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.

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.

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.

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.

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.

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