Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-12-01T00:50:58.319Z Has data issue: false hasContentIssue false
  • English
  • Français

A new form of Jensen's inequality and its application to statistical experiments

Published online by Cambridge University Press:  17 February 2009

D. J. F. Nonnenmacher  and
R. Zagst
R. Zagst
Affiliation:
Dept of Mathematics & Economics, Universität Ulm, Helmholtzstr. 18, D-89069 Ulm, Germany.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Jensen's inequality for the expectation of a convex function of a random variable is proved for a wide class of convex functions defined on a space of probability measures. The result is applied to statistical experiments using the concept of Blackwell-sufficiency. In particular, we show a monotonicity result for the expected information of Poisson-experiments. As an application to economics we consider the introduction of new production technologies.

Type
Research Article
Information
The ANZIAM Journal , Volume 36 , Issue 4 , April 1995 , pp. 389 - 398
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Araujo, A. and Giné, E., The central limit theorem for real and Banach valued random variables (John Wiley & Sons, New York, 1980).Google Scholar
[2]Bauer, H., Probability theory and elements of measure theory (Academic Press, London, 1981).Google Scholar
[3]Billingsley, P., Convergence and probability measures (John Wiley & Sons, New York, 1968).Google Scholar
[4]Billingsley, P., Probability and measure (John Wiley & Sons, New York, 1986).Google Scholar
[5]Blackwell, D., “Equivalent comparisons of experiments”, Ann. Math. Stat. 24 (1953) 265–272.CrossRefGoogle Scholar
[6]DeGroot, M. H., Optimal statistical decisions (McGraw-Hill, New York, 1970).Google Scholar
[7]DeGroot, M. H., “Changes in utility as information”, Theory and Decision 17 (1984) 287–303.CrossRefGoogle Scholar
[8]Giesy, D. P., “Strong law of large numbers for independent sequences of Banach space-valued random variables”, Lecture Notes in Mathematics, No. 526, (Springer, New York, 1976).CrossRefGoogle Scholar
[9]Heyer, H., Theory of statistical experiments (Springer, New York, 1982).CrossRefGoogle Scholar
[10]Hille, E. and Phillips, R. S., “Functional analysis and semi-groups”, AMS Colloquium Publications Vol. 31, (AMS, Providence, Rhode Island, 1957).Google Scholar
[11]Hinderer, K., Foundations of non-stationary dynamic programming with discrete time parameter (Springer, Berlin, 1970).CrossRefGoogle Scholar
[12]Hinderer, K., Grundbegriffe der wahrscheinlichkeitstheorie (Springer, Berlin, 1985).Google Scholar
[13]Cam, L. Le, Asymptotic methods in statistical decision theory (Springer, New York, 1986).CrossRefGoogle Scholar
[14]Marschak, J., “Economics of information systems”, J. Amer. Statist. Assoc. 66 (1971) 192–219.CrossRefGoogle Scholar
[15]Marschak, J. and Miyasawa, K., “Economic comparability of information systems”, Internat. Eco-nom. Rev. (1968) 137–174.CrossRefGoogle Scholar
[16]Perlman, M. D., “Jensen's inequality for a convex vector-valued function on an infinite-dimensional space”, J. Multivariate Anal. 4 (1974) 52–65.CrossRefGoogle Scholar
[17]Rieder, U. and Wagner, H., “Structured policies in the sequential design of experiments”, Ann. Oper. Res. 32 (1991) 165–188.CrossRefGoogle Scholar
[18]Rieder, U. and Zagst, R., “Monotonicity and bounds for convex stochastic control models”, ZOR - Methods and Models of Operations Research 39 (1994).Google Scholar
[19]Strasser, H., Mathematical theory of statistics (De Gruyter, Berlin, 1985).CrossRefGoogle Scholar
[20]Tonks, I., “Bayesian learning and the optimal investment decision of the firm”, The Economic Journal 93 (1983) 87–98.CrossRefGoogle Scholar
[21]Zagst, R., “Blackwell-Informativität in stochastischen Kontrollmodellen”, Ph. D. Thesis, University of Ulm, 1991.Google Scholar