[商管] [統計]-政大98-風管所精算組統計第二題
題目：
Let Yn denote the nth order statistic of a random sample from a distribution
of the continuous type that has distribution function F(x) and p.d.f
f(x) = F'(x) . Find the limiting distribution of Zn = n[1-F(Yn)].
我的解法如下：
F(Z) = P[Zn <= z] = P(n[1-F(Yn)] <= z) = P(F(Yn) >= 1-z/n) = P(Yn >= G(1-z/n))
= P(max(x1,...,xn) >= G(1-z/n)) = 1 - P(max(x1,...,xn) < G(1-z/n))
= 1 - ( P(x1 < G(1-z/n)) )^n = 1- [F(G(1-z/n))]^n = 1 - (1-z/n)^n
∴ f(z) = F'(Z) = (1 - z/n)^(n-1)
=> lim f(z) = exp{-z}
n→∞
註：G為F的反函數
這樣寫對嗎？因為我看高點的解答答案是n，但我又看不出我哪裡寫錯 >"<
請高手指點一下
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