Adaptive estimation of survival function in the convolution model on R + Gwenna¨ elle MABON CREST - ENSAE & Universit´ e Paris Descartes April, 20th 2016 G. MABON (CREST & MAP5) Survival function in the convolution model April, 20th 2016 1 / 15
Framework Motivation : additive processes Motivations : one-sided error in convolution models (a.k.a. additive measurement errors). → Application to back calculation problems in AIDS research − Groeneboom and Wellner (1992), van Es et al. (1998), Jongbloed (1998), Groeneboom and Jongbloed (2003). → Application in finance (nonparametric regression) − Jirak, Meister and Reiß (2014), Reiß & Selk (2015) . G. MABON (CREST & MAP5) Survival function in the convolution model April, 20th 2016 2 / 15
Framework Motivation : additive processes Motivations : one-sided error in convolution models (a.k.a. additive measurement errors). → Application to back calculation problems in AIDS research − Groeneboom and Wellner (1992), van Es et al. (1998), Jongbloed (1998), Groeneboom and Jongbloed (2003). → Application in finance (nonparametric regression) − Jirak, Meister and Reiß (2014), Reiß & Selk (2015) . G. MABON (CREST & MAP5) Survival function in the convolution model April, 20th 2016 2 / 15
Framework Motivation : additive processes Motivations : one-sided error in convolution models (a.k.a. additive measurement errors). → Application to back calculation problems in AIDS research − Groeneboom and Wellner (1992), van Es et al. (1998), Jongbloed (1998), Groeneboom and Jongbloed (2003). → Application in finance (nonparametric regression) − Jirak, Meister and Reiß (2014), Reiß & Selk (2015) . G. MABON (CREST & MAP5) Survival function in the convolution model April, 20th 2016 2 / 15
Statistical model � We study the following model: Z i = X i + Y i , i = 1 , . . . , n , (1) � X i ’s i.i.d. nonnegative variables with unknown density f , survival function S X . � Y i ’s i.i.d. nonnegative variables with known density g , survival function S Y . � ( X i ) i = ( Y i ) i , Z i ∼ h , survival function S Z . | Target : estimation of S X when the Z i ’s are observed and g is known. G. MABON (CREST & MAP5) Survival function in the convolution model April, 20th 2016 3 / 15
Statistical model Steps � Assumptions : S X , S Y and g belong to L 2 ( R + ). � Find an appropriate orthonormal basis of L 2 ( R + ) ,( ϕ k ) k ≥ 0 , � S X ( x ) = a k ( S X ) ϕ k ( x ) . k ≥ 0 a k ( S X ): k -th component of S X in the orthonormal basis. � Study the MISE of the estimator in this basis. S X , m � 2 ≤ ? E � S X − ˆ � Build a model selection procedure ` a la Birg´ e and Massart. m ∈M γ n (ˆ m = arg min � S X , m ) + pen ( m ) . G. MABON (CREST & MAP5) Survival function in the convolution model April, 20th 2016 4 / 15
Statistical model Steps � Assumptions : S X , S Y and g belong to L 2 ( R + ). � Find an appropriate orthonormal basis of L 2 ( R + ) ,( ϕ k ) k ≥ 0 , � S X ( x ) = a k ( S X ) ϕ k ( x ) . k ≥ 0 a k ( S X ): k -th component of S X in the orthonormal basis. � Study the MISE of the estimator in this basis. S X , m � 2 ≤ ? E � S X − ˆ � Build a model selection procedure ` a la Birg´ e and Massart. m ∈M γ n (ˆ m = arg min � S X , m ) + pen ( m ) . G. MABON (CREST & MAP5) Survival function in the convolution model April, 20th 2016 4 / 15
Statistical model Steps � Assumptions : S X , S Y and g belong to L 2 ( R + ). � Find an appropriate orthonormal basis of L 2 ( R + ) ,( ϕ k ) k ≥ 0 , � S X ( x ) = a k ( S X ) ϕ k ( x ) . k ≥ 0 a k ( S X ): k -th component of S X in the orthonormal basis. � Study the MISE of the estimator in this basis. S X , m � 2 ≤ ? E � S X − ˆ � Build a model selection procedure ` a la Birg´ e and Massart. m ∈M γ n (ˆ m = arg min � S X , m ) + pen ( m ) . G. MABON (CREST & MAP5) Survival function in the convolution model April, 20th 2016 4 / 15
Statistical model Steps � Assumptions : S X , S Y and g belong to L 2 ( R + ). � Find an appropriate orthonormal basis of L 2 ( R + ) ,( ϕ k ) k ≥ 0 , � S X ( x ) = a k ( S X ) ϕ k ( x ) . k ≥ 0 a k ( S X ): k -th component of S X in the orthonormal basis. � Study the MISE of the estimator in this basis. S X , m � 2 ≤ ? E � S X − ˆ � Build a model selection procedure ` a la Birg´ e and Massart. m ∈M γ n (ˆ m = arg min � S X , m ) + pen ( m ) . G. MABON (CREST & MAP5) Survival function in the convolution model April, 20th 2016 4 / 15
Survival function estimation Convolution equation Let z ≥ 0, by definition S Z ( z ) = P ( Z > z ), we get �� S Z ( z ) = P ( X + Y > z ) = 1 x + y > z f ( x ) 1 x ≥ 0 g ( y ) 1 y ≥ 0 d x d y � �� + ∞ � = f ( x ) d x g ( y ) 1 y ≥ 0 1 z − y ≥ 0 d y z − y � �� + ∞ � + f ( x ) d x g ( y ) 1 y ≥ 0 1 z − y ≤ 0 d y 0 � z = S X ( z − y ) g ( y ) d y + S Y ( z ) . 0 S Z ( z ) = S X ⋆ g ( z ) + S Y ( z ) (2) G. MABON (CREST & MAP5) Survival function in the convolution model April, 20th 2016 5 / 15
Survival function estimation Convolution equation Let z ≥ 0, by definition S Z ( z ) = P ( Z > z ), we get �� S Z ( z ) = P ( X + Y > z ) = 1 x + y > z f ( x ) 1 x ≥ 0 g ( y ) 1 y ≥ 0 d x d y � �� + ∞ � = f ( x ) d x g ( y ) 1 y ≥ 0 1 z − y ≥ 0 d y z − y � �� + ∞ � + f ( x ) d x g ( y ) 1 y ≥ 0 1 z − y ≤ 0 d y 0 � z = S X ( z − y ) g ( y ) d y + S Y ( z ) . 0 S Z ( z ) = S X ⋆ g ( z ) + S Y ( z ) (2) G. MABON (CREST & MAP5) Survival function in the convolution model April, 20th 2016 5 / 15
Survival function estimation Laguerre procedure � For R + -supported functions, the convolution product writes � z S X ⋆ g ( z ) = S X ( u ) g ( z − u ) d u 0 � z � ∞ � ∞ = a k ( S X ) a j ( g ) ϕ k ( u ) ϕ j ( z − u ) d u . 0 k =0 j =0 � We introduce the Laguerre basis defined for k ∈ N , x ≥ 0 , by � k � ( − x ) j � k √ 2 L k (2 x ) e − x ϕ k ( x ) = with L k ( x ) = . j j ! j =0 The ( ϕ k ) k ’s form an orthonormal basis of L 2 ( R + ). � What makes the Laguerre basis relevant is the relation � x ϕ k ( u ) ϕ j ( x − u ) d u = 2 − 1 / 2 ( ϕ k + j ( x ) − ϕ k + j +1 ( x )) . 0 G. MABON (CREST & MAP5) Survival function in the convolution model April, 20th 2016 6 / 15
Survival function estimation Laguerre procedure � For R + -supported functions, the convolution product writes � z S X ⋆ g ( z ) = S X ( u ) g ( z − u ) d u 0 � z � ∞ � ∞ = a k ( S X ) a j ( g ) ϕ k ( u ) ϕ j ( z − u ) d u . 0 k =0 j =0 � We introduce the Laguerre basis defined for k ∈ N , x ≥ 0 , by � k � ( − x ) j � k √ 2 L k (2 x ) e − x ϕ k ( x ) = with L k ( x ) = . j j ! j =0 The ( ϕ k ) k ’s form an orthonormal basis of L 2 ( R + ). � What makes the Laguerre basis relevant is the relation � x ϕ k ( u ) ϕ j ( x − u ) d u = 2 − 1 / 2 ( ϕ k + j ( x ) − ϕ k + j +1 ( x )) . 0 G. MABON (CREST & MAP5) Survival function in the convolution model April, 20th 2016 6 / 15
Survival function estimation Laguerre procedure � For R + -supported functions, the convolution product writes � z S X ⋆ g ( z ) = S X ( u ) g ( z − u ) d u 0 � z � ∞ � ∞ = a k ( S X ) a j ( g ) ϕ k ( u ) ϕ j ( z − u ) d u . 0 k =0 j =0 � We introduce the Laguerre basis defined for k ∈ N , x ≥ 0 , by � k � ( − x ) j � k √ 2 L k (2 x ) e − x ϕ k ( x ) = with L k ( x ) = . j j ! j =0 The ( ϕ k ) k ’s form an orthonormal basis of L 2 ( R + ). � What makes the Laguerre basis relevant is the relation � x ϕ k ( u ) ϕ j ( x − u ) d u = 2 − 1 / 2 ( ϕ k + j ( x ) − ϕ k + j +1 ( x )) . 0 G. MABON (CREST & MAP5) Survival function in the convolution model April, 20th 2016 6 / 15
Survival function estimation Laguerre procedure � It yields S X ⋆ g ( z ) � � � � ∞ k � � 1 = √ ϕ k ( z ) a k ( S X ) a 0 ( g ) + a k − l ( g ) − a k − l − 1 ( g ) a l ( S X ) . 2 k =0 l =0 � Equation implies (2) � S X ⋆ g ( z ) = S Z ( z ) − S Y ( z ) = ( a k ( S Z ) − a k ( S Y )) ϕ k ( z ) k ≥ 0 � We obtain for any m that G m � S X , m = � S Z , m − � S Y , m � S • , m = t ( a 0 ( S • ) , . . . , a m − 1 ( S • )). � G m is the lower triangular Toeplitz matrix with elements a 0 ( g ) if i = j , 1 G m = √ (3) a i − j ( g ) − a i − j − 1 ( g ) if j < i , 2 0 otherwise . G. MABON (CREST & MAP5) Survival function in the convolution model April, 20th 2016 7 / 15
Survival function estimation Laguerre procedure � It yields S X ⋆ g ( z ) � � � � ∞ k � � 1 = √ ϕ k ( z ) a k ( S X ) a 0 ( g ) + a k − l ( g ) − a k − l − 1 ( g ) a l ( S X ) . 2 k =0 l =0 � Equation implies (2) � S X ⋆ g ( z ) = S Z ( z ) − S Y ( z ) = ( a k ( S Z ) − a k ( S Y )) ϕ k ( z ) k ≥ 0 � We obtain for any m that G m � S X , m = � S Z , m − � S Y , m � S • , m = t ( a 0 ( S • ) , . . . , a m − 1 ( S • )). � G m is the lower triangular Toeplitz matrix with elements a 0 ( g ) if i = j , 1 G m = √ (3) a i − j ( g ) − a i − j − 1 ( g ) if j < i , 2 0 otherwise . G. MABON (CREST & MAP5) Survival function in the convolution model April, 20th 2016 7 / 15
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