3.2 Different parameter estimators
Maximum Likelihood Estimator
If ^ is the ML estimator, then IF = (u2
rank 2), and
— 1)/2, d = A = 0 (implying У has
T=n
(μ3 - 3μ1)2
( μi4 — 6 μ2 + 3)2
24
with limiting χ2 distribution. If, moreover, / is the ML estimator, then
μ1 = 0, μ2 = 1, and T reduces to
T=n
μ2 + (μ4 - 3)2 ^
6 + 24
(12)
the well-known Jarque-Bera (1980) statistic for testing for skewness and
non-normal kurtosis.
Robust Estimators
Some straightforward algebra shows that V has rank 2 only if IF = (u2 —
1)/2. Thus, if ^ is not the ML estimator, then V has full rank and T,
given by (11), has a limiting χ32 distribution. So, the IM test with robust
estimators is sensitive to ‘more’ specification error than the IM test with
ML estimators 1 .
Throughout we use the median as a robust estimator of the location
parameter β. The asymptotic distribution of T, however, does not depend
on this choice. Neither does the local asymptotic power in the examples we
will consider. Alternative M-estimators of location could be used, but we
use the median since it is minimax robust (Huber, 1964).
As robust estimators of scale we consider two robust M-estimators. An
M-estimator of scale, ^, solves, for some chosen c > 0,
n У ʃ ) bc
where bc = Eφ [ρc(u)], β is an equivariant estimator of β, and ρc is an even
function, not identically zero, non-decreasing on [0, ∞[, differentiable a.e.
1 This property is not unique relative to robust estimators.
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