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※ 引述《imgodya (許我一個PhD)》之銘言：
: Find E{R^2/(1-R^2)} under the null hypothesis, where R^2 is the sample squared
: multiple correlation coefficient. Hint: E{[χ^2]^k} = 2^kΓ(v/2 + k)/Γ(v/2),
: whereχ^2 is aχ^2-distributed random variable with v degrees of freedom.
: 翻了很多書 不太懂這個問題的定理
: 想請問要如何推導??
Under the null hypothesis,
R^2 R^2 *SST SSR SSR/ σ^2
------- = -------------- = ------- = ------------
1-R^2 (1-R^2) *SST SSE SSE/ σ^2
(k-1 is the number of explanatory variables)
Since SSR/ σ^2 and SSE/ σ^2 are inpependent χ^2-distributed
random variable with degrees of freedom k-1 and n-k respectively
under null hypothesis.
R^2 SSR/ σ^2 1
E(-------) = E(-----------) = E( SSR/ σ^2 * ------------)
1-R^2 SSE/ σ^2 SSE/ σ^2
1
=E( SSR/ σ^2 )* E ( ------------)
SSE/ σ^2
Compute these expectations with the hint
E{[χ^2]^k} = 2^kΓ(v/2 + k)/Γ(v/2)
SSR/ σ^2 ~ χ^2(k-1) E(SSR/ σ^2)= k-1
SSE/ σ^2 ~ χ^2(n-k) 1
E ( --------- ) = 2^(-1)Γ( (n-k)/2 -1 )/Γ( (n-k)/2 )
SSE/ σ^2
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※ 編輯: jangwei 來自: 203.73.183.95 (02/23 01:04)