Mean-Square Convergence of Recursive Kernel Estimators of Non-Homogeneous Poisson Process Intensity Function and its Derivative

2015 ◽  
Vol 1084 ◽  
pp. 684-688
Author(s):  
Anna V. Kitaeva ◽  
Mikhail V. Kolupaev
Keyword(s):  
Poisson Process ◽  
Fixed Time ◽  
Kernel Estimators ◽  
Time Interval ◽  
Mean Square ◽  
Homogeneous Poisson Process ◽  
Single Realization ◽  
Mean Square Convergence ◽  
Scheme Of Series ◽  
Non Homogeneous Poisson Process

The structure of the estimators is similar to the recursive kernel estimators of a density function and its derivative. The estimators have been constructed using a single realization of Poisson process on a fixed time interval. Mean-square convergence has been proved in a scheme of series. Simulation studies have been carried out to illustrate the convergence.

scholarly journals PENDUGAAN KOMPONEN PERIODIK FUNGSI INTENSITAS BERBENTUK FUNGSI PERIODIK KALI TREN LINEAR SUATU PROSES POISSON NON-HOMOGEN

2013 ◽  
Vol 12 (1) ◽  
pp. 49
Author(s):  
W. ISMAYULIA ◽  
I W. MANGKU ◽  
S. SISWANDI
Keyword(s):  
Poisson Process ◽  
Mean Square Error ◽  
Linear Trend ◽  
Periodic Component ◽  
Mean Square ◽  
Homogeneous Poisson Process ◽  
Single Realization ◽  
The Mean ◽  
Bias Variance ◽  
Non Homogeneous Poisson Process

In this manuscript, estimation of the periodic component of intensity having form periodic function multiplied by the linear trend of a non homogeneous Poisson process is discussed. The estimator is constructed using a single realization of the Poisson process observed in the interval  0,𝑛 . It is assumed that the period of the periodic component is known. The convergence of the Mean Square Error (MSE) of the estimator has been proved. In addition, asymptotic approximations to the bias, variance, and Mean Square Error (MSE) of the estimator have been proved. An asymptotic optimal bandwidth is also given.

Weak limit results for the extremes of a class of shot noise processes

1995 ◽  
Vol 32 (03) ◽  
pp. 707-726 ◽  
Author(s):  
Patrick Homble ◽  
William P. McCormick
Keyword(s):  
Poisson Process ◽  
Shot Noise ◽  
Impulse Response Function ◽  
Weak Limit ◽  
Intensity Function ◽  
Important Class ◽  
Homogeneous Poisson Process ◽  
Periodic Intensity ◽  
Non Homogeneous Poisson Process ◽  
Over Time

Shot noise processes form an important class of stochastic processes modeling phenomena which occur as shocks to a system and with effects that diminish over time. In this paper we present extreme value results for two cases — a homogeneous Poisson process of shocks and a non-homogeneous Poisson process with periodic intensity function. Shocks occur with a random amplitude having either a gamma or Weibull density and dissipate via a compactly supported impulse response function. This work continues work of Hsing and Teugels (1989) and Doney and O'Brien (1991) to the case of random amplitudes.

scholarly journals Maximizing the Expected Duration of Owning a Relatively Best Object in a Poisson Process with Rankable Observations

2009 ◽  
Vol 46 (02) ◽  
pp. 402-414
Author(s):  
Aiko Kurushima ◽  
Katsunori Ano
Keyword(s):  
Poisson Process ◽  
Stopping Time ◽  
Fixed Time ◽  
Time Interval ◽  
Prior Density ◽  
Deterministic Equation ◽  
Fixed Time Interval ◽  
Unknown Number ◽  
Random Intensity ◽  
Time Selection

Suppose that an unknown number of objects arrive sequentially according to a Poisson process with random intensity λ on some fixed time interval [0,T]. We assume a gamma prior density G λ(r, 1/a) for λ. Furthermore, we suppose that all arriving objects can be ranked uniquely among all preceding arrivals. Exactly one object can be selected. Our aim is to find a stopping time (selection time) which maximizes the time during which the selected object will stay relatively best. Our main result is the following. It is optimal to select the ith object that is relatively best and arrives at some time s i (r) onwards. The value of s i (r) can be obtained for each r and i as the unique root of a deterministic equation.

An optimal parking problem

1982 ◽  
Vol 19 (4) ◽  
pp. 803-814 ◽  
Author(s):  
Mitsushi Tamari
Keyword(s):  
Poisson Process ◽  
Decision Maker ◽  
Optimal Stopping ◽  
Renewal Process ◽  
A Priori ◽  
Expected Loss ◽  
Homogeneous Poisson Process ◽  
Optimal Stopping Rule ◽  
Non Homogeneous Poisson Process ◽  
Parking Place

The decision-maker drives a car along a straight highway towards his destination and looks for a parking place. When he finds a parking place, he can either park there and walk the distance to his destination or continue driving. Parking places are assumed to occur in accordance with a Poisson process along the highway. The decision-maker does not know the distance Y to his destination exactly in advance. Only an a priori distribution is assumed for Y and cases of typically important distribution are examined. When we take as loss the distance the decision-maker must walk and wish to minimize the expected loss, the optimal stopping rule and the minimum expected loss are obtained. In Section 3 a generalization to the cases of a non-homogeneous Poisson process and a renewal process is considered.

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