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Exponential models, brownian motion, and independence
Author(s) -
Seshadri V.
Publication year - 1988
Publication title -
canadian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.804
H-Index - 51
eISSN - 1708-945X
pISSN - 0319-5724
DOI - 10.2307/3314728
Subject(s) - exponential function , foliation (geology) , mathematics , independence (probability theory) , brownian motion , exponential family , combinatorics , mathematical analysis , mathematical physics , statistics , geochemistry , metamorphic rock , geology
Some examples of steep, reproductive exponential models are considered. These models are shown to possess a τ‐parallel foliation in the terminology of Barndorff‐Nielsen and Blaesild. The independence of certain functions follows directly from the foliation. Suppose X(t) is a Wiener process with drift where X(t) = W(t) + ct , 0 < t < T. Furthermore let Y = max [ X(s), 0 < s < T ]. The joint density of Y and X = X(T) , the end value, is studied within the framework of an exponential model, and it is shown that Y(Y – X ) is independent of X . It is further shown that Y(Y – X) suitably scaled has an exponential distribution. Further examples are considered by randomizing on T .