A class of non-identifiable stochastic models

1977 ◽  
Vol 14 (3) ◽  
pp. 475-482 ◽  
Author(s):  
Violet R. Cane
Keyword(s):  
Stochastic Process ◽  
Poisson Process ◽  
Stochastic Models ◽  
Birth Process ◽  
De Finetti’S Theorem ◽  
Polya Process ◽  
To Come ◽  
Pólya Process ◽  
De Finetti's Theorem ◽  
Random Times

If events occur in time according to a stochastic process then, under not very restrictive conditions, each realization will appear to come from a Poisson process with its own rate provided that the events in the realization occur at effectively random times. This result is related to de Finetti's theorem on exchangeable events. Particular applications are to the Pólya process describing accidents and the pure birth process.

A class of non-identifiable stochastic models

1977 ◽  
Vol 14 (03) ◽  
pp. 475-482 ◽  
Author(s):  
Violet R. Cane
Keyword(s):  
Stochastic Process ◽  
Poisson Process ◽  
Stochastic Models ◽  
Birth Process ◽  
De Finetti’S Theorem ◽  
Polya Process ◽  
To Come ◽  
Pólya Process ◽  
De Finetti's Theorem ◽  
Random Times

If events occur in time according to a stochastic process then, under not very restrictive conditions, each realization will appear to come from a Poisson process with its own rate provided that the events in the realization occur at effectively random times. This result is related to de Finetti's theorem on exchangeable events. Particular applications are to the Pólya process describing accidents and the pure birth process.

scholarly journals On history-dependent mixed shock models

2021 ◽  
pp. 1-18
Author(s):  
Dheeraj Goyal ◽  
Maxim Finkelstein ◽  
Nil Kamal Hazra
Keyword(s):  
Poisson Process ◽  
Failure Rate ◽  
Survival Function ◽  
Shock Model ◽  
Homogeneous Poisson Process ◽  
Proposed Model ◽  
Mean Lifetime ◽  
Polya Process ◽  
Pólya Process ◽  
Non Homogeneous Poisson Process

In this paper, we consider a history-dependent mixed shock model which is a combination of the history-dependent extreme shock model and the history-dependent $\delta$ -shock model. We assume that shocks occur according to the generalized Pólya process that contains the homogeneous Poisson process, the non-homogeneous Poisson process and the Pólya process as the particular cases. For the defined survival model, we derive the corresponding survival function, the mean lifetime and the failure rate. Further, we study the asymptotic and monotonicity properties of the failure rate. Finally, some applications of the proposed model have also been included with relevant numerical examples.

An analysis of the Pólya point process

1983 ◽  
Vol 15 (01) ◽  
pp. 39-53 ◽  
Author(s):  
Ed Waymire ◽  
Vijay K. Gupta
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Critical Value ◽  
General Representation ◽  
Infinitely Divisible ◽  
Generating Functional ◽  
Representation Problem ◽  
Polya Process ◽  
Pólya Process ◽  
Stochastic Point Processes

The Pólya process is employed to illustrate certain features of the structure of infinitely divisible stochastic point processes in connection with the representation for the probability generating functional introduced by Milne and Westcott in 1972. The Pólya process is used to provide a counterexample to the result of Ammann and Thall which states that the class of stochastic point processes with the Milne and Westcott representation is the class of regular infinitely divisble point processes. So the general representation problem is still unsolved. By carrying the analysis of the Pólya process further it is possible to see the extent to which the general representation is valid. In fact it is shown in the case of the Pólya process that there is a critical value of a parameter above which the representation breaks down. This leads to a proper version of the representation in the case of regular infinitely divisible point processes.

Characterization of the Generalized Pólya Process and its Applications

2014 ◽  
Vol 46 (4) ◽  
pp. 1148-1171 ◽  
Author(s):  
Ji Hwan Cha
Keyword(s):  
Joint Distribution ◽  
Arrival Times ◽  
Stochastic Intensity ◽  
Polya Process ◽  
New Type ◽  
Compound Process ◽  
Pólya Process

In this paper some important properties of the generalized Pólya process are derived and their applications are discussed. The generalized Pólya process is defined based on the stochastic intensity. By interpreting the defined stochastic intensity of the generalized Pólya process, the restarting property of the process is discussed. Based on the restarting property of the process, the joint distribution of the number of events is derived and the conditional joint distribution of the arrival times is also obtained. In addition, some properties of the compound process defined for the generalized Pólya process are derived. Furthermore, a new type of repair is defined based on the process and its application to the area of reliability is discussed. Several examples illustrating the applications of the obtained properties to various areas are suggested.

Univariate and multivariate stochastic comparisons and ageing properties of the generalized Pólya process

2018 ◽  
Vol 55 (1) ◽  
pp. 233-253 ◽  
Author(s):  
F. G. Badía ◽  
C. Sangüesa ◽  
Ji Hwan Cha
Keyword(s):  
Intensity Function ◽  
Stochastic Comparisons ◽  
Polya Process ◽  
Pólya Process ◽  
Intensity Functions ◽  
Birth Processes

Abstract In this work we consider the generalized Pólya process with baseline intensity function r and parameters α and β recently studied by Cha (2014). The aim of this work is to provide both univariate and multivariate stochastic comparisons between two generalized Pólya processes with different baseline intensity functions and the same parameters α and β for the epoch and inter-epoch times of the two processes. The comparisons are analogous to stochastic comparisons in Belzunce et al. (2001) for two nonhomogeneous Poisson or pure-birth processes with different intensity functions. Moreover, we study both univariate and multivariate ageing properties of the epoch and inter-epoch times of the generalized Pólya process.

scholarly journals On intensities of modulated Cox measures

1998 ◽  
Vol 11 (3) ◽  
pp. 411-423 ◽  
Author(s):  
Jewgeni H. Dshalalow
Keyword(s):  
Stochastic Process ◽  
Stochastic Models ◽  
Queueing Systems ◽  
Random Measure ◽  
Random Measures ◽  
Explicit Formulas ◽  
State Dependent

In this paper we introduce and study functionals of the intensities of random measures modulated by a stochastic process ξ, which occur in applications to stochastic models and telecommunications. Modulation of a random measure by ξ is specified for marked Cox measures. Particular cases of modulation by ξ as semi-Markov and semiregenerative processes enabled us to obtain explicit formulas for the named intensities. Examples in queueing (systems with state dependent parameters, Little's and Campbell's formulas) demonstrate the use of the results.

A normal form theorem for Lω1p, with applications

1982 ◽  
Vol 47 (3) ◽  
pp. 605-624 ◽  
Author(s):  
Douglas N. Hoover
Keyword(s):  
Normal Form ◽  
Probability Model ◽  
Random Variables ◽  
Interpolation Theorem ◽  
Complete Theory ◽  
De Finetti’S Theorem ◽  
Form Theorem ◽  
Normal Form Theorem ◽  
De Finetti's Theorem

AbstractWe show that every formula of Lω1P is equivalent to one which is a propositional combination of formulas with only one quantifier. It follows that the complete theory of a probability model is determined by the distribution of a family of random variables induced by the model. We characterize the class of distribution which can arise in such a way. We use these results together with a form of de Finetti’s theorem to prove an almost sure interpolation theorem for Lω1P.

scholarly journals A $q$-Analogue of de Finetti's Theorem

10.37236/167 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Alexander Gnedin ◽  
Grigori Olshanski
Keyword(s):  
Boundary Problem ◽  
Galois Field ◽  
Infinite Group ◽  
Natural Action ◽  
De Finetti’S Theorem ◽  
De Finetti's Theorem ◽  
Invertible Matrices

A $q$-analogue of de Finetti's theorem is obtained in terms of a boundary problem for the $q$-Pascal graph. For $q$ a power of prime this leads to a characterisation of random spaces over the Galois field ${\Bbb F}_q$ that are invariant under the natural action of the infinite group of invertible matrices with coefficients from ${\Bbb F}_q$.

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