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 ) - u^γ, if y>β ;

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

(u∕σ

( u² - 1)/σ

I(y > β)(u² - 1)

Now let F_n be skewed normal with β = 0, σ = 1, and γ_n = e∕√n > 0.
Then, γо_n = 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 ,n,σ 0 ,n, 0)] = -

0
^o⁽ Yn ) .
EFn { [(Y-вП)² - ¹

The third element in parentheses is

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

∙M 0 ,n ∖ ^σ⁰^,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-β ) ⁺⁰ ( ^Yn ).