Polya Process Research Articles

A Multivariate Polya Model of Brand Choice and Purchase Incidence

In this paper we develop an integrated stochastic model of purchase timing and brand selection which incorporates the influence of marketing mix variables, seasonality and trend, and also allows for various individual choice mechanisms. Our approach rests on the assumptions of a zero-order choice process, a Poisson timing process and purchase rates following a multivariate Gamma Distribution over the population, the scale parameters of which vary according to marketing activities and time. The resulting model is a Multivariate Polya Process, and the distribution of brand choice probabilities turns out to be a Generalized Dirichlet Distribution. Thus, most currently used zero-order models can be considered to be special cases of this approach. Furthermore, we derive a number of market diagnostics which provide insights into market structure and demonstrate the model's use for marketing strategy simulation. Based on extensive testing of the underlying hypotheses we finally validate the model using empirical data and show that it fits the market in question.

Data age and information value: a Polya process

The paper relates the value of information in a Polya process to the duration of the process and to the age of the most recently observed data set. A decision process is constructed to measure the impact of obtaining information concerning the true state of the system p periods earlier. The value to the decision-maker of decreasing the age of the observation is studied. A numerical example is presented.

Negative binomial distributions for point processes

Negative binomial point processes are defined for which all finite-dimensional distributions associated with disjoint bounded Borel sets are negative binomial in the usual sense. For these processes we study classical notions such as infinite divisibility, conditional distributions, Palm probabilities, convergence, etc. Negative binomial point processes appear to be of interest because they are mathematically tractable models which can be used in many situations. The general results throw some new light on some well-known special cases like the Polya process and the Yule process.

Spectres d'impulsions à ⪡un photoélectron⪢ des photomultiplicateurs

Analysing the histogram of anode pulse amplitudes allows a discussion of the hypotheses that have been proposed to account for the statistical processes of secondary multiplication in a photomultiplier (PM 56 DVP): a Poisson process, as was suggested by Lombard and Martin (1961), or a Polya process, as suggested by Prescott (1966). Comparison of experimental single-electron pulse spectra with calculated theoretical spectra is first performed by means of moment analysis, from which fractional variance, skewness (a measure of the asymmetry of the distribution), and excess of kurtosis (a measure of the sharpness of the peak and the extent of the tails, as compared with those of a Gaussian distribution), are deduced; then, by directly comparing the shapes of the spectra and minimizing the area between them. On the whole, this study on a focussed dynodes PM makes it clear that though a Polya process is a better approach to reality than a Poisson process, the gain distribution which it leads to does not correspond to the experimental spectrum. A better accord is obtained on assuming a dynode distribution including, in a first approximation, two Poisson distributions of distinct mean value and expressing an inhomogeneity of the secondary multiplication gain value different from that imagined by Prescott.

Stochastic processes and practical working models or why is the Polya process approach defective in modem practice and how cope with its deficiencies?

Abstract 1. One of the principal aims of mathematical statistics is the creation of theoretical models of probabilistically governed events or series of events. These models are mostly constructed by bringing together in suitable arrangements different elements from the tool-kit of general methods and premanufactured simple pieces of work at hand and known by the research worker.

On Esscher transforms of distribution functions defining a compound poisson process for large values of the parameter

Abstract F. Esscher has recently generalized the transformation of the distribution functions defining a Poisson process, which he introduced in 1932 and which Ammeter in 1948 applied to a Polya process. This generalization was first achieved for a positive compound Poisson process with an unconditioned distribution function of the risk (in the sequel called the risk distribution) and with a conditioned distribution relative to the hypothesis that a change has occurred in the random function attached to the process, of the size of such a change (in the sequel called claim distribution), which both are independent of the parameter t. In the following section 2 and throughout this paper the investigation will be restricted in the same way. An extension to cases where the risk distribution and/or the claim distribution depend on t can easily be derived by using the results given by the present author in a report to the congress in London (1964). Also the theory may easily be extended to cases where the changes in the random function attached to the process may be positive or negative. Later Esscher extended the transformation to concern a general distribution function for which the corresponding characteristic function is known (1963). This general case will be considered in section 1 here below.

Note on the Application of Compound Poisson Processes to Sickness and Accident Statistics

In order to fix our ideas an illustration of the theory for (a) a general elementary random process, (b) a compound Poisson process and (c) a Polya process shall be given here below following Ove Lundberg (On Random Processes and Their Application to Accident and Sickness Statistics, Inaug. Diss., Uppsala 1940).Let the continuous parameter t* be measured on an absolute scale from a given point of zero and consider the random function N* (t*) which takes only non-negative and integer values with N* (o) = o. This function constitutes a general elementary random process for which the conditional probability that N* (t*) = n relative to the hypothesis that shall be denoted , while the absolute probability that N* (t) = n i.e. shall be written If quantities of lower order than dt* are neglected, we may write for the conditional probability that N* (t* + dt*) = n + 1 relative to thehyp othesis that N* (t*) = n, i.e. is the intensity function of the process which is assumed to be a continuous function of t* (the condition of existence for the integral over the given interval of t* for every n > m may be substituted for the condition of continuity). The expectations for an arbitrary but fix value of t* of N* (t*) and p* (t*) will be denoted by the corresponding symbol with a bar so thatIf is uniformly bounded for all n in the interval o ≤ t* < T*, where T* is an arbitrary but fix value of t*, we have i.a. that