Estimation of Singularity Location for Poisson Process S. Dachian - PowerPoint PPT Presentation

✬ ✩ Estimation of Singularity Location for Poisson Process S. Dachian Laboratoire de Math´ ematiques Appliqu´ ees Universit´ e Blaise Pascal Clermont-Ferrand, FRANCE ✫ ✪

✬ ✩ Statistique Asymptotique des Processus Stochastiques IV Le Mans, December 19-20, 2002 The model – The process. — Poisson process of intensity function S θ ( · ): X = { X ( t ) , 0 � t � T } . – The observations. — n independent realizations (trajectories) of X : ( X 1 , . . . , X n ) = X n . – The hypotheses on S θ ( · ) . — The intensity function S θ ( · ) is regular every- where on [0 ,T ] except at the point θ , where it has a singularity. – The unknown parameter. — The location (the point) of the singularity: θ ∈ Θ = ( α,β ) ⊆ (0 ,T ) . – The types of singularities. — Three types: “cusp”, singularity of “0”-type and singularity of “ ∞ ”-type. – The asymptotics. — n − → ∞ . ✫ ✪ S. Dachian, Clermont-Ferrand (France) 1

✬ ✩ Statistique Asymptotique des Processus Stochastiques IV Le Mans, December 19-20, 2002 � a | t − θ | p + Ψ( θ, t ) , if t � θ We consider S θ ( t ) of the form S θ ( t ) = . b | t − θ | p + Ψ( θ, t ) , if t � θ We suppose that a 2 + b 2 > 0, S θ ( t ) > 0 for all t � = θ , the function Ψ( θ, t ) is continuous and uniformly in t H¨ older continuous of order µ with respect to θ . “cusp” singularity of “0”-type singularity of “ ∞ ”-type 0 < p < 1 / 2 0 < p < 1 − 1 < p < 0 µ > p + 1 / 2 µ > ( p + 1) / 2 µ > ( p + 1) / 2 Ψ( θ, θ ) > 0 Ψ( θ, θ ) = 0 ✫ ✪ S. Dachian, Clermont-Ferrand (France) 2

✬ ✩ Statistique Asymptotique des Processus Stochastiques IV Le Mans, December 19-20, 2002 The history of the problem – Prakasa Rao, B.L.S. , “Estimation of the location of the cusp of a continuous density”, Annals of Mathematical Statistics , vol. 20, no. 1, pp. 76–87, 1968. – Ibragimov, I.A. et Khasminskii, R.Z. , “ Statistical Estimation. Asymptotic Theory ”, Springer-Verlag, New York, 1981. – Dachian, S. , “Estimation of Cusp Location by Poisson Observations”, Statis- tical Inference for Stochastic Processes , to appear, 2001. – Dachian, S. , “Estimation of Singularity Location by Poisson Observations”, in preparation, 2002. – Dachian, S. et Kutoyants, Yu.A. , “On Cusp Estimation of Ergodic Diffusion Process”, Journal of Statistical Planning and Inference , to appear, 2001. ✫ ✪ S. Dachian, Clermont-Ferrand (France) 3

✬ ✩ Statistique Asymptotique des Processus Stochastiques IV Le Mans, December 19-20, 2002 The likelihood ratio is: � n � � T � T � � � L ( θ, X n ) = exp ln S θ ( t ) dX i ( t ) − n S θ ( t ) − 1 dt . i =1 0 0 The maximum likelihood estimator (MLE) � θ n is defined as one of the solutions of L ( � θ n , X n ) = sup L ( θ, X n ). the maximum likelihood equation θ ∈ Θ The Bayesian estimator (BE) for prior density q ( · ) and quadratic loss function is � β � θ | X n � � ·| X n � defined by � θ n = θ q dθ , where the posterior density q is given by: α   − 1 β � � θ | X n � = L ( θ, X n ) q ( θ )  L ( θ, X n ) q ( θ ) dθ  q . α ✫ ✪ S. Dachian, Clermont-Ferrand (France) 4

✬ ✩ Statistique Asymptotique des Processus Stochastiques IV Le Mans, December 19-20, 2002 The “cusp” case. — We introduce the stochastic process (on R ) � � Γ θ W p +1 / 2 ( u ) − 1 θ | u | 2 p +1 2 Γ 2 Z 1 ( u ) = exp � � � a 2 + b 2 � θ = B p + 1 , p + 1 Γ 2 0 < Γ 2 where cos( πp ) − 2 ab , θ < + ∞ , Ψ(0 , 0) and W H ( · ) is a fractional Brownian motion (fBm) of Hurst parameter H , that is, a centered Gaussian process with covariance � � � | u 1 | 2 H + | u 2 | 2 H − | u 1 − u 2 | 2 H � = 1 W H ( u 1 ) W H ( u 2 ) E . 2 We introduce equally the random variables ξ 1 and ζ 1 by Z 1 ( ξ 1 ) = sup Z 1 ( u ) u ∈ R   − 1 � + ∞ � + ∞   and ζ 1 = u Z 1 ( u ) du Z 1 ( u ) du . −∞ −∞ ✫ ✪ S. Dachian, Clermont-Ferrand (France) 5

✬ ✩ Statistique Asymptotique des Processus Stochastiques IV Le Mans, December 19-20, 2002 Theorem. — In the case of “cusp”, we have the following lower bound on the risks of all the estimators of θ : � �� 2 n 1 / (2 p +1) � � E ζ 2 lim lim inf sup E θ θ n − θ 1 δ → 0 n →∞ θ n | θ − θ 0 | <δ for all θ 0 ∈ Θ, where inf is taken on the set of all the estimators θ n of θ . Definition. — We say that an estimator θ n is asymptotically efficient if � �� 2 n 1 / (2 p +1) � = E ζ 2 lim lim sup E θ θ n − θ 1 n →∞ δ → 0 | θ − θ 0 | <δ for all θ 0 ∈ Θ. ✫ ✪ S. Dachian, Clermont-Ferrand (France) 6

✬ ✩ Statistique Asymptotique des Processus Stochastiques IV Le Mans, December 19-20, 2002 Theorem. — In the case of “cusp”, the BE � θ n and the MLE � θ n have uniformly in θ ∈ K (for any compact K ⊂ Θ) the following properties: • � θ n and � θ n are consistent, that is, P θ P θ � � θ n − → θ and θ n − → θ, • the limit distributions of � θ n and � θ n are given by n 1 / (2 p +1) �� � n 1 / (2 p +1) �� � θ n − θ = ⇒ ζ 1 and θ n − θ = ⇒ ξ 1 , • for any k > 0 the convergence of moments equally holds: � �� � n 1 / (2 p +1) �� k � � = E | ζ 1 | k , n →∞ E θ lim θ n − θ � � �� � n 1 / (2 p +1) �� k � � = E | ξ 1 | k . n →∞ E θ lim θ n − θ � Moreover, the BE � θ n are asymptotically efficient. ✫ ✪ S. Dachian, Clermont-Ferrand (France) 7

✬ ✩ Statistique Asymptotique des Processus Stochastiques IV Le Mans, December 19-20, 2002 The “0”-type and “ ∞ ”-type singularity cases. — We introduce the stochastic process (on R ) � � + ∞ � � � � � 1 − u − a − b � � p + 1 | u | p +1 sign( u ) + Z 2 ( u ) = exp p ln ν ( dz ) − E ν ( dz ) � z −∞ � �� � � u � + ∞ � � � + ln a � 1 − u p � 1 − u � � � � d ( z ) | z | p dz ν ( dz ) − − 1 − p ln , � � b z z 0 −∞ � a, if z � 0 and ν is a Poisson process of intensity d ( z ) | z | p . where d ( z ) = b, if z � 0 We introduce equally the random variables ξ 2 (in the case of “0”-type singularity only) and ζ 2 (in both cases) by Z 2 ( ξ 2 ) = sup Z 2 ( u ) u ∈ R   − 1 � + ∞ � + ∞   and ζ 2 = u Z 2 ( u ) du Z 2 ( u ) du . −∞ −∞ ✫ ✪ S. Dachian, Clermont-Ferrand (France) 8

✬ ✩ Statistique Asymptotique des Processus Stochastiques IV Le Mans, December 19-20, 2002 Theorem. — In the case of “0”-type or “ ∞ ”-type singularity, we have the following lower bound on the risks of all the estimators of θ : � �� 2 n 1 / ( p +1) � � E ζ 2 lim lim inf sup E θ θ n − θ 2 δ → 0 n →∞ θ n | θ − θ 0 | <δ for all θ 0 ∈ Θ, where inf is taken on the set of all the estimators θ n of θ . Definition. — We say that an estimator θ n is asymptotically efficient if � �� 2 n 1 / ( p +1) � = E ζ 2 lim lim sup E θ θ n − θ 2 n →∞ δ → 0 | θ − θ 0 | <δ for all θ 0 ∈ Θ. ✫ ✪ S. Dachian, Clermont-Ferrand (France) 9

✬ ✩ Statistique Asymptotique des Processus Stochastiques IV Le Mans, December 19-20, 2002 The BE � θ n (in both cases) and the MLE � Theorem. — θ n (in the case of “0”-type singularity only) have uniformly in θ ∈ K (for any compact K ⊂ Θ) the following properties: • � θ n and � θ n are consistent, that is, P θ P θ � � θ n − → θ and θ n − → θ, • the limit distributions of � θ n and � θ n are given by n 1 / ( p +1) �� � n 1 / ( p +1) �� � θ n − θ = ⇒ ζ 2 and θ n − θ = ⇒ ξ 2 , • for any k > 0 the convergence of moments equally holds: � �� � n 1 / ( p +1) �� k � � = E | ζ 2 | k , n →∞ E θ lim θ n − θ � � �� � n 1 / ( p +1) �� k � � = E | ξ 2 | k . n →∞ E θ lim θ n − θ � Moreover, the BE � θ n are asymptotically efficient. ✫ ✪ S. Dachian, Clermont-Ferrand (France) 10

✬ ✩ Statistique Asymptotique des Processus Stochastiques IV Le Mans, December 19-20, 2002 Ideas of the proof We use the Ibragimov and Khasminskii method which consist in studying the nor- malized likelihood ratio process Z n ( u ) = L ( θ u , X n ) L ( θ, X n ) , u ∈ U n , where we denote θ u = θ + u n − 1 /ν (with ν = 2 p + 1 or ν = p + 1) and the set � � n 1 /ν ( α − θ ) , n 1 /ν ( β − θ ) U n = , and establishing the three following properties: – The finite-dimensional distributions of Z n ( u ) converge to those of Z ( u ) (with Z = Z 1 or Z = Z 2 ) uniformly in θ ∈ K . � � 2 � � � C | u 1 − u 2 | ν uniformly in θ ∈ K . � Z 1 / 2 ( u 1 ) − Z 1 / 2 – E θ ( u 2 ) � n n � − c | u | ν � – E θ Z 1 / 2 ( u ) � exp uniformly in θ ∈ K . n ✫ ✪ S. Dachian, Clermont-Ferrand (France) 11