[機統] Hogg 6th Edition Exercise 4.3.3

[yueayase (scrya)]

Question: Let Yn denote the maximum of a random sample from a distribution

          of the continuous type that has cdf F(x) and pdf f(x) = F'(x).

          Find the limiting distribution of Zn = n[1-F(Yn)].



過程:

(1) Find the cdf of Yn:

    F  (t) = P(Yn=max{X1,...,Xn} ≦ t)
     Yn
           = P(X1≦t, ... , Xn≦t)

           = P(X1≦t)...P(Xn≦t)
                   n
           = [F(t)]


(2) Find the cdf of Zn = n[1-F(Yn)]


    F  (t) = P(Zn≦t) = P(F(Yn)≧1-t/n)
     Zn
                               -1
                      = P(Yn≧F  (1-t/n))

                                    -1
                      = 1 - P(Yn < F  (1-t/n))

                                -1         n
                      = 1 - [F(F  (1-t/n))]
                                   n
                      = 1 - (1-t/n)
                             -t
                      -> 1- e   , n->∞


    Zn -> Z ~ exp(1)


但我覺得這樣做有瑕疵, 因為並沒有說F(x)的反函數存在,

雖然F(x)一定是nondecreasing, 所以取反函數不等式不會改變方向

所以就不知道反函數不存在怎麼做



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※ 發信站: 批踢踢實業坊(ptt.cc) 
◆ From: 111.251.185.27
→ yhliu       :不需反函數存在, F(Xi) i.i.d. U(0,1).
→ yhliu       :F(Yn) = max{F(X1),...,F(Xn)}
→ yhliu       :關於 F(X) 的轉換的分布, 參見:
→ yhliu       :telnet://bs2.twbbs.org之Statistics精華區→9→4→3
→ yhliu       :或 telnet://bbs.ncku.edu.tw 之 Statistics版精華區
→ yhliu       :(按z進入) 10→2→6 或 11→5→3.
→ yueayase    :OK, 考慮inverse image做夾擠, 好方法