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EXPLOITING THE WAITING TIME PARADOX: APPLICATIONS OF THE SIZE-BIASING TRANSFORMATION

Published online by Cambridge University Press:  06 March 2006

Mark Brown
Mark Brown
Affiliation:
Department of Mathematics, The City College, CUNY, New York, NY, E-mail: CYBERGARF@aol.com
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Abstract

We consider the transformation T that takes a distribution F into the distribution of the length of the interval covering a fixed point in the stationary renewal process corresponding to F. This transformation has been referred to as size-biasing, length-biasing, the renewal length transformation, and the stationary lifetime operator. We review and develop properties of this transformation and apply it to diverse areas.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 20 , Issue 2 , April 2006 , pp. 195 - 230
Copyright
© 2006 Cambridge University Press

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References

REFERENCES

Abate, J. & Whitt, W. (1996). An operational calculus for probability distributions via Laplace transforms. Advances in Applied Probability 28: 75–113.Google Scholar
Aldous, D. & Pitman, J. (1998). Tree-valued Markov chains derived from Galton-Watson processes. Annales de l'Instute Henri Poincaré: Probabilités et statistiques 34: 637–686.Google Scholar
Athreya, K.B. (2000). Change of measures for Markov chains and the L log L theorem for branching processes. Bernoulli 6: 323–338.Google Scholar
Bahadur, R.R. (1971). Some limit theorems in statistics. Philadelphia: SIAM.
Barlow, R.E. & Marshall, A.W. (1964). Bounds for distributions with monotone hazard rate, II. Annals of Mathematical Statistics 35: 1258–1274.Google Scholar
Barlow, R.E. & Proschan, F. (1975). Statistical theory of reliability and life testing: Probability models. New York: Holt, Rhinehart and Winston.
Bremaud, P. (1981). Point processes and queues: Martingale dynamics. New York: Springer-Verlag.
Brown, M. (1983). Approximating IMRL distributions by exponential distributions with applications to first passage times. Annals of Probability 11: 419–427.Google Scholar
Brown, M. (1985). A measure of variability based on the harmonic mean and its use in approximations. Annals of Statistics 13: 1239–1243.Google Scholar
Brown, M. (1987). Inequalities for distributions with increasing failure rate. In A.E. Gelfand, Contributions to the theory and applications of statistics. A volume in honor of Herbert Solomon. New York: Academic Press, pp. 3–17.
Brown, M. (1999). Interlacing eigenvalues in time reversible Markov chains. Mathematics of Operations Research 24: 847–864.Google Scholar
Brown, M. (2001). Exploiting the waiting time paradox: Applications of the renewal length transformation. City College, CUNY Technical Report, City College, New York.
Chernoff, H. (1972). Sequential analysis and optimal design. Philadelphia: SIAM.
Cox, D.R. & Lewis, P.A.W. (1966). The statistical analysis of series of events. New York: Wiley.
Daley, D.J. (1988). Tight bounds on the exponential approximation of some aging distributions. Annals of Probability 16: 414–423.Google Scholar
Dembo, A. & Rinott, Y. (1996). Some examples of normal approximations by Stein's method. In Random Discrete Structures, IMA Vol. Math. 76. New York: Springer-Verlag.
Diaconis, P. & Fill, J.A. (1990). Strong stationary times via a new form of duality. Annals of Probability 18: 1483–1522.Google Scholar
Donnelly, P. (1991). The heaps process, libraries and size-biased permutations. Journal of Applied Probability 28: 321–335.Google Scholar
Efron, B. & Morris, C. (1973). Combining possibly related estimation problems (with discussion). Journal of the Royal Statistical Society B 35: 379–421.Google Scholar
Ethier, S.N. (1990). The distribution of the frequencies of age-ordered alleles in a diffusion model. Advances in Applied Probability 22: 519–532.Google Scholar
Ethier, S.N. (1992). Equivalence of two descriptions of the age of alleles. Journal of Applied Probability 29: 185–189.Google Scholar
Feller, W. (1971). An introduction to probability theory and its applications, Vol. II, 2nd ed. New York: Wiley.
Geiger, J. (1996). Size-biased and conditioned random splitting trees. Stochastic Processes and their Applications 2: 187–207.Google Scholar
Gnedin, A.V. (1998). On convergence and extensions of size-biased permutations. Journal of Applied Probability 35: 642–650.Google Scholar
Goldstein, L. & Reinart, G. (1997). Stein's method and the zero bias transformation with application to simple random sampling. Annals of Applied Probability 4: 935–952.Google Scholar
Goldstein, L. & Rinott, Y. (1996). Multivariate normal approximations by Stein's method and size bias couplings. Journal of Applied Probability 33: 1–17.Google Scholar
Harkness, W.L. & Shataram, R. (1969). Convergence of a sequence of transformations of distribution functions. Pacific Journal of Mathematics 31: 403–415.Google Scholar
Hille, E. & Phillips, R.S. (1957). Functional analysis and semi-groups. American Mathematical Society Colloquium Publications, Vol. XXXI. Providence, RI: American Mathematical Society.
Hudson, H.M. (1978). A natural identity for exponential families with applications in multiparameter estimation. Annals of Statistics 6: 473–484.Google Scholar
Lyons, R., Pemantle, R., & Peres, Y. (1995). Conceptual proofs of L log L criteria for mean behavior of branching processes. Annals of Probability 23: 1125–1138.Google Scholar
Olofsson, P. (1998). The L log L condition for general branching processes. Journal of Applied Probability 3: 537–544.Google Scholar
Peng, J.C. (1975). Simultaneous estimation of the parameters of independent Poisson distributions. Stanford University Technical Report 78, Stanford, CA.
Phillips, R.S. (1954). An inversion formula for Laplace transforms and semi-groups of linear operators. Annals of Mathematics 59: 325–356.Google Scholar
Pitman, J. (1996). Random discrete distributions invariant under size-biased permutations. Advances in Applied Probability 28: 525–539.Google Scholar
Reinert, G. (1998). Stein's method and application to empirical measure. Sociedad Mathematica Mexicana 14: 65–120.Google Scholar
Rinott, Y. & Rotar, V. (2000). Normal approximations by Stein's method. Decisions in Economics and Finance 23: 15–29.Google Scholar
Ross, S.M. (1996). Stochastic processes, 2nd ed. New York: Wiley.
Shorrock, R.W. (1972). A limit theorem for inter-record times. Journal of Applied Probability 9: 219–223.Google Scholar
Stein, C. (1955). Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. In Proceedings of the third Berkeley symposium on mathematical statistics and probability. Berkeley: University of California Press, pp. 197–209.
Stein, C. (1973). Estimation of the mean of a multivariate normal distribution. Stanford University Technical Report 48, Stanford, CA.
Stein, C. (1974). Estimation of the mean of a multivariate normal distribution: I. Estimation of the means. Stanford University Technical Report 63, Stanford, CA.
Thorisson, H. (2000). Coupling, stationarity and regeneration. New York: Springer-Verlag.