Summary
For n=1,2, ... let P n be a set of probability measures on the measurable space (X, A) containing a fixed nulldistribution P 0. We study approximations of P n n ={P n: P∈P n } for large n. Degeneracy is avoided by postulating that
-
(1)
no P n can be distinguished from P n 0 with probability almost 1 (P∈P n );
-
(2)
events of small P 0-probability contain little information.
The second condition turns out to be equivalent to the existence of nondegenerating multinomial approximations to P n n obtained from partitions of the sample space X. In particular we consider the case that (X, A) is the real axis endowed with the field of Borel sets and that dP/dP 0 (x) are monotonic functions of x. If (1) and (2) are satisfied, the family P n n can be approximated by a multinomial family obtained from an interval partition of the line which depends on the accuracy of approximation, but not on n.
Article PDF
Similar content being viewed by others
References
LeCam, L.: Sufficiency and approximate sufficiency. Ann. Math. Statist. 35, 1419–1455 (1964)
LeCam, L.: Likelihood functions for large numbers of independent observations. Festschrift for J. Neyman, F.N. David editor, pp. 167–187. London-New York: Wiley and Sons 1966
LeCam, L.: Théorie asymptotique de la décision statistique. L'université de Montréal (1969)
LeCam, L.: On the assumptions used to prove asymptotic normality of maximum likelihood estimates. Ann. Math. Statist. 41, 802–828 (1970)
Huber, P.J.: Robust estimation of a location parameter. Ann. Math. Statist. 35, 73–101 (1964)
Wald, A.: Tests of statistical hypotheses concerning several parameters when the number of observations is large. Trans. Amer. Math. Soc. 54, 426–482 (1943)
Author information
Authors and Affiliations
Additional information
To Prof. L. Schmetterer on the occasion of his 60th birthday
Rights and permissions
About this article
Cite this article
Müller, D.W. Asymptotically multinomial experiments and the extension of a theorem of wald. Z. Wahrscheinlichkeitstheorie verw Gebiete 50, 179–204 (1979). https://doi.org/10.1007/BF00533639
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00533639