Estimation of the Intensity Function of a Periodic Poisson Process with Power Function Trend

2012 ◽  
Author(s):  
R. Rachmawati ◽  
W. Budiharto
Keyword(s):  
Poisson Process ◽  
Power Function ◽  
Intensity Function

scholarly journals The Confidence Interval of the Estimator of the Periodic Intensity Function in the Presence of Power Function Trend on the Nonhomogeneous Poisson Process

CAUCHY ◽  
2021 ◽  
Vol 7 (1) ◽  
pp. 73-83
Author(s):  
Ikhsan Maulidi ◽  
Bonno Andri Wibowo ◽  
Nina Valentika ◽  
Muhammad Syazali ◽  
Vina Apriliani
Keyword(s):  
Confidence Interval ◽  
Stochastic Processes ◽  
Poisson Process ◽  
Power Function ◽  
Intensity Function ◽  
Nonhomogeneous Poisson Process ◽  
Constructive Proof ◽  
Power Functions ◽  
Periodic Intensity ◽  
Intensity Estimator

The nonhomogeneous Poisson process is one of the most widely applied stochastic processes. In this article, we provide a confidence interval of the intensity estimator in the presence of a periodic multiplied by trend power function. This estimator's confidence interval is an application of the formulation of the estimator asymptotic distribution that has been given in previous studies. In addition, constructive proof of the convergent in probability has been provided for all power functions.

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.

Misspecified change-point estimation problem for a Poisson process

2001 ◽  
Vol 38 (A) ◽  
pp. 122-130 ◽  
Author(s):  
Ali S. Dabye ◽  
Yury A. Kutoyants
Keyword(s):  
Maximum Likelihood ◽  
Poisson Process ◽  
Change Point ◽  
Maximum Likelihood Estimate ◽  
Intensity Function ◽  
Point Estimation ◽  
Lower Function ◽  
Inhomogeneous Poisson Process ◽  
Change Point Estimation ◽  
Unknown Point

Consider an inhomogeneous Poisson process X on [0, T] whose unknown intensity function ‘switches' from a lower function g∗ to an upper function h∗ at some unknown point θ∗. What is known are continuous bounding functions g and h such that g∗(t) ≤ g(t) ≤ h(t) ≤ h∗(t) for 0 ≤ t ≤ T. It is shown that on the basis of n observations of the process X the maximum likelihood estimate of θ∗ is consistent for n →∞, and also that converges in law and in pth moment to limits described in terms of the unknown functions g∗ and h∗.

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

1995 ◽  
Vol 32 (3) ◽  
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.

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