Nonparametric estimation in a multiplicative noise model Charlotte - PowerPoint PPT Presentation

Introduction Density estimation strategy Nonparametric estimation in a multiplicative noise model Charlotte Dion (1) , (2) Joint work with Fabienne Comte (2) (1) LJK, UMR CNRS 5224, Université Grenoble Alpes, Grenoble (2) MAP5, UMR CNRS 8145, Université Paris Descartes, Paris Cité Lundi 18 avril 2016 Charlotte Dion JPS 18/04/2016 Les Houches 1 / 22

Introduction Density estimation strategy Motivation: the model Nonnegative random variable X height, weight... time between the first symptom of a disease and the death of the patient → survival data. Interest: nonparametric estimation of the density function f � ∞ the survival function F ( x ) = P ( X > x ) = f ( u ) du � x ( E [ X ] = F , E [ h ( X )] ). Charlotte Dion JPS 18/04/2016 Les Houches 1 / 22

Introduction Density estimation strategy Motivation: the model Classical noise model Y i = X i + ε i , i = 1 , . . . , n with E [ ε i ] = 0 . But often the noise depends on the level of the signal: Y i = X i + αX i ε i , α ∈ R , Y i = X i (1 + αε i ) � �� � ⇓ Y i = X i U i , i = 1 , . . . , n, U i ∼ U [1 − a, 1+ a ] , 0 < a < 1 with E [ U i ] = 1 . Charlotte Dion JPS 18/04/2016 Les Houches 2 / 22

Introduction Density estimation strategy Motivation: the model Classical noise model Y i = X i + ε i , i = 1 , . . . , n with E [ ε i ] = 0 . But often the noise depends on the level of the signal: Y i = X i + αX i ε i , α ∈ R , Y i = X i (1 + αε i ) � �� � ⇓ Y i = X i U i , i = 1 , . . . , n, U i ∼ U [1 − a, 1+ a ] , 0 < a < 1 with E [ U i ] = 1 . → What does it represent? A partial transmission of the information X i up to an error of order ± 100 a % : unintentionally during a survey Charlotte Dion JPS 18/04/2016 Les Houches 2 / 22

Introduction Density estimation strategy What does it represent? deliberately to mask some data. Figure : X , Y , X vs Y , a = 0 . 5 Charlotte Dion JPS 18/04/2016 Les Houches 3 / 22

Introduction Density estimation strategy Motivation: literature case U i ∼ U [0 , 1] Vardi (1989), Vardi and Zhang (1992) asymptotic framework Link with deconvolution method. If ε ∼ E (1) , exp( − ε ) ∼ U [0 , 1] . The density function of a nonnegative random variable Y is decreasing ⇔ Y = XU with U ∼ U [0 , 1] independent of X . Charlotte Dion JPS 18/04/2016 Les Houches 4 / 22

Introduction Density estimation strategy Motivation: literature case U i ∼ U [0 , 1] Vardi (1989), Vardi and Zhang (1992) asymptotic framework Link with deconvolution method. If ε ∼ E (1) , exp( − ε ) ∼ U [0 , 1] . The density function of a nonnegative random variable Y is decreasing ⇔ Y = XU with U ∼ U [0 , 1] independent of X . Asgharian et. al. (2012) asymptotic nonparametric estimation of F . Brunel et. al. (2015) (non-asymptotic) adaptive estimator of f and of survival function F , optimal rates of convergence. Charlotte Dion JPS 18/04/2016 Les Houches 4 / 22

Introduction Density estimation strategy Model Y i = X i U i , i = 1 , . . . , n, U i ∼ U [1 − a, 1+ a ] , 0 < a < 1 the ( X i ) { i =1 ,...,n } and ( U i ) { i =1 ,...,n } are independent the X i are i.i.d. with density f the U i are i.i.d. with density U [1 − a, 1+ a ] , a known the Y i are observed, i.i.d. with density f Y on R + . Issues: how can we estimate the density f and the associated survival function F ? Charlotte Dion JPS 18/04/2016 Les Houches 5 / 22

Introduction Density estimation strategy Notations � ∞ L 2 ( R + ) = { t : R + → R , | t ( x ) | 2 dx < ∞} 0 � + ∞ and the associated scalar product � t, v � = t ( x ) v ( x ) dx and norm � 0 � t � 2 = R + | t ( x ) | 2 dx . If t is bounded: � t � ∞ = sup x ∈ R + | t ( x ) | . Assumption f ∈ L 2 ( R + ) Charlotte Dion JPS 18/04/2016 Les Houches 6 / 22

Procedure Introduction First step Density estimation strategy Second step Density f Y � y f Y ( y ) = 1 f ( x ) 1 − a dx, y ∈ ]0 , + ∞ [ , 2 a x y 1+ a If � f � ∞ < + ∞ , then � f Y � ∞ < ∞ yf Y ( y ) → y → 0 0 and yf Y ( y ) y → + ∞ 0 → Charlotte Dion JPS 18/04/2016 Les Houches 7 / 22

Procedure Introduction First step Density estimation strategy Second step Auxiliary function For a bounded t , derivable and t ′ ∈ L 2 ( R + ) , � + ∞ � � � � �� 1 y y E [ t ( Y 1 ) + Y 1 t ′ ( Y 1 )] = t ( y ) f − f dy 2 a 1 + a 1 − a 0 = � t, g � . � � � � �� g ( x ) := 1 x x , x ∈ R + f − f 2 a 1 + a 1 − a ψ t ( y ) := t ( y ) + yt ′ ( y ) . E [ ψ t ( Y 1 )] = � t, g � with Charlotte Dion JPS 18/04/2016 Les Houches 8 / 22

Procedure Introduction First step Density estimation strategy Second step Auxiliary function For a bounded t , derivable and t ′ ∈ L 2 ( R + ) , � + ∞ � � � � �� 1 y y E [ t ( Y 1 ) + Y 1 t ′ ( Y 1 )] = t ( y ) f − f dy 2 a 1 + a 1 − a 0 = � t, g � . � � � � �� g ( x ) := 1 x x , x ∈ R + f − f 2 a 1 + a 1 − a ψ t ( y ) := t ( y ) + yt ′ ( y ) . E [ ψ t ( Y 1 )] = � t, g � with → Strategy (different from U ∼ U [0 , 1] ) : Build a projection estimator of g . Look for an inversion formula to get f . Charlotte Dion JPS 18/04/2016 Les Houches 8 / 22

Procedure Introduction First step Density estimation strategy Second step � � � � x x f − f = 2 ag ( x ) 1 + a 1 − a Charlotte Dion JPS 18/04/2016 Les Houches 9 / 22

Procedure Introduction First step Density estimation strategy Second step � � � � x x f − f = 2 ag ( x ) 1 + a 1 − a � 1 + a � f ( x ) − f 1 − ax = 2 ag ((1 + a ) x ) Charlotte Dion JPS 18/04/2016 Les Houches 9 / 22

Procedure Introduction First step Density estimation strategy Second step � � � � x x f − f = 2 ag ( x ) 1 + a 1 − a � 1 + a � f ( x ) − f 1 − ax = 2 ag ((1 + a ) x ) �� 1 + a � � 1 + a � � 2 � 1 + a � f 1 − ax − f x = 2 ag 1 − a (1 + a ) x 1 − a . . . �� 1 + a � �� 1 + a � �� 1 + a � � N − 1 � N � N − 1 f x − f x = 2 ag (1 + a ) x 1 − a 1 − a 1 − a Charlotte Dion JPS 18/04/2016 Les Houches 9 / 22

Procedure Introduction First step Density estimation strategy Second step � � � � x x f − f = 2 ag ( x ) 1 + a 1 − a � 1 + a � f ( x ) − f 1 − ax = 2 ag ((1 + a ) x ) �� 1 + a � � 1 + a � � 2 � 1 + a � f 1 − ax − f x = 2 ag 1 − a (1 + a ) x 1 − a . . . �� 1 + a � �� 1 + a � �� 1 + a � � N − 1 � N � N − 1 f x − f x = 2 ag (1 + a ) x 1 − a 1 − a 1 − a �� 1 + a � �� 1 + a � � N � k N − 1 � f ( x ) − f x = 2 a g (1 + a ) x 1 − a 1 − a k =0 Charlotte Dion JPS 18/04/2016 Les Houches 9 / 22

Procedure Introduction First step Density estimation strategy Second step � � � � x x f − f = 2 ag ( x ) 1 + a 1 − a � 1 + a � f ( x ) − f 1 − ax = 2 ag ((1 + a ) x ) �� 1 + a � � 1 + a � � 2 � 1 + a � f 1 − ax − f x = 2 ag 1 − a (1 + a ) x 1 − a . . . �� 1 + a � �� 1 + a � �� 1 + a � � N − 1 � N � N − 1 f x − f x = 2 ag (1 + a ) x 1 − a 1 − a 1 − a �� 1 + a � �� 1 + a � � N � k N − 1 � f ( x ) − f x = 2 a g (1 + a ) x 1 − a 1 − a k =0 �� 1 + a � � k N − 1 � f N ( x ) := 2 a g (1 + a ) x 1 − a k =0 Charlotte Dion JPS 18/04/2016 Les Houches 9 / 22

Procedure Introduction First step Density estimation strategy Second step Projection f ( x ) − f N ( x ) = f (((1 + a ) / (1 − a )) N x ) , gives � f − f N � N →∞ 0 . → Notice that f ∈ L 2 ( R + ) ⇒ g ∈ L 2 ( R + ) . Orthonormal basis of L 2 ( R + ) : ( ϕ j ) j ∈ N , � ∞ g ( x ) = a j ( g ) ϕ j ( x ) , with a j ( g ) = � ϕ j , g � . j =0 For m ∈ M n ⊂ N , m − 1 � g m := a j ( g ) ϕ j projection S m = Vect { ϕ 0 , ϕ 1 , . . . , ϕ m − 1 } j =0 with a j ( g ) = � ϕ j , g � = E [ ϕ j ( Y 1 ) + Y 1 ϕ ′ j ( Y 1 )] = E [ ψ ϕ j ( Y 1 )] . Charlotte Dion JPS 18/04/2016 Les Houches 10 / 22

Procedure Introduction First step Density estimation strategy Second step Estimator of g and f m − 1 � � n � n a j = 1 [ Y i ϕ ′ j ( Y i ) + ϕ j ( Y i )] = n − 1 g m = � a j ϕ j , � � ψ ϕ j ( Y i ) . n j =0 i =1 i =1 Charlotte Dion JPS 18/04/2016 Les Houches 11 / 22

Procedure Introduction First step Density estimation strategy Second step Estimator of g and f m − 1 � � n � n a j = 1 [ Y i ϕ ′ j ( Y i ) + ϕ j ( Y i )] = n − 1 g m = � � a j ϕ j , � ψ ϕ j ( Y i ) . n j =0 i =1 i =1 Then, as: �� 1 + a � � k N − 1 � f N ( x ) = 2 a g (1 + a ) x 1 − a k =0 it comes �� 1 + a � � k N − 1 � � f N,m ( x ) = 2 a � g m (1 + a ) x . 1 − a k =0 Charlotte Dion JPS 18/04/2016 Les Houches 11 / 22