X t m X t Y t Z t Y t l Z t k X t h x Z t h z Z t Y t h y z X t Y t Z t E. G γ. F θ. z Θ Γ. γ F θ

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4 X t m X t Y t Z t Y t l Z t k X t hxz t hzz t Y t hy z X t Y t Z t E F { f( y z; θ); θ Θ R p } θ G { g( y z; γ); γ Γ R q } γ ΘΓ z ΘΓ F θ θ γ F θ G γ G γ E [] = () h( y, z) dydz h( z) () h( y z) dydz () h( y) dy = =

5 θ arg max E [ l f( y z; θ) ], θ Θ γ arg max E [ l g( y z; γ) ] γ Γ ˆθ ˆ γ F θ G γ f L ( θ ˆ ) sup l f( Y Z ; θ), g L ( ˆ γ ) sup l g( Y Z ; γ) t t γ Γ t = 1 t t θ Θ t = 1 A ( θ) E f A ( γ ) E g B ( θ) E f l f( Y Z ; θ) t t, θ θ l gy ( Z; γ ) t t, γ γ l f( Y Z ; θ) l f( Y Z ; θ) t t t t, θ θ l gy ( Z; γ ) l gy ( Z; γ ) t t t t B ( γ ) E g, γ γ l f( Y Z ; θ) l g( Y Z ; γ ) t t t t B ( θγ, ) E fg, θ γ ˆθ θ ( θ ˆ θ ) 1 1 N[, A ( θ ) B ( θ ) A ( θ )] f f f ( ˆ γ γ ) 1 1 N[, A ( γ ) B ( γ ) A ( γ )] ˆθ ˆ γ g g g θ ˆ θ A ( θ ) B ( θ ) A ( θ ) A ( θ ) B ( θ, γ ) A ( θ ) f f f f fg g ~ N , ˆ γ γ A ( γ ) B ( γ, θ ) A ( θ ) A ( γ ) B ( γ ) A ( γ ) g gf f g g g

6 hy z hy z hy ( z) KLIC( h, f ) l hxdx ( ) f( y z; θ ) = E l h( y z) E l g( y z; γ ) H H f H g F θ G γ F θ G γ [ ] [ ] = E l h( y z) E l f( y z; θ ) hy z G γ hy ( z) KLIC( h, g) l hxdx ( ) gy ( z; γ ) f Y Z t t H : E l ( ; θ ) gy ( Z; γ ) t t = f Y Z t t H : E l ( ; ) θ f, gy ( Z; γ ) > t t f Y Z t t H : E l ( ; ) θ g gy ( Z; γ ) < t t [ ] [ ] F θ G γ F θ G γ F θ

7 f g f Y Z t t LR ( ˆ, ˆ ) L (ˆ ) L (ˆ ) l ( ; θ ˆ ) θ γ = θ γ t gy ( Z; ˆ = 1 γ ) t t 1 LR as E f Y Z t t ( ˆ, ˆ ) l ( ; ).. θ θ γ gy ( Z; γ ) t t l[ f( Yt Zt; θ) / g( Yt Zt; γ )] θ θ ω* var l ( ; ) l ( ; ) f Y Z f Y Z t t t t = ( ; γ ) E gy Z ( ; γ ) gy Z t t t t E f θ l ( Y Z ; ) t t gy ( Z; γ ) t t Z = ( Z Z m 1,..., ) m λ = ( λ λ 1,..., m ) m m i =1 λ i Z i ( m, λ) M m (; λ) (f( ; )g( ; γ ) D LR ( θ ˆ, ˆ γ ) M (; λ ) p+ q λ 1 1 B ( θ ) A ( θ ) B ( θ, γ ) A ( γ ) f f fg g W = 1 1 B ( γ, θ ) A ( θ ) B ( γ ) A ( γ ) gf f g g pq

8 ( f( ; ) g( ; γ ) 1 / 1 / f Y Z t t D LR E N gy Z t t ( ˆ, ˆ ) l ( ; θ θ γ ) (, ω* ) ( ; γ ) F θ G γ F G = θ γ F G F G θ γ ad G F G F γ θ θ γ γ θ Y = β + β Z + ε, Y = β + β W + β Z + ε, Y = α + α X + ε, t 1 t t t 1 t t Y = α + α W + α X + ε, t 1 t t t t 1 t t t Y = α + α X + ε, t 1 t t Y = β + β X + β Z + ε, t 1 t t t

9 ( F G = ) E [l f( θ )]) θ γ h E [l g( γ )]) θ γ ˆθ γˆ F θ G γ (F G F G ad G F ) θ γ θ γ γ θ h h E [l f( θ )]) E [l g( γ )]) E [l f( θ )]) E [l g( γ )]) θ = γ θ γ ˆθ γˆ ˆθ γˆ F θ G γ F θ G γ ( G F ) γ θ E [l f( θ )]) h E [l g( γ )]) θ ˆθ γ F θ γˆ G γ θ γ ˆθ ˆ γ

10 E [l f( θ )] E [l g( γ )] f( θ) = g( γ ) f( ; θ) g( ; γ ) E [l f( θ )] = E [l g( γ )] f( θ) = g( γ ) f( ; θ ) g( ; γ ) f( θ) = g( γ ) f( ; θ) g( ; γ ) f( θ) = g( γ ) f( ; θ) g( ; γ ) f( ; θ) g( γ ) f( ; θ) = g( ; γ ) F θ G γ f( ; θ) g( ; γ ) ω * ˆ l ( ; ˆ 1 f Y Z θ ) 1 t t ω ( ; ˆ t 1 gy Z γ ) = t t t = 1 f l ( Y Z ; θ ˆ ) t t gy ( Z; ˆ γ ) t t 1/ D H : LR (ˆ θ, ˆ γ )/ ωˆ N( 1, ), F θ G γ

11 1/ as.. H : LR (ˆ θ, ˆ γ )/ ωˆ +, f 1/ as.. H : LR (ˆ θ, ˆ γ )/ ωˆ g f( ; θ) = g( ; γ ) f( ; θ) = g( ; γ ) ω * = ω H : ω* = ω H A : ω * ω D H : ωˆ M (;ˆ λ ) p+ q ω as.. H : ωˆ + A ˆλ F θ G γ ω H f( ; θ) g( ; γ ) E [l g( γ )] E [l f( θ)] f( ; θ) = g( ; γ ) f( ; θ) = g( ; γ ) f ( ; θ ) = θ g( ; γ ) ω * = H : ω * = θ H A : ω *

12 θ D H : ωˆ M (;ˆ λ ), p θ as.. H : ωˆ + A 1 1 ˆλ W = ( B A + B R A R f f f g ) l gy ( z; γ ) / γ = R l f( y z; θ) / θ F θ G γ H θ

13 α + α X ε 1 t + t β + β Z ε 1 t + t γ + γ + γ Z + ε X 1 t t t γ 1 γ γ 1 γ γ 1 γ

14 F = { f( y x; θ); θ Θ} G = { g( y z; γ); γ Γ} θ γ λ = λ = 1 H f :λ = H g :λ = 1 1 λ λ f( y x; θ) g( y z; λ) C = {( c y x, z; θγ, ); θ Θ, γ Γ} = λ 1 λ λ f y x θ g y z λ dy ( ; ) ( ; ) H : y = Zβ + u, u ~ N(, ω I) C λ H : y = X α + u, u ~ N(, σ I), f f f g g g C λ

15 H : ( 1 λν ) λν y = X Z u, c α + β + λ σ ω u ~ N(, I), = 1 λ λ ν where ν +, λ σ ω κ = λv / ω H : y ( ) X Z u, u ~ N (, = 1 κ α + κ β + ν I), c λ λ / = 1 λ λ where κ = λν ω ad ν +, σ ω ν σ H f :λ = H f :κ = κ κ H : y = X α + κ Zˆ β + u c λ κ = κ κ = κ H : y = κ X αˆ + Zβ + u c λ κ = κ

16 H c : Retur t = α + α x t + γ( ˆ β + ˆ β z t ) + ε 1 1 t : Retur = ( α + γ ˆ β ) + α x + γ ˆ β z + ε H c t 1 t 1 t t = δ + δ x + δ z + ε 1 t t t H x : y t = α + α x t + ε t, 1 H z :y t = β + β z t + ε 1 t ŷ z t H c : y t = α + α x t + α yˆ z t + ε 1 1 t α ˆα ŷ x t H c : y t = β + β z t + β yˆ x t + ε 1 t β ˆβ H c :y t = γ + γ x + γ z t + ε t γ 1 ˆ γ 1 γ ˆ γ ˆα ˆ 1 γ 1 ˆα ˆ γ ˆβ1 ˆ γ ˆβ ˆ γ 1

18 ε

19 ε ε ε

20 ε ε ε ε ε ε

23 No-ested Overlappig Nested Irtercept SIZE it BM it E it E it CF it CF it E it E it CF it CF it.r ε ε AR it 1 E it E it it ε AR it 1 CF it CF it it ε AR it 1 SIZE it BM it 3 E it 4 E it it ε ε AR it 1 SIZE it BM it 3 CF it 4 CF it it AR it 1 E it E it 3 CF it 4 CF it it AR it 1 SIZE it BM it 3 E it 4 E it 5 CF it 6 CF it it AR it i t t E it i t E it i t E it E it 1 CF it t CF it i t CF it CF it 1 SIZE it i t BM it E it E it CF it CF it t White t i t

24 ε ε ε ε ε ε χ

25 χ

28 f g L ( θ ), L ( γ ) θˆ, ˆ γ f f f L ( θ L ( ) L (ˆ ˆ ) θ = θ ) + A o ( θ θˆ ) + (ˆ θ θ ) (ˆ θ θ ) + ( 1), f p θ g g g L ( ˆ γ ) L ( γ ) = L (ˆ γ ) + A ( γ ˆ γ ) + (ˆ γ γ ) ( ˆ γ γ ) +o ( 1) g p γ LR ( θ ˆ, ˆ γ ) = LR ( θ, γ ) (ˆ θ θ ) A (ˆ θ θ ) + f ( ˆ γ γ ) A (ˆ γ γ ) + o ( 1) ( ) g p o A.1 YQY ~ M (; λ), whereλ = QΩ m ( m, λ) χ γ A f LR ( ˆ θˆ θ θˆ θ θ, ˆ γ ) = +, ˆ γ γ A ˆ γ γ A f LR ( ˆ θˆ θ / θ, ˆ γ ) = ˆ γ γ D LR ( θ ˆ, ˆ γ ) M (; λ ) p+ q λ g A 1 1 / g θ ˆ θ ˆ γ γ o

29 Af A B A A B A BA B A f f f f fg g f f fg g W = = 1 1 A g A B A A B A B A B A g gf f g g g gf f g g γ LR ( θ ˆ, ˆ γ ) = LR ( θ, γ ) (ˆ θ θ ) A (ˆ θ θ ) f + ( ˆ γ γ ) A (ˆ γ γ ) + o ( 1) g p 1 1 LR ( θ, γ ) / ( θ ˆ 1 = θ ) A / (ˆ θ θ ) f 1 1 / 1 / + ( ˆ γ γ ) A (ˆ γ γ ) g = LR ( θ, γ ) O ( 1) O ( 1) + O ( 1) O ( 1) p p p p = LR ( θ, γ ) + O ( 1) p o 1 / 1 / 1 / LR ( θ ˆ, ˆ γ ) = LR ( θ, γ ) + o ( 1) p 1 / E [l{ f( Y Z ; θ ) / g( Y Z ; γ )}] t t t t 1 / 1 / f Y Z t t LR ( ˆ, ˆ ) E l ( ; θ θ γ ) gy ( Z; γ ) t t 1 / 1 LR E f Y Z t t = (, ) + o l ( ; θ θ γ ) p gy Z () 1 ( ; γ ) t t ω * 1 / 1 / f Y Z t t D LR E N gy Z t t ( ˆ, ˆ ) l ( ; θ θ γ ) (, ω* ) ( ; γ ) l[ f( Y Z ; θ ) / g( Y Z ; γ )] ω t t t t *

30 ˆ l ( ; ˆ ) ( ; ˆ 1 f Y Z θ 1 f Y Z θ ) t t t t ω ( ; ˆ γ ) ( ; ˆ t gy Z γ ) t t t gy Z = 1 = 1 t t ω * ˆω LR ( θ ˆ, ˆ γ ) = O ( 1) p 1 f Y Z t t = l ( ; θ ˆ ) t 1 gy ( Z; ˆ γ ) = t t ( θ, γ ) B ˆ l ( ; ˆ 1 f Y Z θ ) 1 t t ω = ( θ ˆ, ˆ γ ) ( ; ˆ t gy Z γ ) LR = 1 t t f l ( Y Z ; θ ˆ 1 ) 1 t t = () p ( ; ˆ t gy Z γ ) O 1 = 1 t t ˆ l ( ; ) 1 f Y Z θ t t ω = t gy ( Z; γ ) = 1 t t f l ( Y Z ; θ ) l f( Y Z ; θ ) t t t t 1 + ( θˆ θ ) t = 1 gy ( Z; γ ) θ t t 1 f Y Z gy Z t t t t l ( ; θ ) l ( ; γ ) ( ˆ ) t gy ( Z; γ ) γ γ γ = 1 t t θ ˆ θ B o () B o () f p fg p ˆ + ˆ B o () B γ γ o gf p g p () 1 θ θ ˆ γ γ / 1 / ( ˆ γ γ ) O p () 1 θˆ θ B B f fg θˆ θ ωˆ = o () + 1 p ˆ γ γ B B ˆ gf g γ γ B f gf B B g A B A A B A fg A B A A B A f f f f fg g g gf f g g g = W 1 ( θ ˆ θ )

31 D ωˆ M (;ˆ λ ) p+ q ˆλ l gy ( z; γ ) / γ = R l f( y z; θ ) / θ, 1 1 BA λi BRA f f p f g = det 1 1 RB A RB R A λi f f f g q q 1 = ( λ) det BA λi + BRA 1 1 W = B A + B R A R f f f g D ωˆ M (;ˆ λ ) p BA λi + BRA R BRA = det λiq f f p f g f g 1 [ R f f p f g ] ( ) ˆλ F = { f( y z; θ); θ Θ} G = { g( y z; γ); γ Γ} θ γ

32 1 / D H : LR (ˆ θ, ˆ γ )/ ωˆ N( 1, ), 1 / as.. H : LR (ˆ θ, ˆ γ )/ ωˆ +, f 1 / as.. H : LR (ˆ θ, ˆ γ )/ ωˆ g f Y Z f Y Z t t t t ˆ l ( ; ˆ ) l ( ; ˆ 1 θ 1 θ ) ω = t gy ( Z; ˆ γ ) gy Z t t t ( ; ˆ = 1 = 1 γ t t ) f Y Z t t m l ( ; θ ˆ ) t gy ( Z; ˆ γ ) t t LR ( θ ˆ, ˆ γ ) = ( m + m m ) = m, 1 ˆ ω = 1 1 m ( m) f Y Z t t LR ( ˆ, ˆ ) l ( ; θ ˆ ) θ γ = gy ( Z; ˆ γ ), 1 / t = 1 LR ( θ ˆ, ˆ γ )/ ωˆ = t t 1 / m m = 1 m ( m) m ( m) / y = ( 11,,..., 1), x = ( m, m,..., m ) 1 y = xβ + ε t = b b 1 s ( x x) b = x x 1 1 ( ) x y, s = e e / ( 1) = y { I x( x x) x }{ I 1 xxx ( ) x } y/ ( 1) 1 1 ( xx ) = ( m ), F θ G γ

33 xy = yx = m, yy =, b = m / m, s = { y ( m / m ) x }{ y x( m / m )} / ( 1) = { ( m) / m }/( 1), t = m / m m = b ( m )/ m 1 m ( m) ( m ) 1 1 ( / 1) 1 / 1 / 1 m m ( / 1) t = = b 1 m ( m) m ( m) / 1 F = { f( y z; θ); θ Θ} G θ γ = { gy ( zγ ; ); γ Γ} f Y Z t t m = l ( ; θ ˆ ) t gy ( Z; ˆ γ ) t t ( / 1) 1 / 1/ LR ( θ ˆ, ˆ γ )/ ωˆ F θ G γ F Gγ θ G γ F θ

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