Second order asymptotic efficiency for a Poisson process
aa r X i v : . [ m a t h . S T ] J un Second Order Asymptotic Efficiency for aPoisson Process ∗ Samvel B. GASPARYAN † Universit´e du Maine, Le Mans, FranceYerevan State University, Yerevan, Armenia
Abstract
We consider the problem of the estimation of the mean function of an inhomogeneousPoisson process when its intensity function is periodic. For the mean integrated squarederror (MISE) there is a classical lower bound for all estimators and the empirical meanfunction attains that lower bound, thus it is asymptotically efficient. Following the ideas ofthe work by Golubev and Levit, we compare asymptotically efficient estimators and proposean estimator which is second order asymptotically efficient. Second order efficiency is doneover Sobolev ellipsoids, following the idea of Pinsker.
MSC2010 numbers:
Keywords:
Poisson process, second order estimation, asymptotic efficiency.
We consider the problem of non-parametric estimation of the mean function of an inhomoge-neous Poisson process. We suppose that the unknown intensity function is periodic. It is knownthat the empirical mean function is an asymptotically efficient (in several senses, see e.g. Ku-toyants [7],[8]) estimator. Particularly, we are interested in asymptotic efficiency with respectto the mean integrated squared error (MISE). Note that there are many estimators that areasymptotically efficient in this sense. The goal of the present work is to choose in this class ofasymptotically efficient estimators an estimator which is asymptotically efficient of the secondorder. Such a statement of the problem was considered by Golubev and Levit [6] in the problemof distribution function estimation for the model of independent, identically distributed randomvariables. Then the ideas of this work were applied to the second order asymptotically efficientestimation for different models - Dalalyan, Kutoyants [1] proved second order asymptotic ef-ficiency in the estimation problem of the invariant density of an ergodic diffusion process, inpartial linear models the second order asymptotic efficiency was proved by Golubev, H¨ardle [5].In this paper (combined with the paper [3]) we prove the second order asymptotic efficiencyresult for the mean function of a Poisson process. The main idea that led to development ofthese type of problems was proposed by Pinsker in [10] (more details on the Pinsker bound canbe found in [9], [11]). ∗ Completely revised version of the originally published paper † [email protected] Auxiliary Results
We are given a probability space (Ω , F , P ) and a stochastic process X T = { X t , t ∈ [0 , T ] } . Recallthat X T is an inhomogeneous Poisson process if 1 . X = 0 a.s. 2 . The increments of the process X T on the disjoints intervals are independent random variables. 3. We have P ( X t − X s = k ) = [Λ( t ) − Λ( s )] k k ! e − [Λ( t ) − Λ( s )] , ≤ s < t ≤ T, k ∈ Z + . Here { Λ( t ) , t ∈ [0 , T ] } is a non-decreasing function, and is called the mean function of thePoisson process, because E X ( t ) = Λ( t ). If the mean function is absolutely continuousΛ( t ) = Z t λ ( s )d s, then { λ ( t ) , ≤ t ≤ T } is called the intensity function.Let us consider the problem of estimation Λ( · ) , when its intensity function is a τ -periodicfunction. For simplicity we suppose that T = T n = τ n. Then the observations X T = { X t , t ∈ [0 , τ n ] } , can be written in the form X n = ( X , X , · · · , X n ) , (1)where X j = { X j ( t ) , ≤ t ≤ τ } , X j ( t ) = X ( j − τ + t − X ( j − τ , j = 1 , · · · , n. It is well known that the empirical estimator ˆΛ n ( t ) = 1 n n X j =1 X j ( t ) , t ∈ [0 , τ ]is consistent and asymptotically normal: for all t ∈ [0 , τ ] √ n ( ˆΛ n ( t ) − Λ( t )) = ⇒ N (0 , Λ( t )) . Moreover, this estimator is asymptotically efficient in the sense of the following lower bound:for all estimators { ¯Λ n ( t ) , t ∈ [0 , τ ] } and all t ∗ ∈ [0 , τ ] we havelim δ → lim n → + ∞ sup Λ ∈ V δ n E Λ ( ¯Λ n ( t ∗ ) − Λ( t ∗ )) ≥ Λ ∗ ( t ∗ ) , where V δ = { Λ( · ) : sup t ∈ [0 ,τ ] | Λ( t ) − Λ ∗ ( t ) | ≤ δ } and for the empirical mean function one hasan equality. This is a particular case of a general lower bound given in Kutoyants [7]. Similarinequality holds for MISE ([8])lim δ → lim n → + ∞ sup Λ ∈ V δ n Z τ E Λ ( ¯Λ n ( s ) − Λ( s )) d s ≥ Z τ Λ ∗ ( s )d s. (2) Definition.
The estimators Λ ∗ n ( · ) for which we have an equality in (2) , i.e., lim δ → lim n → + ∞ sup Λ ∈ V δ n Z τ E Λ (Λ ∗ n ( s ) − Λ( s )) d s = Z τ Λ ∗ ( s )d s, are called (first order) asymptotically efficient. The empirical mean function is an asymptotically efficient estimator also in this sense. ([8])The goal of the present work is to find in the class of first order asymptotically efficientestimators an estimator which is second order asymptotically efficient. We follow the mainssteps of the proof of Golubev, Levit [6]. 2
Main Result
For a given integer m > , τ ] such that their ( m − F perm ( R, S ) = (cid:26) Λ( · ) : Z τ [Λ ( m ) ( t )] d t ≤ R, Λ(0) = 0 , Λ( τ ) = S (cid:27) , m > , (3)where R > , S > λ ( · ) is periodic, hence the equality of its values and the values of its derivativeson the endpoints of the interval [0 , τ ] (for estimating a non-periodic function, see, for example,[2]). Introduce as wellΠ = Π m ( R, S ) = (2 m − R (cid:18) SπR m (2 m − m − (cid:19) m m − . (4) Proposition.
Consider Poisson observations X = ( X , X , · · · , X n ) defined in (1) . Then, forall estimators ¯Λ n ( t ) of the mean function Λ( t ) , following lower bound holds lim n → + ∞ sup Λ ∈F m ( R,S ) n m m − (cid:18)Z τ E Λ ( ¯Λ n ( t ) − Λ( t )) d t − n Z τ Λ( t )d t (cid:19) ≥ − Π . This proposition is going to be presented in the forthcoming work [3] (proof relies on amethod developed in [4]). In the next theorem we propose an estimator which attains this lowerbound, thus we prove that this lower bound is sharp. IntroduceΛ ∗ n ( t ) = ˆΛ ,n φ ( t ) + N n X l =1 ˜ K l,n ˆΛ l,n φ l ( t ) , where { φ l } + ∞ l =0 is the trigonometric cosine basis in L [0 , τ ] (see (5) below), ˆΛ l,n are the Fouriercoefficients of the empirical mean function with respect to this basis and˜ K l,n = (cid:18) − (cid:12)(cid:12)(cid:12)(cid:12) πlτ (cid:12)(cid:12)(cid:12)(cid:12) m α ∗ n (cid:19) + , α ∗ n = (cid:20) SnR τπ m (2 m − m − (cid:21) m m − ,N n = τπ ( α ∗ n ) − m ≈ C n m − , x + = max( x, , x ∈ R . The main result of this paper states
Theorem.
The estimator Λ ∗ n ( t ) attains the lower bound described above, that is, lim n → + ∞ sup Λ ∈F m ( R,S ) n m m − (cid:18)Z τ E Λ (Λ ∗ n ( t ) − Λ( t )) d t − n Z τ Λ( t )d t (cid:19) = − Π . Consider the L [0 , τ ] Hilbert space. Evidently, F perm ( R, S ) ⊂ L [0 , τ ] . The main idea of the proofis to replace the estimation problem of the infinite-dimensional (continuum) mean function bythe estimation problem of infinite-dimensional but countable vector of its Fourier coefficients.Recall that the space L [0 , τ ] is isomorphic to the space ℓ = ( θ = ( θ l ) + ∞ l =0 : + ∞ X l =0 θ l < + ∞ ) , || θ || = + ∞ X l =0 θ l ! . ⊂ ℓ of Fourier coefficients of the functions from the set F perm ( R, S ) . Consider a complete, orthonormal system in the space L [0 , τ ] ,φ ( t ) = r τ , φ l ( t ) = r τ cos (cid:18) πlτ t (cid:19) , l ∈ N . (5)Each function f ∈ L [0 , τ ] is a L − limit of its Fourier series f ( t ) = + ∞ X l =0 θ l φ l ( t ) , θ l = Z τ f ( t ) φ l ( t )d t. Suppose that Λ l = Z τ Λ( t ) φ l ( t )d t, λ l = Z τ λ ( t ) φ l ( t )d t. Then
Lemma.
The mean function Λ belongs to the set F perm ( R, S ) (see (3) ) if and only if its Fouriercoefficients w.r.t. the cosine trigonometric basis satisfy + ∞ X l =1 (cid:18) πlτ (cid:19) m Λ l ≤ R, Λ( τ ) = S, (6) or, the Fourier coefficients of its intensity function satisfy + ∞ X l =1 (cid:18) πlτ (cid:19) m − λ l ≤ R, Λ( τ ) = S. (7)For the proof see, for example, [11] (Lemma A.3). To introduce the estimator denote theFourier coefficients of the empirical mean function byˆΛ l,n = Z τ ˆΛ n ( t ) φ l ( t )d t, l ∈ Z + , ˆΛ n ( t ) = 1 n n X j =1 X j ( t ) . Consider the estimator ˜Λ n ( t ) = + ∞ X l =0 ˜Λ l,n φ l ( t ) , ˜Λ l,n = K l,n ˆΛ l,n . Here K l,n are some numbers. Without loss of generality we can take K ,n = 1 , that is ˜Λ ,n = ˆΛ ,n . In this case, using the Parseval’s equality, we get E Λ k ˜Λ n − Λ k − E Λ k ˆΛ n − Λ k = + ∞ X l =1 ( K l,n − σ l,n + + ∞ X l =1 | K l,n − | Λ l . (8)Here σ l,n = E Λ ( ˆΛ l,n − Λ l ) . To compute this quantity, introduce the notation π j ( t ) = X j ( t ) − Λ( t ) . In the sequel, we are going to use the following property of stochastic integrals (see, for example,[7],[8]) E Λ (cid:20) Z τ f ( t )d π j ( t ) Z τ g ( t )d π j ( t ) (cid:21) = Z τ f ( t ) g ( t )dΛ( t ) , f, g ∈ L [0 , τ ] . l,n − Λ l = 1 n n X j =1 Z τ π j ( t ) φ l ( t )d t = 1 n n X j =1 Z τ (cid:18)Z τt φ l ( s )d s (cid:19) d π j ( t ) , which entails that σ l,n = E Λ | ˆΛ l,n − Λ l | = 1 n Z τ (cid:18)Z τt φ l ( s )d s (cid:19) dΛ( t ) . Simple algebra yields σ l,n = 1 n (cid:16) τπl (cid:17) (cid:20) Λ( τ ) − τ Z τ cos (cid:18) πlτ t (cid:19) λ ( t )d t (cid:21) . Combining with (8), this leads to E Λ k ˜Λ n − Λ k − E Λ k ˆΛ n − Λ k = Sn + ∞ X l =1 (cid:16) τπl (cid:17) ( K l,n − + ∞ X l =1 ( K l,n − Λ l + 1 n r τ + ∞ X l =1 (cid:16) τπl (cid:17) (1 − K l,n ) λ l . (9)For the third term in the right-hand side we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n r τ + ∞ X l =1 (cid:16) τπl (cid:17) (1 − K l,n ) λ l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤≤ n r τ max l | − K l,n | (cid:0) πlτ (cid:1) m + ∞ X l =1 (cid:18) πlτ (cid:19) m − λ l (cid:18) πlτ (cid:19) − ≤ n r τ max l | − K l,n | (cid:0) πlτ (cid:1) m + ∞ X l =1 (cid:18) πlτ (cid:19) m − λ l ! + ∞ X l =1 (cid:18) πlτ (cid:19) − ! . Using (7) from the Lemma we obtain + ∞ X l =1 (cid:18) πlτ (cid:19) m − λ l ! ≤ √ R. Hence (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n r τ + ∞ X l =1 (cid:16) τπl (cid:17) (1 − K l,n ) λ l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C n max l | − | K l,n | | (cid:0) πlτ (cid:1) m Now, consider the first two terms of the right-hand side of the equation (9). Introduce a set ofpossible kernels (for all c n > C n = (cid:26) K l,n : | K l,n − | ≤ (cid:12)(cid:12)(cid:12)(cid:12) πlτ (cid:12)(cid:12)(cid:12)(cid:12) m c n (cid:27) . From (6) follows Sn + ∞ X l =1 (cid:16) τπl (cid:17) ( K l,n −
1) + + ∞ X l =1 | K l,n − | Λ l = Sn + ∞ X l =1 (cid:16) τπl (cid:17) ( K l,n − + ∞ X l =1 | K l,n − | (cid:0) πlτ (cid:1) m (cid:18) πlτ (cid:19) m Λ l ≤ Sn + ∞ X l =1 (cid:16) τπl (cid:17) ( K l,n −
1) + c n R. C n ˜ K l,n = arg min C n | K l,n | = (cid:18) − (cid:12)(cid:12)(cid:12)(cid:12) πlτ (cid:12)(cid:12)(cid:12)(cid:12) m c n (cid:19) + , (10)we obtain sup Λ ∈F perm ( R,S ) (cid:16) E Λ k ˜Λ n − Λ k − E Λ k ˆΛ n − Λ k (cid:17) ≤ Sn + ∞ X l =1 (cid:16) τπl (cid:17) ( ˜ K l,n −
1) + c n R + C n max l | − ˜ K l,n | (cid:0) πlτ (cid:1) m . (11)Here ˜Λ n ( t ) is the estimator corresponding to the kernel ˜ K ( u ) . In fact, we have not yet constructedthe estimator. We have to specify the sequence of positive numbers c n in the definition (10).Consider the function H ( c n ) = Sn + ∞ X l =1 (cid:16) τπl (cid:17) ( ˜ K l,n −
1) + c n R and minimize it with respect to the positive sequence c n . Introduce as well N n = τπ c − m n . Then H ( c n ) = Sn " X l ≤ N n (cid:16) τπl (cid:17) c n (cid:18) πlτ (cid:19) m − c n (cid:18) πlτ (cid:19) m ! − X l>N n (cid:16) τπl (cid:17) + c n R. To minimize this function consider its derivative H ′ ( c n ) = Sn X l ≤ N n (cid:16) τπl (cid:17) " c n (cid:18) πlτ (cid:19) m − (cid:18) πlτ (cid:19) m + 2 c n R = 0 . (12)Consider such sums ( β ∈ N ) X l ≤ N n l β = [ N n ] X l =1 (cid:18) l [ N n ] (cid:19) β [ N n ] β = [ N n ] β +1 [ N n ] X l =1 (cid:18) l [ N n ] (cid:19) β N n ] , hence, if c n −→ , as n −→ + ∞ , N n ] β +1 X l ≤ N n l β −→ Z x β d x, that is, X l ≤ N n l β = [ N n ] β +1 β + 1 (1 + o (1)) , n −→ + ∞ . Using this identity we can transform (11) (remembering that N n = τπ c − m n ) Sn c n (cid:16) πτ (cid:17) m − X l ≤ N n l m − − (cid:16) πτ (cid:17) m − X l ≤ N n l m − ! = − c n R,Sn (cid:18) c n (cid:16) πτ (cid:17) m − N m − n m − − (cid:16) πτ (cid:17) m − N m − n m − (cid:19) = − c n R (1 + o (1)) ,Sn τπ c − m − m n (cid:18) m − − m − (cid:19) = − c n R (1 + o (1)) . c ∗ n = α ∗ n (1 + o (1)) , α ∗ n = (cid:20) SnR τπ m (2 m − m − (cid:21) m m − . (13)Now, using the identity ( β ∈ N , β > X l>N n l β = 1 N β − n Z + ∞ x β d x · (1 + o (1)) , n −→ + ∞ , for β = 2 X l>N n l = 1 N n · (1 + o (1)) , n −→ + ∞ , calculate H ( c ∗ n ) = Sn (cid:20) ( c ∗ n ) (cid:16) πτ (cid:17) m − N m − n m − − c ∗ n (cid:16) πτ (cid:17) m − N m − n m − − (cid:16) τπ (cid:17) N n (cid:21) (1 + o (1))+ ( c ∗ n ) R = Sn τπ " ( c ∗ n ) ( c ∗ n ) − m − m m − − c ∗ n ( c ∗ n ) − m − m m − − ( c ∗ n ) m (1 + o (1)) + ( c ∗ n ) R == Sn τπ ( c ∗ n ) m − m (2 m − m −
1) (1 + o (1)) + ( c ∗ n ) R == ( − m ) R ( c ∗ n ) m ( c ∗ n ) m − m (1 + o (1)) + ( c ∗ n ) R == − (2 m − α ∗ n ) R (1 + o (1)) , where we have used the relation (13). Now, choosing the sequence c n = α ∗ n for the definition ofthe estimator in (10), we obtain from (11)sup Λ ∈F perm ( R,S ) (cid:16) E Λ k ˜Λ n − Λ k − E Λ k ˆΛ n − Λ k (cid:17) ≤≤ − (2 m − α ∗ n ) R (1 + o (1)) + C n max l | − | ˜ K l,n | | (cid:0) πlτ (cid:1) m . (14)If we show that 1 n max l | − ˜ K l,n | (cid:0) πlτ (cid:1) m = o ( n − m m − ) , (15)then, since (see (4)) Π = (2 m − α ∗ n ) Rn m m − , we get from (14) lim n → + ∞ n m m − sup Λ ∈F perm ( R,S ) (cid:16) E Λ k ˜Λ n − Λ k − E Λ k ˆΛ n − Λ k (cid:17) ≤ − Π . This combined with the proposition will end the proof. To prove (15) recall that˜ K l,n = (cid:18) − (cid:12)(cid:12)(cid:12)(cid:12) πlτ (cid:12)(cid:12)(cid:12)(cid:12) m α ∗ n (cid:19) + , α ∗ n = (cid:20) SnR τπ m (2 m − m − (cid:21) m m − . Therefore, for m > n max l | − ˜ K l,n | (cid:0) πlτ (cid:1) m ≤ n max l − ˜ K l,n (cid:0) πlτ (cid:1) m = 2 n α ∗ n = C n m − m − = o ( n − m m − ) . Acknowledgements
The author is grateful to Y.A. Kutoyants for his suggestions and in-teresting discussions, also to A.S. Dalalyan and Y.K. Golubev for their fruitful comments.7 eferences [1] Dalalyan A.S. and Kutoyants Y.A., On second order minimax estimation of invariant densityfor ergodic diffusion.
Statistics & Decisions 22(1), 17-42, 2004. [2] Delattre S. and Hoffmann M., The Pinsker Bound in Mixed Gaussian White Noise.
Math.Methods of Statist. 10(3), 283-315, 2001. [3] Gasparyan S.B., Kutoyants Y.A., On the lower bound in second order asymptotically effi-cient estimation for Poisson processes.
Working Paper. [4] Gill R.D. and Levit B.Y., Applications of the van Trees inequality: a Bayesian Cram´er-Raobound,
Bernoulli 1(1-2), 59-79, 1995. [5] Golubev G.K. and H¨ardle W., Second order minimax estimation in partial linear models,
Math. Methods of Statist., 9(2), 160-175, 2000. [6] Golubev G.K. and Levit B.Y., On the second order minimax estimation of distributionfunctions,
Math. Methods of Statist., 5(1), 1-31, 1996. [7] Kutoyants Y.A.,
Statistical Inference for Spatial Poisson Processes , Lecture Notes in Statis-tics 134, (Springer-Verlag, New York, 1998).[8] Kutoyants Y.A.,
Introduction to Statistics of Poisson Processes . To appear.[9] Nussbaum M., Minimax risk: Pinskers bound,
In Encyclopedia of Statistical Sciences, Up-date Volume 3, S. Kotz (Ed), (New York: Wiley, 451-460, 1999). [10] Pinsker M.S., Optimal filtering of square-integrable signals in Gaussian noise,
ProblemsInform. Transmission, 16(2), 120-133, 1980. [11] Tsybakov A.B.,