Dr. Hackney STA Solutions pg 97

Dr. Hackney STA Solutions pg 97 - G 1 also requires φ to...

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Second Edition 6-11 An estimator of the given form is invariant if, for all a and ( x 1 , . . . , x n ), W ( x 1 + a, . . . , x n + a ) = φ ¯ x + a s s 2 = φ ¯ x s s 2 = W ( x 1 , . . . , x n ) . In particular, for a sample point with s = 1 and ¯ x = 0, this implies we must have φ ( a ) = φ (0), for all a ; that is, φ must be constant. On the other hand, if φ is constant, then the estimators are invariant by part a). So we have invariance if and only if φ is constant. Invariance
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Unformatted text preview: G 1 also requires φ to be constant because G 2 is a subgroup of G 1 . Finally, an estimator of σ 2 is invariant with respect to G 3 if W ( cx 1 ,...,cx n ) = c 2 W ( x 1 ,...,x n ). Estimators of the given form are invariant because W ( cx 1 ,...,cx n ) = φ ³ c ¯ x cs ´ c 2 s 2 = c 2 φ ³ ¯ x s ´ s 2 = c 2 W ( x 1 ,...,x n ) ....
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