Testing the Information Matrix Equality with Robust Estimators



To simplify the notations and calculations that follow we transform M in
order to diagonalise
V .Let

1+d 0 -d

W=  010,

-101

where

Eφ [(u4 - 6u2 + 3)(u2 - 1 - 2IF)] _ 1    1    ( 4

Eφ ((u4 - 6u2 + 3)2)       = 2 - 12 Eφ u

(8)


(9)


Then

1

WM = —
σ2

- 1 - d(μ4 - 6μ2 + 3)
ʌ        O .ʌ

μ3 - 3μ1

μ4 — 6 μ2 + 3


1 / u2 - 1 - 2IF (u; σ; Φ) - d (u4 - 6u2 + 3) ʌ

( Y ; θ ) = -2                  u3 - 3 u                  ,

σ               u4 - 6u2 +3

and V = WVW' is given by

1
V = σ4diag[A, 6, 24],

where

A = -8 + 4ASV(^) + 2E (u4IF) - 1 [E (u4IF)]2 .        (10)

We conclude that the IM test statistic in the normal model can be written
as the sum of three (asymptotically independent) statistics,

A+ (μ2 - 1 - d(μ4 - 6μ2 + 3))2 + 6(μ3 - 3μ1)2

(11)


+ 24(μ4 - 6μ2 + 3)2 .



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