Model Building With the Aid of Stochastic Processes

Abstract

A majority of the papers appearing in statistical journals are concerned with development of statistical methods appropriate to particular theoretical models. Careful study of these publications is rewarded by a constantly growing repertory of well-founded techniques. However, for many practising statisticians, a recurrent problem is that of choosing a model appropriate to a given situation. It is important that the model shall be a sufficiently good approximation to reality, and yet be simple enough to yield practicable statistical techniques. Professor Cramer's paper provides an invaluable survey of possibilities open to a statistician faced by the need to choose a suitable model. The paper is based on the concept of 'stochastic processes' involving random variables in 'infinitely-dimensional space'. This does not mean, of course, that situations in which infinitely-many variables are measured are to be considered; rather, the values of random variables dependent on time, which latter variable is supposed capable of being given infinitely many values (in any time interval), are studied. It is possible to gain useful insight into the available models without a deep understanding of the nature of 'infinitelydimensional space' though this might be helpful. The following brief notes are intended as a guide to help gain a rapid appreciation of the useful results described in the paper. Sections 1-5 are generally of a historical nature. Section 6 contains important definitions: it should be noted that weak (second order) stationarity is defined in terms of moments only, not of distributions. The first part of Section 7 contains a lucid introduction to the nature of the Wiener process. The latter part of this Section may be found rather more difficult on first reading. Sections 8 and 9 on the Poisson process and generalizations thereof, are interesting reading. Quite probably the reader will recognize the 'risk process' of Section 9 in settings other than those mentioned by Professor Cram6r. Section 10 goes into little detail but is worth reference by those with problems likely to call for this kind of model. Section 11, and its specialized sequel, Section 12, contain a varied assortment of models. Here will be found the most intense concentration of potentially useful models in the paper, including the most common 'accident-proneness' model (part 3 of section 11) and the 'birth and death' and 'busy time' models of section 12. For most readers really pressed for time Sections 8-12 probably represent the best investment of their time. For those with more time, or with special interests in analysis of data of 'time-series type' the later sections of the paper, which are concerned with more formal 'stochastic process' theory, will well-repay reading. Particular mention may be made of the description of the Ornstein-Uhlenbeck process (section 14), and the integral representation ((23), section 16) of a stochastic process. N. L. Johnson