1.最大似然估计：

xa+1 ; x > c; 0 否则。假设参数 a > 0 是未知的， c > 0 是一个

(a) 找出 a 的最大似然估计并展示您的完整工作原理。
(b) 找到 ^ aML 的观察到的 Fisher 信息和渐近方差。
(c) ^ aML 的渐近分布是什么？
(d) 举一个最大似然估计失效的例子，不应该使用和

2. 最大似然检验：

(a) 写下参数 的对数似然函数和得分函数。
(b) 证明参数的 MLE 是 ^ ML = 1
×。
(c) 找出 ^ ML 的观察到的 Fisher 信息和 ^ ML 的渐近方差。
(d) 重写对数似然函数和得分函数，以便仅表示数据 xi

(e) 求出检验 = 0 的 Wald 统计量 t( 0) 并用 0 和 ^ ML 而不是

(f) 找出检验的 Wilks 似然比统计量 W( 0) = 0（再次表示为 0

(g) 解释和证明 W(0) 和 t(0) 之间的联系。提示：使用泰勒级数开发-

1. Maximum likelihood estimation:
Let x1; : : : ; xn be i.i.d. observations from a Pareto(a,c) distribution, which has probability density
function f(x) = aca
xa+1 ; x > c; 0 otherwise. Suppose that the parameter a > 0 is unknown and c > 0 is a
known constant.
(a) Find the maximum likelihood estimate of a and show your complete workings.
(b) Find the observed Fisher information for ^ aML and the asymptotic variance.
(c) What is the asymptotic distribution of ^ aML?
(d) Give an example where maximum likelihood estimation breaks down and should NOT be used and
explain why.

2. Maximum likelihood testing:
Let x1; x2; : : : ; xn be a random sample from the exponential distribution with parameter  > 0 and
probability density function f(x) = e x. The support is x > 0.
(a) Write down the log-likelihood function and the score function for the parameter .
(b) Show that the MLE for the parameter  is ^ ML = 1
 x .
(c) Find the observed Fisher information for ^ ML and the asymptotic variance of ^ ML.
(d) Rewrite the log-likelihood function and the score function so that the data xi is represented only
via ^ ML (so that the resulting functions do not contain xi).
(e) Find the Wald statistic t(0) for testing  = 0 and express it in terms of 0 and ^ ML rather than
xi.
(f) Find the Wilks likelihood ratio statistic W(0) for testing  = 0 (again expressed in terms of 0
and ^ ML).
(g) Explain and demonstrate the connection between W(0) and t(0). Hint: Use Taylor series devel-
opment of the logarithm log(x)  log(x0) + x x0