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



More intriguing information

1. For Whom is MAI? A theoretical Perspective on Multilateral Agreements on Investments
2. The name is absent
3. APPLYING BIOSOLIDS: ISSUES FOR VIRGINIA AGRICULTURE
4. Testing the Information Matrix Equality with Robust Estimators
5. Education Responses to Climate Change and Quality: Two Parts of the Same Agenda?
6. The name is absent
7. Second Order Filter Distribution Approximations for Financial Time Series with Extreme Outlier
8. TRADE NEGOTIATIONS AND THE FUTURE OF AMERICAN AGRICULTURE
9. Licensing Schemes in Endogenous Entry
10. Survey of Literature on Covered and Uncovered Interest Parities