Strong Consistency of a Kernel-type Estimator for the Intensity Obtained as the Product of a Periodic Function with the Power Function Trend of a Non-homogeneous Poisson Process

2015 ◽  
Vol 9 (4) ◽  
pp. 383-387 ◽  
Author(s):  
Ikhsan Maulidi ◽  
I Mangku ◽  
Hadi Sumarno
Keyword(s):  
Periodic Function ◽  
Poisson Process ◽  
Power Function ◽  
Strong Consistency ◽  
Homogeneous Poisson Process ◽  
Non Homogeneous Poisson Process ◽  
Kernel Type

scholarly journals Strong Consistency and Asymptotic Distribution of Estimator for the Intensity Function Having Form of Periodic Function Multiplied by Power Function Trend of a Poisson Process

2018 ◽  
Vol 8 (02) ◽  
Author(s):  
Nina Valentika ◽  
Wayan Mangku ◽  
Windiani Erliana
Keyword(s):  
Periodic Function ◽  
Poisson Process ◽  
Asymptotic Distribution ◽  
Power Function ◽  
Strong Consistency ◽  
Observation Interval ◽  
Intensity Function ◽  
Periodic Component ◽  
Single Realization ◽  
Non Homogeneous Poisson Process

This manuscript discusses the strong consistency and the asymptotic distribution of an estimator for a periodic component of the intensity function having a form of periodic function multiplied by power function trend of a non-homogeneous Poisson process by using a uniform kernel function. It is assumed that the period of the periodic component of intensity function is known. An estimator for the periodic component using only a single realization of a Poisson process observed at a certain interval has been constructed. This estimator has been proved to be strongly consistent if the length of the observation interval indefinitely expands. Computer simulation also showed the asymptotic normality of this estimator.

scholarly journals The Numerical Simulation for Asymptotic Normality of the Intensity Obtained as a Product of a Periodic Function with the Power Trend Function of a Nonhomogeneous Poisson Process

2020 ◽  
Vol 3 (3) ◽  
pp. 271-278
Author(s):  
Ikhsan Maulidi ◽  
Mahyus Ihsan ◽  
Vina Apriliani
Keyword(s):  
Numerical Simulation ◽  
Periodic Function ◽  
Asymptotic Normality ◽  
Poisson Process ◽  
Power Function ◽  
Observation Interval ◽  
Intensity Function ◽  
Nonhomogeneous Poisson Process ◽  
Trend Function ◽  
Kernel Type

In this article, we provided a numerical simulation for asymptotic normality of a kernel type estimator for the intensity obtained as a product of a periodic function with the power trend function of a nonhomogeneous Poisson Process. The aim of this simulation is to observe how convergence the variance and bias of the estimator. The simulation shows that the larger the value of power function in intensity function, it is required the length of the observation interval to obtain the convergent of the estimator.

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.

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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