Semiparametric Estimation Theory for Discretely Observed L´ evy Processes Chris A.J. Klaassen Enno Veerman Korteweg-de Vries Institute for Mathematics University of Amsterdam EURANDOM August 29, 2011
Basics Semiparametrics Efficient Estimation for Discretely Observed L´ evy Processes Further comments Discretely observed L´ evy processes Let { Y t : t ≥ 0 } be a L´ evy process; sample paths are c` adl` ag; stationary independent increments. Observe this process at times t = 0 , 1 , 2 , . . . and base inference on X i = Y i − Y i − 1 , i = 1 , . . . , n . Since { Y t : t ≥ 0 } is a L´ evy process, the observations X 1 , . . . , X n are i.i.d. with infinitely divisible distribution. Chris A.J. Klaassen Enno Veerman Semiparametric Estimation Theory for Discretely Observed L´ evy
Basics Semiparametrics Efficient Estimation for Discretely Observed L´ evy Processes Further comments Discretely observed L´ evy processes Infinitely divisible The observations X 1 , . . . , X n are i.i.d. with infinitely divisible distribution P µ,σ,ν and characteristic function � i µ t − 1 � � � � e itX � 2 σ 2 t 2 + e itx − 1 − itx 1 [ | x | < 1] � E = exp d ν ( x ) , where µ ∈ R , σ ≥ 0 , and the L´ evy measure ν ( · ) is a measure on R \ { 0 } satisfying � [ x 2 ∧ 1] d ν ( x ) < ∞ . Chris A.J. Klaassen Enno Veerman Semiparametric Estimation Theory for Discretely Observed L´ evy
Basics Semiparametrics Efficient Estimation for Discretely Observed L´ evy Processes Further comments Discretely observed L´ evy processes Infinitely divisible The observations X 1 , . . . , X n are i.i.d. with infinitely divisible distribution in P = { P µ,σ,ν : µ ∈ R , σ ≥ 0 , ν ( · ) L´ evy measure } . P defines a semiparametric model with µ and σ as Euclidean parameters, and ν ( · ) as Banach parameter. Parameter of interest θ : P → R k Chris A.J. Klaassen Enno Veerman Semiparametric Estimation Theory for Discretely Observed L´ evy
Basics Semiparametrics Efficient Estimation for Discretely Observed L´ evy Processes Further comments Outline 1 Basics Semiparametrics 2 Efficient Estimation for Discretely Observed L´ evy Processes 3 Further comments Chris A.J. Klaassen Enno Veerman Semiparametric Estimation Theory for Discretely Observed L´ evy
Basics Semiparametrics Efficient Estimation for Discretely Observed L´ evy Processes Further comments Outline 1 Basics Semiparametrics 2 Efficient Estimation for Discretely Observed L´ evy Processes 3 Further comments Chris A.J. Klaassen Enno Veerman Semiparametric Estimation Theory for Discretely Observed L´ evy
Basics Semiparametrics Efficient Estimation for Discretely Observed L´ evy Processes Further comments Crash Course Semiparametrically Efficient Estimation 1 Asymptotic bound on performance of estimators in a regular parametric model (Local Asymptotic Normality): H´ ajek-LeCam Convolution Theorem Local Asymptotic Minimax Theorem Local Asymptotic Spread Theorem 2 Regular parametric submodels of semiparametric model 3 Least favorable parametric submodel ⇒ semiparametric bound Techniques to obtain semiparam. efficient influence function: Projection of influence function on tangent space Projection of score function on subspace of tangent space determined by nuisance parameters 4 Construction of estimator attaining bounds; i.e., of estimator that is asymptotically linear in the efficient influence function Chris A.J. Klaassen Enno Veerman Semiparametric Estimation Theory for Discretely Observed L´ evy
Basics Semiparametrics Efficient Estimation for Discretely Observed L´ evy Processes Further comments H´ ajek-LeCam Convolution Theorem In a regular parametric model one has Local Asymptotic Normality n n � p ( X i ; θ n ) � ℓ θ 0 ( X i ) − 1 h � � ˙ 2 h T I ( θ 0 ) h + o P (1) √ n log = p ( X i ; θ 0 ) i =1 i =1 under θ 0 with θ n = θ 0 + h / √ n , where ˙ ℓ θ 0 ( · ) is the score function. Convolution theorem; under LAN ∀ h √ n ( T n − q ( θ n )) D � � q ( θ 0 ) I − 1 ( θ 0 )˙ q T ( θ 0 ) → θ n L ⇒ L = N 0 , ˙ ∗ M � q ( θ 0 ) I − 1 ( θ 0 )˙ q T ( θ 0 ) � and L = N 0 , ˙ iff n � � �� √ n q ( θ 0 ) + 1 P � q ( θ 0 ) I − 1 ( θ 0 ) ˙ T n − ˙ ℓ θ 0 ( X i ) → θ 0 0 n i =1 Chris A.J. Klaassen Enno Veerman Semiparametric Estimation Theory for Discretely Observed L´ evy
Basics Semiparametrics Efficient Estimation for Discretely Observed L´ evy Processes Further comments H´ ajek-LeCam Convolution Theorem Efficiency ( T n ) is called (asymptotically) efficient iff n � � �� √ n q ( θ 0 ) + 1 P � q ( θ 0 ) I − 1 ( θ 0 ) ˙ T n − → θ 0 0 ˙ ℓ θ 0 ( X i ) n i =1 Taking q ( θ ) = ( I , 0) θ one can study efficiency in presence of nuisance parameters. Taking regular parametric submodels of semiparametric models one can study efficiency in presence of infinite-dimensional nuisance parameters; try to get q ( θ 0 ) I − 1 ( θ 0 )˙ q T ( θ 0 ) as large as possible. ˙ Chris A.J. Klaassen Enno Veerman Semiparametric Estimation Theory for Discretely Observed L´ evy
Basics Semiparametrics Efficient Estimation for Discretely Observed L´ evy Processes Further comments Geometric Interpretation Efficiency ( T n ) is called (asymptotically) efficient iff � � n �� √ n q ( θ 0 ) + 1 P � ˜ T n − ℓ ( X i ) → θ 0 0 n i =1 with the efficient influence function being ˜ q ( θ 0 ) I − 1 ( θ 0 ) ˙ ℓ ( · ) = ˙ ℓ θ 0 ( · ) ℓ ∈ [ ˙ ˜ ℓ ] = ˙ P ⊂ L 0 ℓ = ˙ ˙ E P 0 ˙ 2 ( P 0 ) , P 0 � θ 0 , ℓ θ 0 , ℓ = 0 The closed linear span of the components of ˙ ℓ (stemming from all regular parametric submodels) is denoted by [ ˙ ℓ ] = ˙ P and is called the tangent space of P at P 0 . Chris A.J. Klaassen Enno Veerman Semiparametric Estimation Theory for Discretely Observed L´ evy
Basics Semiparametrics Efficient Estimation for Discretely Observed L´ evy Processes Further comments Geometric Interpretation Efficiency and linearity ( T n ) is called (asymptotically) linear iff � � n �� √ n q ( θ 0 ) + 1 P � T n − ψ ( X i ) → θ 0 0 n i =1 with ψ ( · ) the influence function . ( T n ) is called (asymptotically) efficient iff ψ = ˜ q ( θ 0 ) I − 1 ( θ 0 ) ˙ ℓ = ˙ ℓ θ 0 the efficient influence function. ( θ ( P ) � q ( θ ) pathwise diff.) For any model P with tangent space ˙ P at P 0 , and ∀ ψ Theorem � � � � ψ − ˜ ℓ ⊥ ˙ ˜ � ˙ P or ℓ = ψ P � Chris A.J. Klaassen Enno Veerman Semiparametric Estimation Theory for Discretely Observed L´ evy
Basics Semiparametrics Efficient Estimation for Discretely Observed L´ evy Processes Further comments Geometric Interpretation Efficient influence function and tangent space ˜ ℓ ∈ [ ˙ ℓ ] = ˙ P ⊂ L 0 2 ( P 0 ) Let P be a nonparametric, semiparametric, or parametric model. Let P 0 ∈ P and let ˜ ℓ ∈ ˙ P be the corresponding efficient influence function. Let P s be a submodel, parametric or not, with P 0 ∈ P s , and let ˜ ℓ s ∈ ˙ P s denote the corresponding efficient influence function. Geometry P s ⊂ ˙ ˙ P 0 ∈ P s ⊂ P , P Chris A.J. Klaassen Enno Veerman Semiparametric Estimation Theory for Discretely Observed L´ evy
Basics Semiparametrics Efficient Estimation for Discretely Observed L´ evy Processes Further comments Geometric Interpretation Projection efficient influence functions ˜ ˜ ℓ s ∈ ˙ P s ⊂ ˙ ℓ ∈ ˙ P ⊂ L 0 P 0 ∈ P s ⊂ P , P , 2 ( P 0 ) Theorem � � � � ˜ ˜ � ˙ P s ℓ s = ℓ � Proof From the preceding Theorem we know � � � � ˜ � ˙ ∀ ψ ℓ = ψ P � and hence in view of ˙ P s ⊂ ˙ P � � � � � �� � � � � � � � � � ˜ � ˙ � ˙ � ˙ ˜ � ˙ P s P P s P s ℓ s = ψ = ψ = ℓ � � � � � Chris A.J. Klaassen Enno Veerman Semiparametric Estimation Theory for Discretely Observed L´ evy
Basics Semiparametrics Efficient Estimation for Discretely Observed L´ evy Processes Further comments Geometric Interpretation Projection efficient influence functions ˜ ℓ s ∈ ˙ P s ⊂ ˙ ℓ ∈ ˙ ˜ P ⊂ L 0 P 0 ∈ P s ⊂ P , P , 2 ( P 0 ) Theorem � � � � ˜ ˜ � ˙ P s ℓ s = ℓ � Increments L´ evy process P 0 some infinitely divisible distribution P s all infinitely divisible distributions P all distributions � g dP , F − 1 θ : P → R k , θ ( P ) = P ( u ) Chris A.J. Klaassen Enno Veerman Semiparametric Estimation Theory for Discretely Observed L´ evy
Basics Semiparametrics Efficient Estimation for Discretely Observed L´ evy Processes Further comments Geometric Interpretation Nonparametric tangent space P 0 ∈ P , all distributions. Lemma P = L 0 ˙ 2 ( P 0 ) Proof Let h ∈ L 0 2 ( P 0 ) , and choose χ : R → (0 , 2) , χ (0) = χ ′ (0) = 1 , 0 < χ ′ /χ < 2 . E.g. χ ( x ) = 2 / (1 + e − x ) . η �→ dP η χ ( η h ( · )) ( · ) = � dP 0 χ ( η h ( x )) dP 0 ( x ) defines a regular parametric submodel with score function η =0 = χ ′ χ ′ ( η h ) h dP 0 � � � ˙ ℓ η ( x ) χ ( η h ( x )) h ( x ) − η =0 = h ( x ) . � � � � χ ( η h ) dP 0 � � Chris A.J. Klaassen Enno Veerman Semiparametric Estimation Theory for Discretely Observed L´ evy
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