Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-05-23T02:20:25.759Z Has data issue: false hasContentIssue false
  • English
  • Français

General convex stochastic orderings and related martingale-type structures

Part of: Distribution theory - Probability Functions of several variables Stochastic processes General theory of linear operators

Published online by Cambridge University Press:  01 July 2016

Francisco Vera  and
James Lynch
Francisco Vera*
Affiliation:
National Institute of Statistical Sciences
James Lynch*
Affiliation:
University of South Carolina
*
∗ Postal address: National Institute of Statistical Sciences, PO Box 14006, Research Triangle Park, NC 27709-4006, USA.
∗∗ Postal address: Department of Statistics, University of South Carolina, Columbia, SC 29208, USA. Email address: lynch@math.sc.edu
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.

Blackwell (1951), in his seminal work on comparison of experiments, ordered two experiments using a dilation ordering: one experiment, Y, is ‘more spread out’ in the sense of dilation than another one, X, if E(c(Y))≥E(c(X)) for all convex functions c. He showed that this ordering is equivalent to two other orderings, namely (i) a total time on test ordering and (ii) a martingale relationship E(Yʹ | Xʹ)=Xʹ, where (Xʹ,Yʹ) has a joint distribution with the same marginals as X and Y. These comparisons are generalized to balayage orderings that are defined in terms of generalized convex functions. These balayage orderings are equivalent to (i) iterated total integral of survival orderings and (ii) martingale-type orderings which we refer to as k-mart orderings. These comparisons can arise naturally in model fitting and data confidentiality contexts.

Keywords

Generalized convexitystochastic orderingiterated total time on testk-martbalayage

MSC classification

Primary: 60E15: Inequalities; stochastic orderings 60G48: Generalizations of martingales
Secondary: 26B25: Convexity, generalizations 47A20: Dilations, extensions, compressions
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 39 , Issue 1 , March 2007 , pp. 105 - 127
Copyright
Copyright © Applied Probability Trust 2007 

References

Blackwell, D. (1951). Comparison of experiments. In Proc. 2nd Berkeley Symp. Math. Statist. Prob., University of California Press, Berkeley, CA, pp. 93–102.Google Scholar
Blackwell, D. (1953). Equivalent comparisons of experiments. Ann. Math. Statist. 24, 265–272.Google Scholar
Dalenius, T. and Reiss, S. P. (2002). Data-swapping: a technique for disclosure control. J. Statist. Planning Infer. 6, 73–85.CrossRefGoogle Scholar
Defays, D. and Anwar, N. (1995). Microaggregation: a generic method. In Proc. 2nd Internat. Symp. Statist. Confidentiality, Office for Official Publications of the European Community, Luxembourg, pp. 69–78.Google Scholar
Denuit, M., Lefèvre, C. and Shaked, M. (1998). The s-convex orders among real random variables, with applications. Math. Inequalities Appl. 1, 585–613.Google Scholar
Denuit, M., Lefèvre, C. and Shaked, M. (2000). On s-convex approximations. Adv. Appl. Prob. 32, 994–1010.Google Scholar
Jaynes, E. T. (1957a). Information theory and statistical mechanics. Phys. Rev. 106, 620–630.CrossRefGoogle Scholar
Jaynes, E. T. (1957b). Information theory and statistical mechanics. II. Phys. Rev. 108, 171–190.Google Scholar
Karlin, S. (1968). Total Positivity. Stanford University Press.Google Scholar
Karlin, S. and Studden, W. J. (1966). Tchebycheff Systems: With Applications in Analysis and Statistics. John Wiley, New York.Google Scholar
Lindsay, B. G. (1989). Moment matrices: application in mixtures. Ann. Statist. 17, 722–740.Google Scholar
Lynch, J. (1988). Mixtures, generalized convexity and balayages. Scand. J. Statist. 15, 203–210.Google Scholar
Meyer, P. A. (1966). Probability and Potentials. Blaisdell, London.Google Scholar
Ross, S. M. (1983). Stochastic Processes. John Wiley, New York.Google Scholar
Sethuraman, J. (2002). Some extensions of the Skorohod representation theorem. Sankhyā A 64, 884–893.Google Scholar
Shaked, M. (1980). On mixtures from exponential families. J. R. Statist. Soc. 42, 192–198.Google Scholar
Shaked, M. and Shanthikumar, J. G. (1994). Stochastic Orders and Their Applications. Academic Press, San Diego, CA.Google Scholar
Strassen, V. (1965). The existence of probability measures with given marginals. Ann. Math. Statist. 36, 423–439.Google Scholar
Vera, F. and Lynch, J. (2005a). K-mart stochastic modeling using iterated total time on test transforms. In Modern Statistical and Mathematical Methods in Reliability (Ser. Qual. Reliab. Eng. Statist. 10), World Scientific, Singapore, pp. 395–409.Google Scholar
Vera, F. and Lynch, J. (2005b). Generalized convex stochastic orderings and related martingale-type structures. Tech. Rep., Department of Tech. Rep. Available at http://www.stat.sc.edu/∼veraf/.Google Scholar
Willenborg, L. and de Waal, T. (2001a). Elements of Statistical Disclosure Control (Lecture Notes Statist. 155). Springer, New York.Google Scholar
Willenborg, L. and de Waal, T. (2001b). Introduction to Biostatistics. Freeman and Company, San Francisco, CA.Google Scholar