0000000000007597
AUTHOR
Wolfgang J. Bühler
Testen von Homogenität bei sehr Seltenen Ereignissen
Der Begriff des sehr seltenen Ereignisses stammt von J. Neyman. Er tritt zuerst auf in der Arbeit [2] von W.J.BUHLER, H. FEIN, D. GOLDSMITH, J.NEYMAN und P.S.PURI. Sei M die Haufigkeit, mit der ein bestimmtes Ereignis bei N unabhangigen Wiederholungen eines Experiments oder einer Beobachtung auftritt. Ist dann die relative Haufigkeit M/N so klein, das die ublichen Approximationen der Verteilung durch die Normalverteilung oder die Poisson-Verteilung nicht moglich sind, aber gleichzeitig Mr/N von der Grosenordnung 1 fur ein r > 1 , so sprechen wir von sehr seltenen Ereignissen (die ursprungliche Definition in [2] mit r = 2 wurde von P.S.PURI [4] zu der hier verwendeten modifiziert).
Some Remarks on Exponential Families
Abstract The following facts may serve to provide a feeling about how restrictive the assumption of an exponential family is. (a) A one-parameter exponential family in standard form with respect to Lebesgue measure is a location parameter family iff it is normal with fixed variance. (b) It is a scale parameter family iff it is gamma with fixed shape parameter. Both facts are known (see Borges and Pfanzagl 1965; Ferguson 1962; Lindley 1958) but may not have received as much attention as they deserve. Under the assumption of differentiable densities, short and elementary proofs are given.
Characterizing extreme points of polyhedra an extension of a result by Wolfgang Bühler
This paper reconsiders the characterization given by Buhler admitting convex polyhedra of probability distributions on a finite or countable set which are given by systems of linear inequalities more complex than those considered before.
Hitting straight lines by compound Poisson process paths
In a recent article Mallows and Nair (1989,Ann. Inst. Statist. Math.,41, 1–8) determined the probability of intersectionP{X(t)=αt for somet≥0} between a compound Poisson process {X(t), t≥0} and a straight line through the origin. Using four different approaches (direct probabilistic, via differential equations and via Laplace transforms) we extend their results to obtain the probability of intersection between {X(t), t≥0} and arbitrary lines. Also, we display a relationship with the theory of Galton-Watson processes. Additional results concern the intersections with two (or more) parallel lines.
On the spatial spread of a pattern
A simple process is considered for the spread of a pattern in a spatially distributed population. Expressions are given for the stochastic means, variances and covariances. Central limit theorems are obtained for the number of individuals that have the pattern, and the time needed for the pattern to reach the n-th subpopulation.