A kernel estimator and associated confidence bands in the - PowerPoint PPT Presentation

A kernel estimator and associated confidence bands in the Spektor-Lord-Willis problem Bogdan Ćmiel, Zbigniew Szkutnik and Jakub Wojdyła Faculty of Applied Mathematics AGH University of Science & Technology, Kraków Be ¸dlewo, 2016 B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH) Confidence bands in the SLW problem Be ¸dlewo, 2016 1 / 16

Wicksell’s corpuscule problem and SLW problem balls of random radii randomly placed in an opaque 3-d medium B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH) Confidence bands in the SLW problem Be ¸dlewo, 2016 2 / 16

Wicksell’s corpuscule problem and SLW problem balls of random radii randomly placed in an opaque 3-d medium random 2-d planar or 1-d linear section through the medium unfold the distribution of spheres’ radii (stereological inverse problem) B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH) Confidence bands in the SLW problem Be ¸dlewo, 2016 2 / 16

Wicksell’s corpuscule problem and SLW problem balls of random radii randomly placed in an opaque 3-d medium random 2-d planar or 1-d linear section through the medium unfold the distribution of spheres’ radii (stereological inverse problem) ⇒ from measured planar sections radii: Wicksell’s problem (1925) B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH) Confidence bands in the SLW problem Be ¸dlewo, 2016 2 / 16

Wicksell’s corpuscule problem and SLW problem balls of random radii randomly placed in an opaque 3-d medium random 2-d planar or 1-d linear section through the medium unfold the distribution of spheres’ radii (stereological inverse problem) ⇒ from measured planar sections radii: Wicksell’s problem (1925) ⇒ from measured linear sections radii: Spektor-Lord-Willis problem (1950/51) B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH) Confidence bands in the SLW problem Be ¸dlewo, 2016 2 / 16

Applications medicine (microtumors in a tissue — original Wicksell’s motivation) B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH) Confidence bands in the SLW problem Be ¸dlewo, 2016 3 / 16

Applications medicine (microtumors in a tissue — original Wicksell’s motivation) geology (mineral deposits in a rock) B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH) Confidence bands in the SLW problem Be ¸dlewo, 2016 3 / 16

Applications medicine (microtumors in a tissue — original Wicksell’s motivation) geology (mineral deposits in a rock) metallurgy (graphite grains in spheroidal graphite cast iron) B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH) Confidence bands in the SLW problem Be ¸dlewo, 2016 3 / 16

Applications medicine (microtumors in a tissue — original Wicksell’s motivation) geology (mineral deposits in a rock) metallurgy (graphite grains in spheroidal graphite cast iron) B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH) Confidence bands in the SLW problem Be ¸dlewo, 2016 3 / 16

SLW problem — Poisson process framework homogeneous Poisson process of ball centers (”low” intensity c ) B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH) Confidence bands in the SLW problem Be ¸dlewo, 2016 4 / 16

SLW problem — Poisson process framework homogeneous Poisson process of ball centers (”low” intensity c ) balls’ radii density ρ on [ 0 ; 1 ] and f := c ρ B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH) Confidence bands in the SLW problem Be ¸dlewo, 2016 4 / 16

SLW problem — Poisson process framework homogeneous Poisson process of ball centers (”low” intensity c ) balls’ radii density ρ on [ 0 ; 1 ] and f := c ρ n - known size of the experiment (related to line probe length) B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH) Confidence bands in the SLW problem Be ¸dlewo, 2016 4 / 16

SLW problem — Poisson process framework homogeneous Poisson process of ball centers (”low” intensity c ) balls’ radii density ρ on [ 0 ; 1 ] and f := c ρ n - known size of the experiment (related to line probe length) X i - observed ”radii” (half-lengths) of intersections B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH) Confidence bands in the SLW problem Be ¸dlewo, 2016 4 / 16

SLW problem — Poisson process framework homogeneous Poisson process of ball centers (”low” intensity c ) balls’ radii density ρ on [ 0 ; 1 ] and f := c ρ n - known size of the experiment (related to line probe length) X i - observed ”radii” (half-lengths) of intersections then X 2 1 , . . . , X 2 N ( n ) form a Poisson process on [ 0 ; 1 ] with intensity ng � 1 g ( u ) = f ( x ) dx =: ( Gf )( u ) √ u B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH) Confidence bands in the SLW problem Be ¸dlewo, 2016 4 / 16

SLW problem — Poisson process framework homogeneous Poisson process of ball centers (”low” intensity c ) balls’ radii density ρ on [ 0 ; 1 ] and f := c ρ n - known size of the experiment (related to line probe length) X i - observed ”radii” (half-lengths) of intersections then X 2 1 , . . . , X 2 N ( n ) form a Poisson process on [ 0 ; 1 ] with intensity ng � 1 g ( u ) = f ( x ) dx =: ( Gf )( u ) √ u � 1 N ( n ) - Poissonian rv, E N ( n ) = n 0 g ( u ) du B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH) Confidence bands in the SLW problem Be ¸dlewo, 2016 4 / 16

SLW problem — Poisson process framework homogeneous Poisson process of ball centers (”low” intensity c ) balls’ radii density ρ on [ 0 ; 1 ] and f := c ρ n - known size of the experiment (related to line probe length) X i - observed ”radii” (half-lengths) of intersections then X 2 1 , . . . , X 2 N ( n ) form a Poisson process on [ 0 ; 1 ] with intensity ng � 1 g ( u ) = f ( x ) dx =: ( Gf )( u ) √ u � 1 N ( n ) - Poissonian rv, E N ( n ) = n 0 g ( u ) du f ( x ) = ( G − 1 g )( x ) = − 2 xg ′ ( x 2 ) - ill-posed inverse problem Goal: Confidence bands for f B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH) Confidence bands in the SLW problem Be ¸dlewo, 2016 4 / 16

Some existing constructions of asymptotic confidence bands In direct problems: many results starting from seminal Bickel & Rosenblatt (AS, 1973) — for standard kernel density estimator B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH) Confidence bands in the SLW problem Be ¸dlewo, 2016 5 / 16

Some existing constructions of asymptotic confidence bands In direct problems: many results starting from seminal Bickel & Rosenblatt (AS, 1973) — for standard kernel density estimator In inverse problems: Bissantz et al. (JRSS, 2007), Bissantz & Holzmann (IP, 2008) — smooth density deconvolution B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH) Confidence bands in the SLW problem Be ¸dlewo, 2016 5 / 16

Some existing constructions of asymptotic confidence bands In direct problems: many results starting from seminal Bickel & Rosenblatt (AS, 1973) — for standard kernel density estimator In inverse problems: Bissantz et al. (JRSS, 2007), Bissantz & Holzmann (IP, 2008) — smooth density deconvolution Birke et al. (IP, 2010) — univariate inverse regression with convolution operator Lounici & Nickl (AS, 2011) — deconvolution Delaigle et al. (JRSS, 2015) — nonparametric regression with errors in explanatory variables Proksch et al. (Bernoulli, 2015) — multivariate inverse regression with convolution operator B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH) Confidence bands in the SLW problem Be ¸dlewo, 2016 5 / 16

Some existing constructions of asymptotic confidence bands In direct problems: many results starting from seminal Bickel & Rosenblatt (AS, 1973) — for standard kernel density estimator In inverse problems: Bissantz et al. (JRSS, 2007), Bissantz & Holzmann (IP, 2008) — smooth density deconvolution Birke et al. (IP, 2010) — univariate inverse regression with convolution operator Lounici & Nickl (AS, 2011) — deconvolution Delaigle et al. (JRSS, 2015) — nonparametric regression with errors in explanatory variables Proksch et al. (Bernoulli, 2015) — multivariate inverse regression with convolution operator In stereological inverse problems: Wojdyła, Szkutnik (SS, 2017) — Wicksell’s problem B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH) Confidence bands in the SLW problem Be ¸dlewo, 2016 5 / 16

SLW problem — the ”central” estimator idea: apply G − 1 to a kernel estimator of the intensity g B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH) Confidence bands in the SLW problem Be ¸dlewo, 2016 6 / 16

SLW problem — the ”central” estimator idea: apply G − 1 to a kernel estimator of the intensity g � f > 0 = problem: g ( 0 ) = ⇒ g discontinuous at zero, boundary effects B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH) Confidence bands in the SLW problem Be ¸dlewo, 2016 6 / 16

SLW problem — the ”central” estimator idea: apply G − 1 to a kernel estimator of the intensity g � f > 0 = problem: g ( 0 ) = ⇒ g discontinuous at zero, boundary effects remedy: reflection device N ( n ) x 2 − X 2 x 2 + X 2 � � � � �� f n ( x ) = − 2 x ˆ � i i K ′ + K ′ 1 [ 0 ; 1 ] ( x ) nh 2 h h i = 1 B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH) Confidence bands in the SLW problem Be ¸dlewo, 2016 6 / 16