where
δG = e2 Eg [ξ ( Y ; θ)] ' V+Eg [ξ ( Y ; θ)].
The expression for the non-centrality parameter is valid for any Fθ and G,
provided that EG[ξ(y; θ)] exists. Note that, for G = Fθ, δG = 0 as it should
be.
When Fθ is the normal distribution, then
δG = e2 A+ (Eσξι)2 + 1 (EGξ'2)2 + 24 (Egёз)2
(14)
where
ξι = u2 — 1 — 2IF — dξ3,
ξ2 = u3 — 3 u,
ξ3 = u4 — 6 u2 + 3,
IF = IF(u; σ; Φ) and u = Y-β. We see from (14) that, for any G, δG is
minimal when σ is the ML estimator, because then A = d = ξι = 0, which
makes the first term in brackets vanish, and because the second and third
term are independent of σ. When σ is not the ML estimator, the first
term vanishes only if EGξ1 = 0. Thus, replacing the ML estimator with
another estimator has two opposite effects on power: it increases (actually,
non-decreases) the non-centrality parameter, but also increases the degrees
of freedom from 2 to 3.
Taking G =∆x , where ∆x is the Dirac measure with all mass at x
(representing outliers in the data at x), the non-centrality parameter is
δχ = e 2( ξ ( x ; θ )) 'V+ξ ( x ; θ ),
while for G = 11 (∆x + ∆-x) (representing symmetric outliers), it is
e2
δ-x,x = Y (ξ(X; θ) + ξ(—X; θ)) V +(ξ(X; θ) + ξ(—x; θ)).
Figure 1 gives the non-centrality parameter for G =∆x and the three es-
timators discussed above (ML, MAD, and TB with 25% breakdown point).
The non-centrality parameter for G = 2(∆x + ∆-x) is plotted in Figure 2.
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