Testing the Information Matrix Equality with Robust Estimators

and for which ρ_c(0) = 0. This estimator is consistent for σ and has influence
function (Huber, 1981)

IF(u; σ∙,Φ) = Pc⁽^> - bc .

E φ [ p_c ( u ) u ]

For ^ to be robust, ρ_c has to be bounded and / has to be a robust estimator
of location (e.g. the median). The ML estimator of σ, for example, is an
M-estimator of scale for which ρ_c(u) = u² (unbounded), b_c = 1, and / is the
sample average, and hence is non-robust.

Consider the robust M-estimator
(TB) function,

The choice of c is governed by the desired breakdown point of ^. The details
of how to compute T , for any choice of c, are given in Appendix A. Table 1
gives the numerical results for 10%, 25% and 50% breakdown points that
are needed to compute T using the TB estimator.

Another, simpler, robust M-estimator of scale is the (standardized) Me-
dian Absolute Deviation (MAD),

^σ = _Ф-1(3/₄) medi{^lYi ^- medjYj|}’

for which ρ_c(u) = I(|u| ≥ c) with c = Φ^-1(3/4) = 0.6745 and b_c = 1 /2. The
breakdown point of the MAD is 50%. Table 1 gives the constants, derived
in Appendix A, that are needed to compute T using the MAD.

4 Local asymptotic power

Let Fn be a sequence of local alternatives tending to the null distribution,
i.e. Fn → F_θ. Under a sequence of local alternatives,

Hn : Y ~ Fn,

the IM indicator M is, given some regularity conditions, asymptotically
normally distributed,

n(M ( M - mn ) 4 N(0 ,v ),

<<   8   9   10   11   12   13   14   >>

More intriguing information