## The Annals of Statistics

### Super-resolution estimation of cyclic arrival rates

#### Abstract

Exploiting the fact that most arrival processes exhibit cyclic behaviour, we propose a simple procedure for estimating the intensity of a nonhomogeneous Poisson process. The estimator is the super-resolution analogue to Shao (2010) and Shao and Lii [J. R. Stat. Soc. Ser. B. Stat. Methodol. 73 (2011) 99–122], which is a sum of $p$ sinusoids where $p$ and the amplitude and phase of each wave are not known and need to be estimated. This results in an interpretable yet flexible specification that is suitable for use in modelling as well as in high resolution simulations.

Our estimation procedure sits in between classic periodogram methods and atomic/total variation norm thresholding. Through a novel use of window functions in the point process domain, our approach attains super-resolution without semidefinite programming. Under suitable conditions, finite sample guarantees can be derived for our procedure. These resolve some open questions and expand existing results in spectral estimation literature.

#### Article information

Source
Ann. Statist., Volume 47, Number 3 (2019), 1754-1775.

Dates
Revised: June 2018
First available in Project Euclid: 13 February 2019

https://projecteuclid.org/euclid.aos/1550026856

Digital Object Identifier
doi:10.1214/18-AOS1736

Mathematical Reviews number (MathSciNet)
MR3911129

Zentralblatt MATH identifier
07053525

Subjects
Primary: 62M15: Spectral analysis 90B22: Queues and service [See also 60K25, 68M20]
Secondary: 60G55: Point processes

#### Citation

Chen, Ningyuan; Lee, Donald K. K.; Negahban, Sahand N. Super-resolution estimation of cyclic arrival rates. Ann. Statist. 47 (2019), no. 3, 1754--1775. doi:10.1214/18-AOS1736. https://projecteuclid.org/euclid.aos/1550026856

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#### Supplemental materials

• Proofs and asymptotic normality. The proofs of all results presented in this paper are provided in Appendix A of the supplement. Appendix B establishes the asymptotic normality of the windowed periodogram estimator.