作者idphobia5566 (idphobia5566)
看板Statistics
标题[分析] converge in probability
时间Sun Feb 13 22:08:13 2011
※ [本文转录自 Math 看板 #1DL-N5M2 ]
作者: idphobia5566 (idphobia5566) 看板: Math
标题: [分析] converge in probability
时间: Sun Feb 13 22:08:02 2011
iid
X1,...,Xn ~ UNIF(a,b) , a<b , let X(n) be the largest order statistic.
(a)What does exp[X(n)] converge in probability? Show your work.
(b)Find the limiting distribution of exp[-n(b-X(n))/(b-a)] . Show your work.
最大顺序统计量之pdf: f_x:n(x) = n(x-a)^n-1/(b-a)^n , a<x<b
想法是
let y = exp(x) , 找出y=exp[X(n)]的pdf，然後用mgf法取极限来求
但这样不知道怎麽积分...因为式子很难看
f(y)= n{[ln(y)-a]^(n-1)}/[y*(b-a)^n] , exp(a) < y < exp(b)
M_y(t)=E[exp(ty)] 超级难积分
请问有没有更好的方法可以做，谢谢
--
※ 发信站: 批踢踢实业坊(ptt.cc)
◆ From: 123.193.73.183
--
※ 发信站: 批踢踢实业坊(ptt.cc)
◆ From: 123.193.73.183
1F：推 FreakyTune:第一题因为X(n)为b的MLE 用continous mapping thm可以 02/14 18:51
2F：→ FreakyTune:可以知道exp(X(n))机率收敛到exp(b) 02/14 18:52
3F：→ FreakyTune:第二题应该是delta method 02/14 18:53
4F：→ idphobia5566:对齁MLE的不变性...谢谢 02/14 21:05