c_1 < c_2$ and

$$T = \log\left[\left(\prod_{i=1}^n X_i\right)/\left(\min_i X_i \right)^n \right].$$

2

Suppose that we have two independent random samples: $X_1,\ldots,X_{100}$ are exponential(θ), and $Y_1,\ldots,Y_{200}$ are exponential(μ). (i.e. The probability density function of $X_1$ is $\theta^{-1} \exp(-x/\theta)$.) Statistician A is asked to perform the test of

$$H_0: \theta = \mu\;\;\mbox{versus}\;\;H_1: \theta \not = \mu.$$

Since he only have a standard normal table, he proposes the following $0.95$ level test with critical region

$$\left\vert T- \frac{1}{3}\right\vert > \frac{1}{10} \frac{\sqrt{5}}{9\theta^{1/2}} z_{0.025}.$$

Here

$$T = \frac{\sum_{i=1}^{100} X_i}{\sum_{i=1}^{100} X_i + \sum_{i=1}^{200} Y_i}.$$

Do you think that it is a reasonable proposal? If your answer is YES, give a theoretical justification. Otherwise, propose an alternative and give a theoretical justification.

3

Let $X_1,\ldots,X_n$ be independent and identically distributed random variables with one of two probability density functions. If $\theta = 0$ then

$$f(x\vert\theta) = \left\{\begin{array}{ll} 1 & \mbox{if $0 < x < 1$} \\ 0 & \mbox{otherwise} \end{array} \right. ,$$

while if $\theta = 1$

$$f(x\vert\theta) = \left\{\begin{array}{ll} 1/(2\sqrt{x}) & \... ... $0 < x < 1$} \\ 0 & \mbox{otherwise} \end{array} \right. .$$

Find the MLE of θ and show that it is consistent.

4

For two factors-starchy or sugary and green base leaf of white base leaf- the following counts for the progeny of self-fertilized heterozygotes were observed:

$$\begin{tabular}{cc} \hline \mbox{Type} & \mbox{Count} \\ \hl... ... green} & 904 \\ \mbox{Sugary white} & 32 \\ \end{tabular}$$

According to genetic theory, the cell probabilities are $0.25(2+\theta)$, $0.25(1-\theta)$, $0.25(1-\theta)$, and $0.25 \theta$, where θ ( $0 < \theta < 1$) is a parameter related to the linkage of the factors.

(41) Find the mle of θ and its asymptotic variance.

(42) Form an approximate $95\%$ confidence interval for θ based on part (41).

5

Let $X_1,\ldots,X_n$ be a random sample from a $N(0,1)$ population. Define

$$Y_1 = \left\vert\frac{1}{n}\sum_{i=1}^n X_i\right\vert,\;\;\; Y_2 = \frac{1}{n} \sum_{i=1}^n \vert X_i\vert.$$

　

Calculate $EY_1$ and $EY_2$.

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