Testing the Information Matrix Equality with Robust Estimators



C.2 Skewed normal alternative

The skewed-normal log-density is

-2 log(2π) - log σ - u2,               if y ≤ β;

log f (У ; β,σ,γ) = <                                      2

[ - 1 log(2 π ) - log σ + log(1 + γ ) - uγ, if y>β ;

where u = (y - β) and uγ = u(1 + γ). The score function, evaluated at
Y = 0, is

(u∕σ

( u2 - 1)

I(y > β)(u2 - 1)

Now let Fn be skewed normal with β = 0, σ = 1, and γn = e∕√n > 0.
Then,
γоn = 0, and by the results of Appendix B.2,

β0 ,n  = EFn ( Y ) = γnφ (0),

σ о ,n  = 1+^2^+o ( γn ) '

It follows that

I(Y>βо,n)} j


EFn [ s ( Y ; β0 ,n0 ,n, 0)] = -


0
o ( Yn ) .
EFn { [(Y-вП)2 - 1

The third element in parentheses is

∞ ,     Yn). - β0n ʌ 2 dφw - 1 + Φ(βоn)

∙M 0 ,n σ 0 ,n /

= (----—) (ββ0,nφ(β0,n) + 1 - ф(β0,n)-------2 0П ’ φ(β0,n) - 1

σ 0 ,n J                                          σ 0 ,n

+H β0 ,n ) + o ( γη )

- 2Yn (φ(0)) - 1 + Ф(β0,n)


= (1 + γn )( γn (φ (0)) +1 - Ф( β0 ,n ))

+O ( Yn )

= Yn (I - (φ(0))2) + 0(Yn)

= Yn ( π ) + 0 ( Yn ).

34



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