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Basu's Theorem
Basu's Theorem
1 questions
Difficulty 6-6
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intermediate (6/10)
conceptual
For
X
1
,
…
,
X
n
∼
N
(
μ
,
σ
2
)
iid, the sample mean
X
ˉ
and the sample variance
S
2
are independent. Which justification is cleanest and most general?
Hide and think first
A.
By Cochran, any two quadratic forms in a Gaussian vector with orthogonal projection matrices are automatically independent here
B.
Basu: the sample mean is complete sufficient for the mean and the sample variance is ancillary, so they are independent
C.
Symmetry of the normal distribution around the mean forces the centered residuals to be uncorrelated with the sample mean itself
D.
The covariance between the sample mean and the sample variance equals zero, and zero covariance implies independence for Gaussians
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