scholarly journals CONSISTENCY OF A UNIFORM KERNEL ESTIMATOR FOR INTENSITY OF A PERIODIC POISSON PROCESS WITH UNKNOWN PERIOD

2008 ◽  
Vol 7 (2) ◽  
pp. 31
Author(s):  
I W. MANGKU
Keyword(s):  
Poisson Process ◽  
Kernel Estimator ◽  
Joint Work ◽  
Classi Cation ◽  
Special Case ◽  
Unknown Period

<p>A uniform kernel estimator for intensity of a periodic Poisson process with unknowm period is presented and a proof of its consistency is discussed. The result presented in this paper is a special case of that in [3]. The aim of discussing a uniform kernel estimator is in order to be able to present a relatively simpler proof of consistency compared to that in [3]. This is a joint work with R. Helmers and R. Zitikis.<br />1991 Mathematics Subject Classi¯cation: 60G55, 62G05, 62G20.</p>

scholarly journals STRONG CONVERGENCE OF A UNIFORM KERNEL ESTIMATOR FOR INTENSITY OF A PERIODIC POISSON PROCESS WITH UNKNOWN PERIOD

2009 ◽  
Vol 8 (1) ◽  
pp. 1
Author(s):  
I W. MANGKU
Keyword(s):  
Strong Convergence ◽  
Poisson Process ◽  
Kernel Estimator ◽  
Joint Work ◽  
The One ◽  
Special Case ◽  
Unknown Period

<p>Strong convergence of a uniform kernel estimator for intensity of a periodic Poisson process with unknowm period is presented and proved. The result presented here is a special case of the one in [3]. The aim of this paper is to present an alternative and a relatively simpler proof of strong convergence compared to the one in [3]. This is a joint work with R. Helmers and R. Zitikis.<br />1991 Mathematics Subject Classication: 60G55, 62G05, 62G20.</p>

scholarly journals CONVERGENCE OF MSE OF A UNIFORM KERNEL ESTIMATOR FOR INTENSITY OF A PERIODIC POISSON PROCESS WITH UNKNOWN PERIOD

2009 ◽  
Vol 8 (2) ◽  
pp. 1
Author(s):  
I W. MANGKU
Keyword(s):  
Poisson Process ◽  
Mean Squared Error ◽  
Kernel Estimator ◽  
Joint Work ◽  
Squared Error ◽  
The One ◽  
Proof Of Convergence ◽  
Special Case ◽  
Unknown Period

Convergence of MSE (Mean-Squared-Error) of a uniform kernel estimator for intensity of a periodic Poisson process with unknowm period is presented and proved. The result presented here is a special case of the one in [3]. The aim of this paper is to present an alternative and a relatively simpler proof of convergence for the MSE of the estimator compared to the one in [3]. This is a joint work with R. Helmers and R. Zitikis.

A model for interaction of a Poisson and a renewal process and its relation with queuing theory

1972 ◽  
Vol 9 (2) ◽  
pp. 451-456 ◽  
Author(s):  
Lennart Råde
Keyword(s):  
Poisson Process ◽  
Renewal Process ◽  
Queuing Theory ◽  
Random Number ◽  
Sufficient Conditions ◽  
Stieltjes Transform ◽  
Time Intervals ◽  
Response Process ◽  
Necessary And Sufficient ◽  
Special Case

This paper discusses the response process when a Poisson process interacts with a renewal process in such a way that one or more points of the Poisson process eliminate a random number of consecutive points of the renewal process. A queuing situation is devised such that the c.d.f. of the length of the busy period is the same as the c.d.f. of the length of time intervals of the renewal response process. The Laplace-Stieltjes transform is obtained and from this the expectation of the time intervals of the response process is derived. For a special case necessary and sufficient conditions for the response process to be a Poisson process are found.

Equilibrium probability distributions for low density highway traffic

1966 ◽  
Vol 3 (1) ◽  
pp. 247-260 ◽  
Author(s):  
G. F. Newell
Keyword(s):  
Poisson Process ◽  
Probability Distributions ◽  
Initial Distribution ◽  
Order Effects ◽  
Highway Traffic ◽  
Wide Range ◽  
Headway Distribution ◽  
Equilibrium Distributions ◽  
Initial Distributions ◽  
Special Case

If on a long homogeneous highway there is no interaction between cars, then, under a wide range of conditions, an initial distribution of cars will in the course of time tend toward that of a Poisson process with statistically independent velocities for the cars in any finite interval of highway. Here we will generalize this known property to obtain the following. Suppose cars do interact in such a way as to delay a car when it passes another, but the density of cars is so low that we can neglect simultaneous interactions between three or more cars. There will again be equilibrium distributions of cars to which general classes of initial distributions will converge. These equilibrium distributions are superpositions of two statistically independent processes, one a Poisson process of single free cars with statistically independent velocities, and the other a Poisson process of interacting pairs of cars with various velocities. In the limit of zero interaction, the density of pairs vanishes leaving only the Poisson process of single cars as a special case. To the same order of approximation, including the first order effects of interactions, the headway distribution between consecutive cars will still have exponential tail outside the range of interaction.

A Risk Model with Delayed Claims

2013 ◽  
Vol 50 (03) ◽  
pp. 686-702 ◽  
Author(s):  
Angelos Dassios ◽  
Hongbiao Zhao
Keyword(s):  
Asymptotic Formula ◽  
Poisson Process ◽  
Ruin Probability ◽  
Risk Model ◽  
Poisson Model ◽  
Exact Formula ◽  
Poisson Models ◽  
Asymptotic Expressions ◽  
Special Case

In this paper we introduce a simple risk model with delayed claims, an extension of the classical Poisson model. The claims are assumed to arrive according to a Poisson process and claims follow a light-tailed distribution, and each loss payment of the claims will be settled with a random period of delay. We obtain asymptotic expressions for the ruin probability by exploiting a connection to Poisson models that are not time homogeneous. A finer asymptotic formula is obtained for the special case of exponentially delayed claims and an exact formula is obtained when the claims are also exponentially distributed.

scholarly journals Kekonvergenan MSE Penduga Kernel Seragam Fungsi Intensitas Proses Poisson Periodik dengan Tren Fungsi Pangkat

2012 ◽  
Vol 3 (1) ◽  
pp. 204
Author(s):  
Ro’fah Nur Rachmawati
Keyword(s):  
Stochastic Process ◽  
Poisson Process ◽  
Continuous Time ◽  
Kernel Estimator ◽  
Intensity Function ◽  
Rank Function ◽  
Service Center ◽  
Discrete State ◽  
Number Of Customers ◽  
Discrete State Space

The number of customers who come to a service center will be different for each particular time. However, it can be modeled by a stochastic process. One particular form of stochastic process with continuous time and discrete state space is a periodic Poisson process. The intensity function of the process is generally unknown, so we need a method to estimate it. In this paper an estimator of kernel uniform of a periodic Poisson process is formulated with a trend component in a rank function (rank coefficient 0 <b <1 is known, and the slope coefficient of the power function (trend) a> 0 is known). It is also demonstrated the convergenity of the estimators obtained. The result of this paper is a formulation of a uniform kernel estimator for the intensity function of a periodic Poisson process with rank function trends (for the case “a” is known) and the convergenity proof of the estimators obtained.

A finite dam with variable release rate

1975 ◽  
Vol 12 (01) ◽  
pp. 205-211 ◽  
Author(s):  
G. F. Yeo
Keyword(s):  
Poisson Process ◽  
Release Rate ◽  
Numerical Results ◽  
Basic Method ◽  
Carry Over ◽  
Special Case ◽  
Instantaneous Release ◽  
Finite Dam

This note considers a finite dam fed by independently and identically distributed (i.i.d.) inputs, being either (i) of at least size β (&gt; 0) or (ii) negative exponentially distributed, occurring in a Poisson process. The instantaneous release rate may be a function r(·) of the content; additional and numerical results are given for the special case where r(x) = µxα (0 ≦ α&lt;∞, 0 &lt; µ &lt;∞) is proportional to the αth power of the content. The basic method used in [7] for the special case r(x) = µx for obtaining the distribution of the number of steps and of the time to first overflowing is shown to carry over almost completely in case (i), but only partially so in case (ii).

A finite dam with variable release rate

1975 ◽  
Vol 12 (1) ◽  
pp. 205-211 ◽  
Author(s):  
G. F. Yeo
Keyword(s):  
Poisson Process ◽  
Release Rate ◽  
Numerical Results ◽  
Basic Method ◽  
Carry Over ◽  
Special Case ◽  
Instantaneous Release ◽  
Finite Dam

This note considers a finite dam fed by independently and identically distributed (i.i.d.) inputs, being either (i) of at least size β (> 0) or (ii) negative exponentially distributed, occurring in a Poisson process. The instantaneous release rate may be a function r(·) of the content; additional and numerical results are given for the special case where r(x) = µxα (0 ≦ α<∞, 0 < µ <∞) is proportional to the αth power of the content. The basic method used in [7] for the special case r(x) = µx for obtaining the distribution of the number of steps and of the time to first overflowing is shown to carry over almost completely in case (i), but only partially so in case (ii).

A model for interaction of a Poisson and a renewal process and its relation with queuing theory

1972 ◽  
Vol 9 (02) ◽  
pp. 451-456 ◽  
Author(s):  
Lennart Råde
Keyword(s):  
Poisson Process ◽  
Renewal Process ◽  
Queuing Theory ◽  
Random Number ◽  
Sufficient Conditions ◽  
Stieltjes Transform ◽  
Time Intervals ◽  
Response Process ◽  
Necessary And Sufficient ◽  
Special Case

This paper discusses the response process when a Poisson process interacts with a renewal process in such a way that one or more points of the Poisson process eliminate a random number of consecutive points of the renewal process. A queuing situation is devised such that the c.d.f. of the length of the busy period is the same as the c.d.f. of the length of time intervals of the renewal response process. The Laplace-Stieltjes transform is obtained and from this the expectation of the time intervals of the response process is derived. For a special case necessary and sufficient conditions for the response process to be a Poisson process are found.

A simple proof of the multivariate random time change theorem for point processes

1988 ◽  
Vol 25 (01) ◽  
pp. 210-214 ◽  
Author(s):  
Timothy C. Brown ◽  
M. Gopalan Nair
Keyword(s):  
Poisson Process ◽  
Simple Proof ◽  
Point Processes ◽  
Time Change ◽  
Random Time ◽  
Random Time Change ◽  
Special Case

A simple proof of the multivariate random time change theorem of Meyer (1971) is given. This result includes Watanabe's (1964) characterization of the Poisson process; even in this special case the present proof is simpler than existing proofs.

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