An Estimation Problem for the Intensity Density of Poisson Processes
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AN ESTIMATION PROBLEM FOR THE INTENSITY DENSITY OF POISSON PROCESSES I. A. Ibragimov∗
UDC 519.2
A Poisson process Xε (t) with intensity density function ε−1 λ(t) is observed on interval [a, b]. The problem is to estimate the function λ(t). It is assumed that the unknown function λ(t) belongs to a given class of functions analytic in a given region G ⊃ [a, b] and is bounded there by a given constant M . The parameter ε is supposed to be known and the problem is considered as ε → 0. Bibliography: 7 titles.
1. Problem and results Below we consider the following nonparametric estimation problem. A Poisson process Xε (t) with intensity density ε−1 λ(t) is observed on an interval [a, b], −∞ < a < b < ∞, of the real axis. The function λ(t) is unknown and need to be estimated based on the observations Xε (t). The parameter ε is supposed to be known. We consider the asymptotic formulation of the problem as ε → 0. Example. Let n independent Poisson processes Y1 (t), Y2 (t), . . . , Yn (t) with the same intensity density λ(t) are observed on an interval [a, b]. In the problem of estimation of λ(t) based on n Yj (t) is sufficient. The observation Xε (t) is the sample (Y1 , . . . , Yn ), the statistics Xε (t) = 1
a Poisson process with intensity density λε (t) = nλ(t) = ε−1 λ(t), ε = n−1 . Denote by A(G, M ) the class of functions f (t) defined on the interval [a, b], that admit analytic continuation to the domain G ⊃ [a, b] of complex density and bounded there by a constant M . Below we consider our estimation problem under the assumption that the unknown function λ belongs to A(G, M ) = A with known G and M . ε of the unknown function λ, such that Theorem 1. There exist estimates λ 1/2 ε − λp ≤ cp ε ln 1 , 1 ≤ p < 4, sup Eλ λ ε λ∈A 1/2 1 1/4 ε − λ4 ≤ c4 ε ln 1 , ln ln sup Eλ λ ε ε λ∈A 1− 2 √ p ε − λp ≤ cp ε ln 1 , 4 < p ≤ ∞. sup Eλ λ ε λ∈A
(1.1) (1.2) (1.3)
ε does not depend on p. The constants cp , except for p, depend The choice of the estimate λ on M and G.
∗
St.Petersburg Department of the Steklov Mathematical Institute, St.Petersburg State University, St.Petersburg, Russia, e-mail: [email protected].
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 474, 2018, pp. 139–148. Original article submitted November 23, 2018. 88 1072-3374/20/2511-0088 ©2020 Springer Science+Business Media, LLC
Theorem 2. The following inequalities hold:
1 ε ln , 1 ≤ p < 4, ε ε λ∈A λ 1 ε − λ4 ≥ C4 ε ln 4 ln ln 1 , Δ4 (A) = inf sup Eλ λ ε ε λε λ∈A 2 1− √ p ε − λp ≥ Cp ε ln 1 , 4 < p ≤ ∞. Δp (A) = inf sup Eλ λ ε ε λ∈A λ ε − λp ≥ Cp Δp (A) = inf sup Eλ λ
(1.4) (1.5) (1.6)
The constants Cp > 0 depend on A and p. Theorem 3. The following asymptotic relations hold as ε → 0: 1 Δp (A) ε ln , 1 ≤ p < 4, ε 14 1 ln ln , Δ4 (A) ε ln ε ε 2 1− √ 1 p , 4 < p ≤ ∞. (1.7) Δp (A) ε ln ε Theorem 3 is an obvious consequence of Theorems 1 and 2. Theorem 1 and 2 are proved in Secs. 2 and 3, respectively. In [1], the author considered similar problems of evaluating functions belonging to the class
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