Testing the Information Matrix Equality with Robust Estimators

asv( σ) =

EΦ ^ρPcu^u)] - b²
{EΦ [^pc(^u)^{u] }2}

EΦ(u⁴IF) = _f ɪ ₁

Eφ [ p_c ( u ) u ]

{f(₁ - νc(4)) +   6

Vc (8) + ν_c (10)

2 c ²      6 c ⁴

3bc .

The computation of T , from (11), is now straightforward.

For the MAD, E_φ [p_c(u)u] = 2cφ(c) (with φ the standard normal pdf)
and Eφ [ρ2(u)] = 2, resulting in

and

ASV ( ^) =-----¹----= = 1.361

16c² φ(c) ²

Eφ(u⁴IF) = _r, _γ ɪ ₁ I³ - Vc(4)I .
^φv ’   Eφ [Pc(u)uH 2   c^vJ

Appendix B

Local asymptotic power of IM test

B.1 Student’s t alternative

Under Hn, Y ~ Fn = Ft₍p_n ) with pn = Пп/е. So

Z

U' n ^,

where U ~ χ2_n and Z ~ N(0, 1), with U and Z independent. By the Central
Limit Theorem, as pn →∞,

U=¹⁺'∕pn ^w+R

where W ~ N(0, 1), W and Z are independent, the remainder term R is
Op p_n^-1 , and E(R) = 0. Therefore σn solves

Z

^ρc ɪ I

∖√1 +72/PnW + Rσn

26

= b_c,

(16)