Supere ffi cient estimation of the intensity of a stationary Poisson point process via the Stein method Marianne Clausel Joint work with Jean-Franc ¸ois Coeurjolly and J´ erˆ ome Lelong Laboratoire Jean Kuntzmann (LJK), Grenoble Alpes University
Back to the initial ideas of Charles Stein Considered as the father of di ff erent problems related to optimal estimation, stochastic calculus,.. .
Back to the initial ideas of Charles Stein Considered as the father of di ff erent problems related to optimal estimation, stochastic calculus,.. . Everything started in the following context Let X ∼ N ( θ, σ 2 I d ) where I d is the d -dimensional identity matrix. Objective: estimate θ based on a single (for simplicity) observation X .
Back to the initial ideas of Charles Stein Considered as the father of di ff erent problems related to optimal estimation, stochastic calculus,.. . Everything started in the following context Let X ∼ N ( θ, σ 2 I d ) where I d is the d -dimensional identity matrix. Objective: estimate θ based on a single (for simplicity) observation X . � θ − θ � 2 � θ mle = X minimizes MSE( � � � � θ ) = E among unbiased estimators.
Back to the initial ideas of Charles Stein Considered as the father of di ff erent problems related to optimal estimation, stochastic calculus,.. . Everything started in the following context Let X ∼ N ( θ, σ 2 I d ) where I d is the d -dimensional identity matrix. Objective: estimate θ based on a single (for simplicity) observation X . � θ − θ � 2 � θ mle = X minimizes MSE( � � � � θ ) = E among unbiased estimators. Stein (1956) � � � (1 − b ( a + X 2 i ) − 1 ) X i i = 1 ,..., d ⇒ MSE( � θ S ) ≤ MSE( � θ mle ) when d ≥ 3 θ S = James-Stein (1961) � θ JS = X (1 − ( d − 2) / � X � 2 ) ⇒ MSE( � θ JS ) ≤ MSE( � θ mle ) when d ≥ 3 θ = X + g ( X ), g : R d → R d . Stein (1981) key-ingredients for the class: �
θ = X + g ( X ) ( X ∼ N ( θ, σ 2 I d , σ 2 known) MSE of � MSE( � θ ) =
θ = X + g ( X ) ( X ∼ N ( θ, σ 2 I d , σ 2 known) MSE of � � d θ ) = E � X − θ � 2 + E � g ( X ) � 2 + 2 MSE( � E (( X i − θ i ) g i ( X )) i = 1
θ = X + g ( X ) ( X ∼ N ( θ, σ 2 I d , σ 2 known) MSE of � � d θ ) = E � X − θ � 2 + E � g ( X ) � 2 + 2 MSE( � E (( X i − θ i ) g i ( X )) i = 1 E[ Zh ( Z )] = E[ h ′ ( Z )], Z ∼ N (0 , 1) Using an IbP for Gaussian r.v. 1 � d θ mle ) + E � g ( X ) � 2 + 2 σ 2 MSE( � θ ) = MSE( � E ∇ g i ( X ) i = 1
θ = X + g ( X ) ( X ∼ N ( θ, σ 2 I d , σ 2 known) MSE of � � d θ ) = E � X − θ � 2 + E � g ( X ) � 2 + 2 MSE( � E (( X i − θ i ) g i ( X )) i = 1 E[ Zh ( Z )] = E[ h ′ ( Z )], Z ∼ N (0 , 1) Using an IbP for Gaussian r.v. 1 � d θ mle ) + E � g ( X ) � 2 + 2 σ 2 MSE( � θ ) = MSE( � E ∇ g i ( X ) i = 1 Now choose g = σ 2 ∇ log f . Use the well-known fact [based on 2 product and chain–rules ] that for h : R → R , (log( h ) ′ ) 2 + 2(log h ) ′′ =
θ = X + g ( X ) ( X ∼ N ( θ, σ 2 I d , σ 2 known) MSE of � � d θ ) = E � X − θ � 2 + E � g ( X ) � 2 + 2 MSE( � E (( X i − θ i ) g i ( X )) i = 1 E[ Zh ( Z )] = E[ h ′ ( Z )], Z ∼ N (0 , 1) Using an IbP for Gaussian r.v. 1 � d θ mle ) + E � g ( X ) � 2 + 2 σ 2 MSE( � θ ) = MSE( � E ∇ g i ( X ) i = 1 Now choose g = σ 2 ∇ log f . Use the well-known fact [based on 2 product and chain–rules ] that for h : R → R , √ h ) ′′ (log( h ) ′ ) 2 + 2(log h ) ′′ = 4 ( h . Get √ � � ∇∇ f ( X ) MSE( � θ ) = MSE( � ≤ MSE( � θ mle ) + 4 σ 2 E θ mle ) � if ∇∇ f ≤ 0 . f ( X )
θ = X + g ( X ) ( X ∼ N ( θ, σ 2 I d , σ 2 known) MSE of � � d θ ) = E � X − θ � 2 + E � g ( X ) � 2 + 2 MSE( � E (( X i − θ i ) g i ( X )) i = 1 E[ Zh ( Z )] = E[ h ′ ( Z )], Z ∼ N (0 , 1) Using an IbP for Gaussian r.v. 1 � d θ mle ) + E � g ( X ) � 2 + 2 σ 2 MSE( � θ ) = MSE( � E ∇ g i ( X ) i = 1 Now choose g = σ 2 ∇ log f . Use the well-known fact [based on 2 product and chain–rules ] that for h : R → R , √ h ) ′′ (log( h ) ′ ) 2 + 2(log h ) ′′ = 4 ( h . Get √ � � ∇∇ f ( X ) MSE( � θ ) = MSE( � ≤ MSE( � θ mle ) + 4 σ 2 E θ mle ) � if ∇∇ f ≤ 0 . f ( X ) Goal : mimic both steps, derive a Stein estimator for the intensity of a Poisson point process [extension Privault-R´ eveillac (2009), d = 1]
Point processes S : Polish state space of the point process (equipped with the σ -algebra of Borel sets B ). A configuration of points is denoted x = { x 1 , · · · , x n , · · · } . For B ⊂ S : x B = x ∩ B . N lf : space of locally finite configurations, i.e. { x , n ( x B ) = | x B | < ∞} , ∀ B bounded ∈ S } equipped with N lf = σ ( { x ∈ N lf , n ( x B ) = m } ; B ∈ B , B bounded , m ≥ 1). Definition A point process X defined on S is a measurable application defined on some probability space ( Ω , F , P ) with values on N lf . Measurability of X ⇐⇒ N ( B ) = | X B | is a r.v. for any bounded B ∈ B .
Poisson point processes Definition of a Poisson point process with intensity ρ ( · ) . ∀ m ≥ 1, ∀ bounded and disjoint B 1 , · · · , B m ⊂ S , the r.v. X B 1 , · · · , X B m are independent. � N ( B ) ∼ P ( B ρ ( u ) du ) ∀ B ⊂ S , ∀ F ∈ N lf � � � � � n e − B ρ ( u ) du P ( X B ∈ F ) = · · · 1( { ( x 1 , · · · , x n ) ∈ F } ) ρ ( x i ) dx i n ! B B n ≥ 0 i = 1 Notation : X ∼ Poisson( S , ρ ).
Our framework Case considered here ρ ( u ) ≡ θ . X homogeneous Poisson point process with intensity θ . We assume observing X on W ⊂ R d . Given N ( W ) = n , we denote X 1 , . . . , X n the n points in W . The MLE estimate of the intensity θ is � θ = N ( W ) / | W | . Construction of a Stein estimator of θ ?
Poisson functionals S : space of Poisson functionals F defined on Ω by � F = f 0 1 ( N ( W ) = 0) + 1 ( N ( W ) = n ) f n ( X 1 , . . . , X n ) , n ≥ 1 f 0 ∈ R , f n : W n → R d measurable symmetric functions called form functions of F .
Towards a Stein estimator (1) θ mle = N ( W ) / | W | . MLE is defined by � θ mle + Aim : define � θ of the form � θ = � 1 | W | ζ where ζ = ∇ log( F ) . Relation MSE( � θ ) < MSE( � θ mle ) satisfied? � 2 � θ mle + 1 MSE( � � θ ) = E | W |∇ log F − θ � � 1 = MSE( � θ mle ) + E[( ∇ log F ) 2 ] + 2E[( ∇ log F )( N ( W ) − θ | W | )] | W | 2 = ⇒ Need to transform 2E[ G ( N ( W ) − θ | W | )] with G = ∇ log F using a IbP formula. Notion of derivative ?
Malliavin derivatives (1) Di ff erential operator: let π : W 2 → R d � � n D π x F = − 1 ( N ( W ) = n ) ( ∇ x i f n )( X 1 , . . . , X n ) π ( X i , x ) , n ≥ 1 i = 1 where S ′ = � F ∈ S : ’the f n are cont. di ff . in any variable x i ’ � and where ∇ x i f n gradient of x i �→ f n ( . . . , x i , . . . ).
Malliavin derivatives (2) Lemma [product and chain rules] For any x ∈ W , for all F , G ∈ S ′ , g ∈ C 1 b ( R ) we have D π x ( FG ) = ( D π x F ) G + F ( D π D π x g ( F ) = g ′ ( F ) D π x G ) and x F .
Malliavin derivatives (2) Lemma [product and chain rules] For any x ∈ W , for all F , G ∈ S ′ , g ∈ C 1 b ( R ) we have D π x ( FG ) = ( D π x F ) G + F ( D π D π x g ( F ) = g ′ ( F ) D π x G ) and x F . To get an IbP type formula, we need to introduce Dom( D π ) of S ′ as � F ∈ S ′ : ∀ n ≥ 1 and z 1 , . . . , z n ∈ R d Dom( D π ) = � f n + 1 � � z n + 1 ∈ ∂ W ( z 1 , . . . , z n + 1 ) = f n ( z 1 , . . . , z n ) , f 1 � � � � z ∈ ∂ W ( z ) = 0 , (1) Remark: compatibility conditions important to derive a correct Stein estimator.
Integration by parts formula Theorem π ), V : R d → R , V ∈ C 1 ( W , R d ) Let G ∈ Dom( D �� � � � D π E x G · V ( x )d x = E F ∇ · V ( u ) − θ ∇ · V ( u )d u W W � ������������������ �� ������������������ � u ∈ X W : = ∇ π, V F � where V : W → R d is defined by V ( u ) = W V ( x ) π ( u , x )d x .
Integration by parts formula Theorem π ), V : R d → R , V ∈ C 1 ( W , R d ) Let G ∈ Dom( D �� � � � D π E x G · V ( x )d x = E F ∇ · V ( u ) − θ ∇ · V ( u )d u W W � ������������������ �� ������������������ � u ∈ X W : = ∇ π, V F � where V : W → R d is defined by V ( u ) = W V ( x ) π ( u , x )d x . Main application: let π ( u , x ) = u ⊤ V ( x ), we can find some V ( omit details ) such that V ( u ) = u / d and ∇ · V ( u ) = 1. Then � � n ∇ G = ∇ π, V G = − 1 1 ( N ( W ) = n ) ∇ x i g n ( X 1 , . . . , X n ) · X i d n ≥ 1 i = 1 ⇒ E[ ∇ G ] = E [ G ( N ( W ) − θ | W | )] .
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