Poisson source localization on the plane: cusp case

Chernoyarov, Oleg V.; Dachian, Sergueı̈; Kutoyants, Yury A.

doi:10.1007/s10463-019-00722-6

Cited by 9 publications

(5 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Mathematical statements are less known. This work is a continuation of a study initiated in [9] and then developed in [1,[4][5][6]15], where it was always supposed that there is only one source on the plane. In the works [5,6,9] it was supposed that the moment of the beginning of the emission τ 0 is known and the unknown parameter was just the position of the source.…”

Section: Introductionmentioning

confidence: 84%

“…This work is a continuation of a study initiated in [9] and then developed in [1,[4][5][6]15], where it was always supposed that there is only one source on the plane. In the works [5,6,9] it was supposed that the moment of the beginning of the emission τ 0 is known and the unknown parameter was just the position of the source. The case where τ 0 is unknown too and we have to estimate both τ 0 and the position of the source was treated in [1,4].…”

Section: Introductionmentioning

confidence: 84%

See 1 more Smart Citation

Localization of two radioactive sources on the plane

Chernoyarov,

Dachian,

Farinetto

et al. 2022

Preprint

View full text Add to dashboard Cite

The problem of localization on the plane of two radioactive sources by K detectors is considered. Each detector records a realization of an inhomogeneous Poisson process whose intensity function is the sum of signals arriving from the sources and of a constant Poisson noise of known intensity. The time of the beginning of emission of the sources is known, and the main problem is the estimation of the positions of the sources. The properties of the maximum likelihood and Bayesian estimators are described in the asymptotics of large signals in three situations of different regularities of the fronts of the signals: smooth, cusp-type and change-point type.

show abstract

Section: Introductionmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 84%

Localization of two radioactive sources on the plane

Chernoyarov,

Dachian,

Farinetto

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…where τ j = τ j (ϑ) is of course smooth function of ϑ. Recall that if κ ∈ 0, 1 2 , then we have cusp case [2] and if κ > 1 2 , then we have the smooth case considered in this work. We start with the problem of estimation the parameter τ = (τ 1 , .…”

Section: Simple Consistent Estimatormentioning

confidence: 95%

“…The case b) is discussed below in the Section 4. For the Cusp case c) see [4], [2]. The change-point case d) is studied in [6].…”

Section: Introductionmentioning

confidence: 99%

Poisson source localization on the plane: the smooth case

Chernoyarov

Kutoyants

2019

Metrika

Self Cite

View full text Add to dashboard Cite

We consider the problem of localization of Poisson source by the observations of inhomogeneous Poisson processes. We suppose that there are k detectors on the plane and each detector provides the observations of Poisson processes whose intensity functions depend on the position of the emitter. We describe the properties of the maximum likelihood and Bayesian estimators. We show that under regularity conditions these estimators are consistent, asymptotically normal and asymptotically efficient. Then we propose some simple consistent estimators and this estimators are further used to construct asymptotically efficient One-step MLE-process.

show abstract

“…Note also that recently the problem of source localization on the plane by observations of Poissonian signals from several detectors was considered in all the three cases (smooth, change-point and cusp) in Kutoyants (2020), Farinetto, Kutoyants andTop (2020) and Chernoyarov, Dachian and Kutoyants (2020), respectively.…”

Section: Introductionmentioning

confidence: 99%

On smooth change-point location estimation for Poisson Processes

Amiri

Dachian

2021

Stat Inference Stoch Process

View full text Add to dashboard Cite

We are interested in estimating the location of what we call “smooth change-point” from n independent observations of an inhomogeneous Poisson process. The smooth change-point is a transition of the intensity function of the process from one level to another which happens smoothly, but over such a small interval, that its length $$\delta _n$$ δ n is considered to be decreasing to 0 as $$n\rightarrow +\infty $$ n → + ∞ . We show that if $$\delta _n$$ δ n goes to zero slower than 1/n, our model is locally asymptotically normal (with a rather unusual rate $$\sqrt{\delta _n/n}$$ δ n / n ), and the maximum likelihood and Bayesian estimators are consistent, asymptotically normal and asymptotically efficient. If, on the contrary, $$\delta _n$$ δ n goes to zero faster than 1/n, our model is non-regular and behaves like a change-point model. More precisely, in this case we show that the Bayesian estimators are consistent, converge at rate 1/n, have non-Gaussian limit distributions and are asymptotically efficient. All these results are obtained using the likelihood ratio analysis method of Ibragimov and Khasminskii, which equally yields the convergence of polynomial moments of the considered estimators. However, in order to study the maximum likelihood estimator in the case where $$\delta _n$$ δ n goes to zero faster than 1/n, this method cannot be applied using the usual topologies of convergence in functional spaces. So, this study should go through the use of an alternative topology and will be considered in a future work.

show abstract

Poisson source localization on the plane: cusp case

Cited by 9 publications

References 12 publications

Localization of two radioactive sources on the plane

Localization of two radioactive sources on the plane

Poisson source localization on the plane: the smooth case

On smooth change-point location estimation for Poisson Processes

Contact Info

Product

Resources

About