[機統] variance unbiased
我最近看一本書, 但覺得他的敘述怪怪的,所以想要請教一個問題
2
給定隨機變數x1, x2, ... xk, 都是normal distribution N(μ, σ)
2
μ is known,但σ unknown, 用maximum likelyhood 找出 estimator:
2 N 2
L(x1,x2,...,xk; σ) = Σ ln p(xk; σ)
k=1
2 2
2 1 -(xk-μ)/(2σ)
p(xk; σ) = -------- e
√2π σ
2
L對σ 偏微分 = 0 得到
2 1 N 2
σ = ----- Σ (x - μ)
ML N k=1 k
2
他說 σ is biased since
ML
2 1 N 2 ??? N-1 2
E[σ ] = --- Σ E[(xk-μ)] ======== ----- σ
ML N k=1 N
2 2
但我覺得 E[(xk-μ)] = σ 根據variance的定義
2 1 2 2
所以E[σ ] = ----- Nσ = σ , 明明就unbiased
ML N
後來我想到一個東西:
2 1 N _ 2 _ 1 N 2
隨機變數 S = ---- Σ (x- x) , x = --- Σ xk 是 unbiased esitmate of σ
N-1 k=1 k N k=1
N _ 2 N _ 2 N 2 _ 2
(N-1) Σ (x - x) = Σ (xk - μ + μ - x) = ... = Σ (xk-μ) - N (x-μ)
k=1 k k=1 k=1
2 N _ 2 2 2
=> (N-1)E(S ) = Σ var(xk) - N var(x) = Nσ - σ = (N-1)σ
k=1
2 2
=> E(S ) = σ => unbiased
就想說會不會那個μ is known, 不是指真正的平均值??
N 2 2
還是我的推導(Σ E[(xk-μ)] = Nσ)是錯的?
k=1
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