Testing the Information Matrix Equality with Robust Estimators

Hence

b = lim nEE_Fs [ s ( Y ; β0 ,n,σ 0 ,_n, 0)]
n→∞

=e 0

\ 1 -∏
\ 2 π

The information matrix is

J = E [s (y ;0, 1, 0) s (y ;0, 1, 0)' ]

1 0 Eφ[u(u² — 1)I(u > 0)] \

■ 2 Eφ[(u² — 1)²1(u> 0)]

■ ■ Eφ[(u² — 1)²1(u> 0)] J

10φ(0) ^\

021 ,

φ(0) 1 1

wherefrom δ = b^,J ¹ b = ^π-¹ e².

2π

C.3 Tilted normal alternative

Expanding f(y; β, σ, κ, λ) around κ =0andλ = 0 gives

f⁽ У ;^β,σ,κ,λ ) = ¹^φ⁽ u )
σ

κλ

1 + ^( u — 3 u ) + 24( u — 6 u + 3)

+ o(κ, λ),

from which the moments given in (15) follow. The score function, evaluated
at κ = λ =0, is

s(y; β, σ, 0, 0)

σ u \

u² -1
σ
u³ — 3 u
6
u⁴ — 6u 2+3
\ 24 /

The information matrix is

J = E [s ( y ; β, 1, 0, 0) s ( y ; β, 1, 0, 0) ']

/ 1 0 0 0 \

0200

= 0 0 1 0 ’

V ^{0 0 0} 24 J