Asymptotic Expansions under degeneracy Yuji Sakamoto and Nakahiro - PowerPoint PPT Presentation

Asymptotic Expansions under degeneracy Yuji Sakamoto and Nakahiro Yoshida Hiroshima International Univ. and University of Tokyo SAPS4 , Universit´ e du Maine , Le Mans, France December 19-20, 2002 1

Stationary Ergodic Diffusion Model ✷ Observations : X = ( X t ) t ∈ [0 ,T ] , dim( X t ) = d , Continuous θ ∈ Θ ⊂ R p , ✷ Parameter : dim( θ ) = p , Θ : open bounded convex ✷ Model : dX t = V 0 ( X t , θ ) dt + V ( X t ) dw t , X 0 ∼ ν θ V 0 : R d × Θ → R d , V : R d → R d ⊗ R r w = ( w t ) : r -dimensional standard Wiener process ν θ : stationary distribution with a positive density dν θ dx ✷ Log-Likelihood : � T � T ℓ ( θ ) = log dν θ 0 ( V V ′ ) − 1 ( X t , θ ) dX t − 1 0 V ′ 0 V ′ 0 ( V V ′ ) − 1 V 0 ( X t , θ ) dt dx + 2 θ T : ( X t ) t ∈ [0 ,T ] → ˆ ˆ ✷ Estimator : θ T ∈ Θ ✷ Asymptotics : T → ∞ 2

Estimation for unknown parameter θ in drift function V 0 ✷ Parameter of interest : θ = ( θ 1 , . . . , θ p ) ∈ Θ ⊂ R p , dX t = V 0 ( X t , θ ) dt + V ( X t ) dw t , X 0 ∼ ν θ , t ∈ [0 , T ]. ✷ True value : θ 0 ∈ Θ. θ ( c ) θ ( c ) δ a ℓ ( c ) (ˆ T ) = 0, a = 1 , . . . , p , δ a = ∂/∂θ a . ✷ Conditional MLE ˆ : T � T � T 0 ( V V ′ ) − 1 ( X t , θ ) dX t − 1 ℓ ( c ) ( θ ) = 0 V ′ 0 V ′ 0 ( V V ′ ) − 1 V 0 ( X t , θ ) dt. 2 δ a = ∂/∂θ a . ✷ (Exact) MLE ˆ δ a ℓ (ˆ θ T : θ T ) = 0, a = 1 , . . . , p , ℓ ( θ ) = log dν θ dx ( X 0 ) + ℓ ( c ) ( θ ) . θ ψ θ ψ ✷ M -estimator ˆ ψ a ; (ˆ T : T ) = 0, a = 1 , . . . , p , for an estimating fuction ψ = ( ψ 1; , . . . , ψ p ; ). 3

Literature on distributional expansion for ergodic diffusion ✷ Second order � Yoshida(1997) : martingale expansion, (global approach) (conditional) MLE, d = 1, p = 1 � Sakamoto-Yoshida(1998) : M -estimator, d = 1, p = 1, Numerical studies on (conditional) MLE for OU, etc � Uchida-Yoshida(2001), (Mykland(1992, 1993)) ✷ Third or higher order � Kusuoka-Yoshida(2000) : ǫ -Markov mixing, (local approach) Diffusion functional having stoch. exp. , arbitrary d , p � S-Y(1998) : third order MLE, arbitrary d , p � S-Y(1999) : representation of expansion, third order M -estimator, arbitrary d , p � S-Y(2000) : under degeneracy � Sakamoto(2000), Kutoyants-Yoshida(2001) 4

Second order expansion of MLE for univariate ergodic diffusion Theorem 1 (Yoshida(P.T.R.F.,1997)). For any θ ∈ Θ ⊂ R , Let X = ( X t ) t ∈ [0 ,T ] be a one-dimensional stationary ergodic diffusion pro- cess satisfying dX t = V 0 ( X t , θ ) dt + dw t with a stationary distribution ν θ given by �� x � n θ ( x ) ν θ ( dx ) = n θ ( x ) = exp 0 2 V 0 ( u, θ ) du −∞ n θ ( u ) dudx, . � ∞ Suppose that sup x ∈ R ∂ x V 0 ( x, θ 0 ) < 0 and that | δ l ∂ j V 0 ( x, θ ) | ≤ C j,l (1 + | x | C j,l ) , ∀ x, θ . Then it holds that √ 1 θ ( c ) T ( A − Bx 2 ) φ ( x ) + ¯ o ( T − 1 / 2 ) , IT (ˆ √ P ( − θ 0 ) ≤ x ) = Φ( x ) + T where I = ν θ 0 (( δV 0 ) 2 ) , A = − ν θ 0 ( δV 0 · k ) / (2 I 3 / 2 ) , B = −{ ν θ 0 ( δV 0 · δ 2 V 0 ) − ν θ 0 ( δV 0 · k ) } / (2 I 3 / 2 ) , � ∞ 2 n θ 0 ( u )(( δV 0 ( u, θ 0 )) 2 − I ( θ 0 )) du. k = − n θ 0 ( x ) − 1 x 5

θ ( c ) Numerical study on expansion of ˆ for OU in S-Y(1998) T 1.0 1.0 0.8 0.8 θ=1 θ=2 T=3, T=3, 0.6 0.6 0.4 0.4 Normal Normal Expansion Expansion 0.2 0.2 MonteCarlo MonteCarlo 0.0 0.0 -4 -2 0 2 4 -4 -2 0 2 4 1.0 1.0 0.8 0.8 θ=1 θ=2 T=6, T=6, 0.6 0.6 0.4 0.4 Normal Normal Expansion Expansion 0.2 MonteCarlo MonteCarlo 0.2 0.0 0.0 -4 -2 0 2 4 -4 -2 0 2 4 6

Third order expansion of MLE for multidimensional diffusion ρ > ( ρ ab ) . Theorem 2 (S-Y(1999)). Let M , γ > 0 , and ˆ Assume For any β ∈ C 2 that [L] , [DM1], [DM2], [DM3] hold true. B (Θ) , let θ ∗ ˆ T = ˆ θ T − β (ˆ θ T ) /T . Moreover assume that the diffusion process X has the geometrically strong mixing property. Then there exist positive constants c, ˜ C, ˜ ǫ such that for any f ∈ E ( M, γ ) √ � � � T (ˆ θ ∗ dy (0) f ( y (0) ) q T, 2 ( y (0) ) CT − (˜ ǫ +2) / 2 , ˆ ρ ab ) + o ( T − 1 ) , � � � ≤ cω ( f, ˜ � E [ f ( T − θ ))] − (1) � � where 1 1 ρ aa ′ ( µ a ′ − ˜ � q T, 2 ( y (0) ) = φ ( y (0) ; ρ ab ) c ∗ abc h abc ( y (0) ; ρ ab ) + β a ′ ) h a ( y (0) ; ρ ab ) 1 + √ √ 6 T T + 1 1 2 T A ∗ ab h ab ( y (0) ; ρ ab ) + 24 T c ∗ abcd h abcd ( y (0) ; ρ ab ) 1 � 72 T c ∗ abc c ∗ def h abcdef ( y (0) ; ρ ab ) + . 7

Coefficients in asymptotic expansion(1) µ a = − 1 β a = β a − ∆ a , abc = − 3Γ ( − 1 / 3) 2 ρ aa ′ ρ bc Γ ( − 1) ( ρ ab ) = ( ρ ab ) − 1 , c ∗ ˜ ρ ab = F a,b , , bc,a ′ , ab,c � A ∗ ab = − τ ab − ρ cd F bcd,a + F ab,cd − F ac,bd − F [ a,c ] , [ b,d ] + 2 F [ ab,c ] ,d + 2 F [ ac,b ] ,d + 4 F [ b,d ] ,ac � + F [ cd,b ] ,a + 2 F [[ b,c ] ,a ] ,d + 2 F [[ b,c ] ,d ] ,a � 1 � 2Γ ( − 1) ce,b Γ ( − 1) ac,e Γ (1) bd,f + Γ ( − 1) cd,e (Γ (1) ab,f + Γ ( − 1) ce,a (Γ (1) bd,f + Γ ( − 1) f,a − Γ (1) fb,a ) + Γ ( − 1) + ρ cd ρ ef bd,f ) d + ρ aa ′ ρ bb ′ ( µ a ′ − ˜ β a ′ )( µ b ′ − ˜ β b ′ ) + 2 ρ aa ′ (∆ c η ∗ a ′ c,b − δ b β a ′ ) , c ∗ abcd = − 12( F [[ a,b ] ,c ] ,d + F [ a,b ] ,cd + F [ ab,c ] ,d ) + 3 F [ a,b ] , [ c,d ] − 4 F abc,d β d ′ − µ d ′ ) + 12 ρ ef (Γ ( − 1) + 12Γ ( − 1 / 3) ab,e + Γ (1) ae,b )Γ ( − 1) ρ dd ′ (˜ cf,d , ab,c [3] ab,c = F ab,c − F [ a,b ] ,c + 1 − α Γ ( α ) b,c = − ρ aa ′ (Γ (1) a ′ c,b + Γ ( − 1) � η ∗ a F [ a,b ] ,c , bc,a ′ ) . 2 ( ab,c ) abcd , and ρ aa ′ ( µ a ′ − ˜ β a ′ ) are functions of F , The coefficients c ∗ abc , A ∗ ab , c ∗ τ , and ∆. 8

Coefficients in asymptotic expansion(2) dν θ 0 dν θ 0 dν θ 0 ∆ a = ρ aa ′ ν θ 0 ( δ a ′ dx ) , τ ab = Cov[ δ a dx ] , dx , δ b F A 1 ,A 2 = ν θ 0 ( B A 1 · B A 2 ) , F A 1 , [ A 2 ,A 3 ] = ν θ 0 ( B A 1 · [ B A 2 · B A 3 ]) , F [ A 1 ,A 2 ] , [ A 3 ,A 4 ] = ν θ 0 ([ B A 1 · B A 2 ] · [ B A 3 · B A 4 ]) , F [[ A 1 ,A 2 ] ,A 3 ] ,A 4 ] = ν θ 0 ([[ B A 1 · B A 2 ] · B A 3 ] · B A 4 ]) , 0 ( V V ′ ) − 1 V 0 ( x, θ ), B A ( x, θ ) = δ a 1 · · · δ a k B ( x, θ ), where B ( x, θ ) = V ′ δ a j = ∂/∂θ a j , A = a 1 · · · a k , [ f ] = − V ′ ∇ G f − ν ( f ) , A G f − ν ( f ) = f − ν ( f ) , d d r ∂ 2 ∂x i + 1 0 ( x, θ 0 ) ∂ k ( x ) V j V i V i � � � A = k ( x ) ∂x i ∂x j 2 i =1 i,j =1 k =1 9

Third order expansion of MLE for CIR model � dX t = ( p − qX t ) dt + r X t dw t , θ = ( p, q ) , Θ = { ( p, q ) : 2 p > r > 0 , q > 0 } 2 q F 1 , 2 = F 2 , 1 = − 1 p F 1 , 1 = r 2 (2 p − r 2 ) , r 2 , F 2 , 2 = qr 2 , 4 q 2 F [1 , 1] , 1 = − r 2 (2 p − r 2 ) 2 , F [1 , 1] , 2 = (2 p − r 2 ) , F [1 , 2] , 1 = F [1 , 2] , 2 = 0 F [2 , 2] , 1 = 1 F [2 , 2] , 2 = − p 8 q r 2 q, F [1 , 1] , [1 , 1] = r 2 (2 p − r 2 ) 3 , F [1 , 1] , [1 , 2] = 0 r 2 q 2 2 F [1 , 1] , [2 , 2] = − r 2 (2 p − r 2 ) q, F [1 , 2] , [1 , 2] = F [1 , 2] , [2 , 2] = 0 F [2 , 2] , [2 , 2] = p 8 q 4 r 2 q 3 , F [[1 , 1] , 1] , 1 = r 2 (2 p − r 2 ) 3 , F [[1 , 1] , 1] , 2 = − r 2 (2 p − r 2 ) 2 F [[1 , 1] , 2] , 1 = F [[1 , 1] , 2] , 2 = F [[1 , 2] , 1] , 1 = F [[1 , 2] , 1] , 2 = 0 F [[1 , 2] , 2] , 1 = F [[1 , 2] , 2] , 2 = F [[2 , 2] , 1] , 1 = F [[2 , 2] , 1] , 2 = 0 1 p F [[2 , 2] , 2] , 1 = − r 2 q 2 , F [[2 , 2] , 2] , 2 = r 2 q 3 , 2 τ 22 = 2 p ∆ 1 =∆ 2 = 0 , τ 11 = 4PolyGamma[1 , 1] , τ 12 = τ 21 = r 2 q, q 2 r 2 . 10

Stochastic Expansion of MLE ˆ θ T ( p = 1 ) MLE ˆ δℓ T (ˆ • θ T θ T ) = 0, δ = ∂/∂θ , : Likelihood eq. • Third order stochastic expansion √ 1 T (ˆ θ T − θ 0 ) = S T ( Z 1 , Z 2 , Z 3 ) + √ R 3 , T T � � 1 z 1 z 2 + 1 υ − 1 ν 3 z 2 S T ( z 1 , z 2 , z 3 ) = z 1 + √ 2¯ 2 ¯ 1 T � 1 � + 1 1 + 3 2 + 1 ν 2 3 ) z 3 ν 3 z 2 1 z 2 + z 1 z 2 2 z 2 6(¯ ν 4 + 3¯ 2¯ 1 z 3 , T υ − 1 υ − 1 where ¯ υ 2 = − 1 ν 3 = 1 ν 4 = 1 T E θ 0 [ δ 2 ℓ T ( θ 0 )], 2 E θ 0 [ δ 3 ℓ T ( θ 0 )], 2 E θ 0 [ δ 4 ℓ T ( θ 0 )], ¯ T ¯ ¯ T ¯ Z 1 = 1 υ − 1 √ ¯ 2 δℓ T ( θ 0 ) , T � � Z 2 = 1 υ − 1 δ 2 ℓ T ( θ 0 ) − E θ 0 [ δ 2 ℓ T ( θ 0 )] √ ¯ , 2 T � � Z 3 = 1 υ − 1 δ 3 ℓ T ( θ 0 ) − E θ 0 [ δ 3 ℓ T ( θ 0 )] √ ¯ . 2 T 11

Derivation of valid distributional asymptotic expansion √ T (ˆ E [ f ( θ T − θ 0 ))] ≈ E [ f ( S T ( Z 1 , Z 2 , Z 3 ))] , S T : stoch. exp., ⇐ (Delta method) � ≈ f ( S T ( z 1 , z 2 , z 3 )) p T ( z 1 , z 2 , z 3 ) dz 1 dz 2 dz 3 , p T : valid asymptotic exp. of ( Z 1 , Z 2 , Z 3 ) � = f ( z 1 + Q T ( z 1 , z 2 , z 3 )) p T ( z 1 , z 2 , z 3 ) dz 1 dz 2 dz 3 S T = z 1 + Q T ( f ( z 1 ) + ∂f ( z 1 ) Q T + 1 � 2 ∂ 2 f ( z 1 ) Q 2 ≈ T ) p T ( z ) dz ⇐ (formal) Taylor’s expansion � � � � Q T p T dz 2 dz 3 + 1 � 2( − ∂ ) 2 = f ( z 1 ) ( − ∂ ) Q T p T dz 2 dz 3 dz 1 ⇐ IBP over R For the validity of p T , the regularity of the distribution of ( Z 1 , Z 2 , Z 3 ) ✷ non-degeneracy of covariance matrix ✷ Cram´ er type condition — non-degeneracy of the Malliavin covariance 12