→ yhliu :一個疑問: 在 tail-index a 的條件下, 如何證明 09/18 22:30
→ yhliu :lim x^{a+1} f(x) 存在? 09/18 22:30
※ 引述《LSCAT (秘密客)》之銘言：
: Suppose that X is a continuous random variable with density f and cdf F.
: We say that X has tail-index a if
: limx->∞ x^a．[1 - F(x)] = A;
: for some positive, finite number A.
: Prove that X has tail-index a if and only if
上面這個應該是limx->∞ x^a．[1 - F(x)] = A ?
: limx->∞ x^a+1．f(x) = aA:
: 想請教高人一下!^^ Thanks!
pf: 首先 a必>0, 如果a<=0 則因為 lim(1-F(x)) -> 0 則A不可能為正數
(=>)
以下極限為 x->∞
A = lim x^a．[1 - F(x)]
= lim { x^(a+1)．[1 - F(x)] / x }
注意到上式 分子分母 -> ∞
用L'Hospital
= lim { [(a+1)x^a．[1 - F(x)] + x^(a+1)*(-f(x)) ] / 1 }
= (a+1)A - lim { x^a+1．f(x) }
so lim { x^a+1．f(x) } = aA
(<=)
lim {x^a *[1-F(x)]}
= lim { [1-F(x)] / [1/x^a] }
L'
= lim { -f(x) / [-a* 1/x^(a+1)] }
= (1/a) lim x^(a+1)*f(x)
= A
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