Deconvolution for an atomic distribution Shota Gugushvili Peter Spreij Bert van Es Universiteit van Amsterdam Stochastic processes: theory and applications A conference in honor of the 65th birthday of Wolfgang J. Runggaldier Bressanone, July 16 - 20, 2007
Something completely different Introduction Estimation procedure Conditions Results Outline 1 Something completely different 2 Introduction 3 Estimation procedure when p is known with unknown p 4 Conditions 5 Results Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution
Something completely different Introduction Estimation procedure Conditions Results Outline 1 Something completely different 2 Introduction 3 Estimation procedure when p is known with unknown p 4 Conditions 5 Results Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution
Something completely different Introduction Estimation procedure Conditions Results Conference dinner at the 23rd EMS, Madeira, 2001 Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution
Something completely different Introduction Estimation procedure Conditions Results Hiking, Porto Santo, 2001 Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution
Something completely different Introduction Estimation procedure Conditions Results Swimming, Porto Santo, 2001 Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution
Something completely different Introduction Estimation procedure Conditions Results The ”better” picture, Porto Santo, 2001 Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution
Something completely different Introduction Estimation procedure Conditions Results Outline 1 Something completely different 2 Introduction 3 Estimation procedure when p is known with unknown p 4 Conditions 5 Results Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution
Something completely different Introduction Estimation procedure Conditions Results Filtering The problem Given X = Y + Z observation = signal + noise , find characteristics of Y given X Recursive filtering Update the conditional distribution (expectation) of Y t given X 0 , . . . X t . Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution
Something completely different Introduction Estimation procedure Conditions Results Filtering The problem Given X = Y + Z observation = signal + noise , find characteristics of Y given X Recursive filtering Update the conditional distribution (expectation) of Y t given X 0 , . . . X t . Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution
Something completely different Introduction Estimation procedure Conditions Results Deconvolution The rough problem Given X i = Y i + Z i observation = signal + noise , estimate the distribution of the Y i given X 1 , . . . , X n . Basic assumptions The Y i are iid , the Z i are iid and the Y i are independent from the Z i . Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution
Something completely different Introduction Estimation procedure Conditions Results Deconvolution The rough problem Given X i = Y i + Z i observation = signal + noise , estimate the distribution of the Y i given X 1 , . . . , X n . Basic assumptions The Y i are iid , the Z i are iid and the Y i are independent from the Z i . Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution
Something completely different Introduction Estimation procedure Conditions Results Deconvolution - model assumptions Classical case The Y i have a density f and the noise variables Z i have a known distribution (standard normal). Non-classical case The Y i are distributed according to Y = UV , where U is Bernoulli with P ( U = 0) = p , V has a density f , and U and V are independent. Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution
Something completely different Introduction Estimation procedure Conditions Results Deconvolution - model assumptions Classical case The Y i have a density f and the noise variables Z i have a known distribution (standard normal). Non-classical case The Y i are distributed according to Y = UV , where U is Bernoulli with P ( U = 0) = p , V has a density f , and U and V are independent. Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution
Something completely different Introduction Estimation procedure Conditions Results Estimation problem Aim Estimate p f (infinite dimensional parameter) based on the observations X 1 , . . . , X n . Nonparametric tools Kernel smoothing Fourier inversion Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution
Something completely different Introduction Estimation procedure Conditions Results Estimation problem Aim Estimate p f (infinite dimensional parameter) based on the observations X 1 , . . . , X n . Nonparametric tools Kernel smoothing Fourier inversion Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution
Something completely different Introduction Estimation procedure Conditions Results Estimation problem Aim Estimate p f (infinite dimensional parameter) based on the observations X 1 , . . . , X n . Nonparametric tools Kernel smoothing Fourier inversion Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution
Something completely different Introduction Estimation procedure Conditions Results Motivation Let y t = x t + z t , where x is a compound Poisson process ( x t = � N t k =1 V i ) and z an independent Brownian motion ( y is a L´ evy process). Assume that y is observed at times 1 , 2 , . . . . Let Y i = y i − y i − 1 , then the Y i are of the above type. Related problem Estimate the common density of the V i . Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution
Something completely different Introduction when p is known Estimation procedure with unknown p Conditions Results Outline 1 Something completely different 2 Introduction 3 Estimation procedure when p is known with unknown p 4 Conditions 5 Results Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution
Something completely different Introduction when p is known Estimation procedure with unknown p Conditions Results Characteristic functions For φ X and φ f , the ch.f.’s of X and V , respectively, one has φ X ( t ) = [ p + (1 − p ) φ f ( t )] e − t 2 / 2 , Assuming that φ f is integrable, we have Inversion formula � ∞ e − itx φ X ( t ) − pe − t 2 / 2 f ( x ) = 1 (1 − p ) e − t 2 / 2 dt . 2 π −∞ Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution
Something completely different Introduction when p is known Estimation procedure with unknown p Conditions Results Empirical c.f. and kernel Basic idea Replace in the inversion formula φ X by its empirical counterpart and apply some smoothing. By φ emp we denote the empirical characteristic function, n φ emp ( t ) = 1 � e itX j . n j =1 We also use w , a kernel function with compact support [ − 1 , +1], and h a bandwidth. Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution
Something completely different Introduction when p is known Estimation procedure with unknown p Conditions Results Estimator The basic idea results in Kernel type estimator for f � ∞ e − itx φ emp ( t ) − pe − t 2 / 2 f nh ( x ) = 1 φ w ( ht ) dt , (1 − p ) e − t 2 / 2 2 π −∞ Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution
Something completely different Introduction when p is known Estimation procedure with unknown p Conditions Results Bias We have, with w h ( x ) = 1 h w ( x h ), � ∞ E [ f nh ( x )] − f ( x ) = 1 e − itx φ f ( t )( φ w ( ht ) − 1) dt 2 π −∞ = f ∗ w h ( x ) − f ( x ) , which vanishes for h → 0, similar to ordinary kernel estimation. Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution
Something completely different Introduction when p is known Estimation procedure with unknown p Conditions Results Estimation of p ‘Nonparametric’ estimator � 1 / g p ng = g φ emp ( t ) φ k ( gt ) dt , e − t 2 / 2 2 − 1 / g where the number g > 0 denotes the bandwidth and φ k denotes a Fourier transform of a kernel k . The definition is motivated by another basic idea and the fact � 1 / g � 1 / g g φ X ( t ) g lim e − t 2 / 2 dt = lim φ Y ( t ) dt = p . 2 2 g → 0 g → 0 − 1 / g − 1 / g Shota Gugushvili, Peter Spreij, Bert van Es Deconvolution for an atomic distribution
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