Abstract
Explicit formulas are given to recursively generate the moments of the mean M for Dubins–Freedman random distribution functions with arbitrary base measure μ. Using a standard inversion formula for moments of a distribution on the unit interval, the distribution of M is approximated for several natural choices of μ. The support of the mean is also considered. It is shown that the support of M is connected whenever μ is concentrated on the vertical bisector of the unit square S, but may have arbitrarily many gaps otherwise.
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REFERENCES
Bloomer, L., and Hill, T. P. (2002). Random probability measures with given mean and variance. J. Theoret. Probab. 15(4), 919–937.
Cifarelli, D. M., and Regazzini, E. (1990). Distribution functions and means of Dirichlet processes. Ann. Statist. 18(1), 429–442.
Dubins, L. E., and Freedman, D. A. (1967). Random distribution functions. Proc. Fifth Berkeley Symposium Math. Statist. Probl. 2, 183–214.
Dubins, L. E., and Savage, L. J. (1965). How to Gamble if You Must. Inequalities for Stochastic Processes, McGraw-Hill, New York.
Feller, W. (1966). An Introduction to Probability Theory and Its Applications, Vol. 2, Wiley, New York.
Ferguson, T. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist. 1, 209–230.
Graf, S., Mauldin, R. D., and Williams, S. C. (1986). Random homeomorphisms. Adv. Math. 60, 239–359.
Hill, T. P., and Monticino, M. G. (1998). Constructions of random distributions via sequential barycenter arrays. Ann. Statist. 26(4), 1242–1253.
Kraft, C. H. (1964). A class of distribution function processes which have derivatives. J. Appl. Prob. 1, 385–388.
Mauldin, R. D., Sudderth, W. D., and Williams, S. C. (1992). Polya trees and random distributions. Ann. Statist. 20(3), 1203–1221.
Mauldin, R. D., and Williams, S. C. (1990). Reinforced random walks and random distributions. Proc. Amer. Math. Soc. 110(1), 251–258.
Mauldin, R. D., and Monticino, M. G. (1995). Randomly generated distributions. Israel J. Math. 91, 215–237.
Monticino, M. G. (1995). A Note on the Moments of the Mean for a Dubins-Freedman Prior, Technical Report, Department of Mathematics, University of North Texas.
Monticino, M. G. (1998). Constructing prior distributions with trees of exchangeable processes. J. Statist. Plann. Inference. 73, 113–133.
Shaked, M., and Shanthikumar, J. G. (1994). Stochastic Orders and Their Applications, Academic Press, London.
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Allaart, P. Moments of the Mean of Dubins–Freedman Random Probability Distributions. Journal of Theoretical Probability 16, 471–488 (2003). https://doi.org/10.1023/A:1023582913550
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DOI: https://doi.org/10.1023/A:1023582913550