hi; i kindof got in trouble by taking a wrong class. i have been constantly studying limiting distributions for three days but i have stuck in some questions.
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suppose that X_1,X_2...,X_n be a random sample with Uniform(θ-1/n,θ+1/n) pdf f(x) = n/2, θ-1/n <= x <= θ+1/n
let X(n)=(X_1,X_2,..X_n) . Find the limiting distribution of Z(n)=n-n^2(X(n) - θ)
-> here is my crap thinking; i have found CDF F(x) =n(x-θ)+1)/2
then order statistics Xn:n = [F(X(n)]^n = [n(x-θ)+1)/2]^n
then i put these into Z_n part and took limit as n goes infinity
limit[n - n^2 *(((n(x-θ)+1)/2)^n -θ)], as n -> infinity, Assumptions -> θ-1/n<x<θ+1/n
i have stuck here. i am also trying to figure out how to use wolfram.. if you know a way for me to make it compute this equation.. i will also be pleased.
thanks in advance for your all help