Optimal Robustness for Estimators And Tests

Cover Optimal Robustness for Estimators And Tests
Optimal Robustness for Estimators And Tests
Allan Birnbaum
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Substituting x = G^ (y) in the first integral and x = G^ (y) o o in the last integral, we get finally o ^ Ip^lnf G^ + 2p(l-p)y J(. Jx)jQ^{x)dx + (l-p)2lnf G* 1 [ ° o °i where inf G^ is R. A. Fisher's "information" of Gq evaluated at e = «f . Notice that this is the variance of Ti; under both O JM G^ and G| . To calculate the efficacy v/e need i, E, (TP|G*) and |rE^jTP|G) ^ eo^-^N' A=0 M "e*^"Ni A=0 By the mean value theorem n^ H« (x) = a»(x) + -f (G|(x) - 0»(x)) = GS('^' -^Iro^f^' 'e' for some
...*.
H. = e > 0. Thus 00 Eect>('^|lG*) =/ r Jp, C-(x) N a|. T |a \ dG*(x).
50 Another application of the mean value theorem yields CO ngA CO Jl(G*(x)) M.
=$ P t^ .. DG*{x) G*=G* ^ \x 6 ^^2 ^^u where G* lies between G* and Gg - -||- A -^-f^ e 'e M. =* Recalling that the first Integral has been shown to be zero, we have ^'^ -\ 00 ' -N^ e=

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