文档介绍：ARKIV FOR MATEMATIK Band 1 nr 17
Communicated 25 January 1950 by HARALD CRA~J[~R and TRYGVE NAGELL
Stochastic processes and statistical inference
By ULF GRENANDER
Introduction
The purpose of this thesis is, partly to sho~ the possibility of applying
statistical concepts and methods of inference to stochastic processes, and partly
to obtain practically working methods of this kind by studying special cases
of inference.
Time-series have been subjected to statistical treatment in a more or less
systematical way for a very long time, but unlike the case of finite dimen-
sional samples, there exists no unified theory. The extensive literature on
stochastic processes has but rarely touched upon questions of inference. On
the other hand, the attempts to treat time-series data do not seem to have
been much influenced by the theory of stochastic processes. This is specially
the case when considering a continuous time-parameter, which will be our main
interest in the following chapters. The treatment of the problem" in the present
dissertation is based on the general idea outlined in CRAMI~I~: Mathematical
methods of statistics -- to base statistical methods on the mathematical theory
of probability.
In the first two chapters we shall give a short survey of some fundamental
facts about stochastic processes and statistical inference. The third and fourth
.chapters will deal with the problem of testing hypotheses and the fifth with
estimation. Finally in the sixth chapter we shall show very shortly that prog-
nosis and filtering of time-series are questions similar to testing and estimation
and can be treated on analogous lines.
Some topics in the theory of stochastic processes
. Measure of probability. Let us consider an abstract space ~Q with the
~ollowing properties. The points in ~2 are denoted by co. In ~Q is defined a
Borelfield of sets containing also /2. On this Borelfield there is defined a
comple