A class of non-identifiable stochastic models

Abstract

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.

Predicting future citation behavior

“…This is not surprising. As Cane (1972Cane ( , 1977 demonstrated, in fairly general terms a stochastic counting process modeled as a mixture of Poisson processes has an equivalent formulation as a nonhomogeneous birth process. Even though the two formulations may have very different intuitive descriptions, it is impossible to distinguish between them on the basis of any observable data.…”

On the distribution theory of over-dispersion

“…T − n . In fact, this is the case for any compound Poisson accident distribution whose compounding distribution has finite moments (Cane 1977), hence also for the UGWD(a, k; ρ). This implies that the availability of information on the times of the occurrence of accidents is not sufficient to guide one?…”

“…spells" hypotheses. This is known as the discrimination problem between the compounded, contagion and generalized models for the negative binomial distribution and has been discussed by Arbous and Kerrich (1951); Bates and Neyman (1952);Gurland (1959) and Cane (1974Cane ( , 1977. For an extensive bibliography on the accident hypotheses mentioned, see Kemp (1970).…”

“…s (1971) attempt was on proving or disproving proneness by ranking individual accident performance on a scale based on their average interval between successive accidents. However, the first systematic study on how one can discriminate between the proneness and contagion models of the negative binomial distribution appears to be that by Cane (1974).…”

On individual risk

“…So why should You regard the limiting relative frequency of future failures, dependent as this is on Karl's randomly varying performance in his future attempts, as an appropriate measure of his risk of failing on this, his first attempt? As commented by Cane (1977) in a parallel context: "…if several clones were grown, each under the same conditions, an observer…might feel that the various values (e.g., of a limiting proportion-APD) they showed needed explanation, although these values could in fact be attributed to chance events." This point is relevant to the assessment of risk in the context of an individual's criminal career (cf.…”