Nonparametric Estimation of the Cumulative Intensity Function for a Nonhomogeneous Poisson Process

1991 ◽  
Vol 37 (7) ◽  
pp. 886-900 ◽  
Author(s):  
Lawrence M. Leemis
Keyword(s):  
Poisson Process ◽  
Nonparametric Estimation ◽  
Intensity Function ◽  
Nonhomogeneous Poisson Process

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.

The Exponential Power-Law Process for the Nonhomogeneous Poisson Process and Bathtub Data

2020 ◽  
Vol 27 (04) ◽  
pp. 2050011
Author(s):  
Shaul K. Bar-Lev ◽  
Frank A. van der Duyn Schouten
Keyword(s):  
Poisson Process ◽  
Power Law ◽  
Likelihood Estimation ◽  
Intensity Function ◽  
Nonhomogeneous Poisson Process ◽  
Base Function ◽  
Power Law Process ◽  
Failure Data ◽  
Exponential Power ◽  
Intensity Functions

Recently, Bar-Lev, Bshouty and Van der Duyn Schouten [Math. Methods Stat. 25 (2016) 79–980] developed a systematic method, called operator-based intensity function, for constructing huge classes of nonmonotonic intensity functions (convex or concave) for the nonhomogeneous Poisson process, all of which are suitable for modeling bathtub data. Each class is parametrized by several parameters (as scale and shape parameters) in addition to the operator index parameter [Formula: see text]. For the sake of demonstration only, we focus in this paper on a special subclass called the exponential power law process (EXPLP[Formula: see text]) whose base function is the intensity function of the power-law process. We describe various properties of such a subclass and use one of its special case, namely EXPLP[Formula: see text] intensity function, to analyze failure data which lack monotonicity. Maximum likelihood estimation of the parameters involved and relevant functions thereof is discussed with respect various aspects as existence, uniqueness, asymptotic behavior and statistical inference facets. Using two real datasets from the literature we provide evidence that the EXPLP[Formula: see text] intensity function is well suited to analyze data which exhibit a bathtub behavior.

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 L2-CONVERGENCE OF A NEAREST NEIGHBOR ESTIMATOR OF THE INTENSITY FUNCTION OF A CYCLIC POISSON PROCES

2014 ◽  
Vol 13 (2) ◽  
pp. 49
Author(s):  
I W. MANGKU
Keyword(s):  
Poisson Process ◽  
Key Words ◽  
Nonparametric Estimation ◽  
Nearest Neighbor ◽  
Intensity Function ◽  
Key Words And Phrases ◽  
Subject Classifications ◽  
Single Realization ◽  
Func Tion

<p>Abstract. We consider the problem of estimating the intensity func- tion of a cyclic Poisson process. We suppose that only a single realization of the cyclic Poisson process is observed within a bounded 'window', and our aim is to estimate consistently the intensity function at a given point. A nearest neighbor estimator of the intensity function is proposed, and we show that our estimator is L2-consistent, as the window expands.<br />AMS 2010 subject classifications: 62E20, 62G05, 62G20.<br />Key words and phrases: cyclic Poisson process, cyclic intensity function, nonparametric estimation, nearest neighbor estimator, period, consis- tency, L2-convergence.</p>

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