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

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.

1983 ◽  
Vol 15 (01) ◽  
pp. 39-53 ◽  
Author(s):  
Ed Waymire ◽  
Vijay K. Gupta

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.


2014 ◽  
Vol 46 (4) ◽  
pp. 1148-1171 ◽  
Author(s):  
Ji Hwan Cha

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.


2019 ◽  
Vol 34 (4) ◽  
pp. 484-506
Author(s):  
Ji Hwan Cha ◽  
F.G. Badía

Most of the multivariate counting processes studied in the literature are regular processes, which implies, ignoring the types of the events, the non-occurrence of multiple events. However, in practice, several different types of events may occur simultaneously. In this paper, a new class of multivariate counting processes which allow simultaneous occurrences of multiple types of events is suggested and its stochastic properties are studied. For the modeling of such kind of process, we rely on the tool of superposition of seed counting processes. It will be shown that the stochastic properties of the proposed class of multivariate counting processes are explicitly expressed. Furthermore, the marginal processes are also explicitly obtained. We analyze the multivariate dependence structure of the proposed class of counting processes.


1996 ◽  
Vol 76 (2) ◽  
pp. 770-787 ◽  
Author(s):  
J. C. Saunders ◽  
D. E. Doan ◽  
C. P. Poje ◽  
K. A. Fisher

1. Single-neuron behavior in the cochlear nerve of neonatal (3-day-old) chicks was examined after exposure to a 120-dB SPL pure tone (0.9 kHz) for 48 h. Exposed animals were tested after 0 days or 12 days of recovery. Nonexposed chicks, age-matched to the exposed animals, formed two control groups. 2. Spectral response plots were obtained from each cell. These plots described the neuron discharge rates in response to 1,767 tone burst stimuli, each with a unique frequency-intensity combination. The tone bursts were presented at frequencies between 0.1 and 4.5 kHz and for intensities between 0 and 100 dB SPL. From these plots the characteristic frequency (CF), CF threshold, and sharpness of tuning (Q10 dB) were derived for each cell. Frequency response-area functions at selected stimulus levels and rate-intensity functions at the CF were also constructed from the spectral response plots. In addition, spontaneous activity was determined. Data were obtained from 903 cells. 3. Neuron activity in the control cells revealed no differences between CF thresholds, Q10 dB, or spontaneous activity in the two age groups. However, age differences at all frequencies were noted in the rate-intensity functions. 4. A frequency-dependent loss in CF threshold was observed in the 0-day recovered cells. The threshold shift (relative to age-matched control cells) was 55-65 dB between 0.8 and 1.5 kHz, but only 10-15 dB between 0.1-0.4 kHz and 2.5-3.5 kHz. The exposed cells showed no loss in frequency selectivity (Q10 dB) at < 0.5 kHz, whereas above this frequency an increasing deterioration in tuning was noted. Spontaneous activity in the 0-day cells was suppressed across the entire range of CFs. The rate-intensity function of exposed cells had a steeper growth rate than that of control cells. 5. At 12 days of recovery, CF threshold, Q10 dB, and spontaneous activity all recovered to the levels exhibited by age-matched control cells. However, the rate-intensity function for cells with CFs between 0.8 and 1.0 kHz showed abnormal growth and higher discharge rates at saturation than the control cells. Outside of this frequency range the rate-intensity functions of control and exposed cells were similar to each other. 6. Recovery of function in the sound-damaged chick ear is accompanied by almost complete repair of the basilar papilla. The tectorial membrane, however, retains a major defect and only the lower layer of this membrane regenerates. An important observation in this presentation was the abnormal rate-intensity functions (in the 12-day recovered cells) reported for frequencies served by that region of the sensory epithelium where the tectorial membrane defect was found. This observation may be related to sustained structural damage to the short hair cell region of the papilla and/or alterations in the efferent control of papilla function mediated by the short hair cells.


2016 ◽  
Vol 46 (19) ◽  
pp. 9397-9406 ◽  
Author(s):  
Yarong Feng ◽  
Xing Chen ◽  
Liyi Jia ◽  
Xiruo Song ◽  
Hosam M. Mahmoud
Keyword(s):  

1985 ◽  
Vol 17 (3) ◽  
pp. 623-637 ◽  
Author(s):  
Paul Embrechts ◽  
Jens L. Jensen ◽  
Makoto Maejima ◽  
J. L. Teugels

Suppose Xi≧0 are i.i.d., i = 1, 2, ···. We derive a saddlepoint approximation for P{∑N(t)k=1Xk> y} as y→∞ and t is fixed, where N(t), t≧0, is either a Poisson or a Pólya process. These results are then compared and contrasted with the well-known Esscher approximation.


1977 ◽  
Vol 14 (3) ◽  
pp. 475-482 ◽  
Author(s):  
Violet R. Cane

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.


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