Redaktion: K. Sigmund, G. Greschonig (Univ. Wien, Strudlhofgasse ...

158 Wahrscheinlichkeitstheorie, Statistik
where
r
∑
1 i�
ki � n and
r
∑ pi � 1�
1 i�
The task of this paper is to find all maxima of (1), usually called the modes of the
multinomial distribution.
Two methods for this exist. One is based on Moran’s inequality, which is not
presented in its sharper version in textbooks like Feller [1957] or Johnson– Kotz–
Balakrishnan [1997]. It is slow for large values of the number r of the pi. By far
more effective is the idea of Finucan [1964] to consider the step–function
s� N���
r
∑
1¨ i� N pi© �
The jumps of s� N� can be determined quite easily. If s� N��� n holds, ki �
N pi©
are the components of the unique mode. If there are more than one mode their ¨
components ki can be determined as � ki
¨ N pi© or � ki
¨ N pi© 1. The subscripts
�
i where 1 has to be subtracted are easy to determine. Numerical examples are
given which exhibit the method.
By a similar method the modes of a hypergeometric distribution can be calculated.
[1] W. Feller [1968], An Introduction to Probability Theory and its Applications.
J. Wiley 1950, 1957 and 1968.
[2] H.M. Finucan [1964], The mode of a Multinomial Distribution. Biometrica
51, 513-517.
[3] N.L. Johnson [1997], S. Kotz and N. Balkrishnan, Discrete Multivariate Distributions.
J. Wiley 1997.
Konsistenzaussagen für konvexe Regression und
intervallzensierte Daten
LUTZ DÜMBGEN
(gemeinsam mit Sandra Freitag, Geurt Jongbloed)
Institut für Mathematik, Wallstraße 40, 23560 Lübeck
duembgen@math.mu-luebeck.de
http://www.math.mu-luebeck.de/workers/duembgen/index.shtml
Angenommen für � n N beobachtet man Zufallsvariablen � Yni � xni� � F εni � (1
� i n) mit gegebenen festen Punkten � ����� � xn1 xnn, einer unbekannten konkaven
Regressionsfunktion F auf R und stochastisch unabhängigen Fehlern εni, welche
gewisse Momentenbedingungen erfüllen. Es wird gezeigt, dass der Kleinste-
Quadrate-Schätzer ˆFn für F unter Regularitätsannahmen an die Punkte xni folgende
Eigenschaft hat:
sup
x� J � ˆFn� x� � F� x� � � Op � � log� n��� n� 2� 5� �
(2)