Change in the Mean in the Domain of Attraction of the Normal Law
DOI:
https://doi.org/10.17713/ajs.v35i2&3.358Abstract
Some weighted approximations in probability of self-normalized and Studentized partial sums processes are reviewed and also described in the context of studying the problem of change in the mean of random variables in the domain of attraction of the normal law. This survey of such results constitutes an extended abstract of the talk with the same title that was presented by Miklós Csörg? on July 18, 2005 in Mikulov, based on the joint works M. Csörg?, B. Szyszkowicz, and Q. Wang (2001), (2003) and (2004) by the three of us.References
Brodsky, B. E., and Darkhovsky, B. S. (1993). Nonparametric Methods in Change-Point Problems. Dordrecht: Kluwer.
Chistyakov, G. P., and Götze, F. (2001). Limit distributions of studentized means. Preprint.
Csörgő, M., Csörgő, S., Horváth, L., and Mason, D. M. (1986). Weighted empirical and quantile processes. The Annals of Probability, 14, 31-85.
Csörgő, M., and Horváth, L. (1988). Nonparametric methods for changepoint problems. In P. R. Krishnaiah and C. R. Rao (Eds.), Quality Control and Reliability (Vol. 7, p. 403-425). Amsterdam: Elsevier.
Csörgő, M., and Horváth, L. (1993). Weighted Approximations in Probability and Statistics. New York: Wiley.
Csörgő, M., and Horváth, L. (1997). Limit Theorems in Change-Point Analysis. New York: Wiley.
Csörgő, M., and Norvaiša, R. (2004). Weighted invariance principle for Banach space valued random variables. Lietuvos Matematikos Rinkinis, 44, 139–175.
Csörgő, M., Norvaiša, R., and Szyszkowicz, B. (1999). Convergence of weighted partial sums when the limiting distribution is not necessarily Radon. Stochastic Processes and their Applications, 81, 81-101.
Csörgő, M., Szyszkowicz, B., and Wang, Q. (2001). Donsker’s theorem and weighted approximations for self-normalized partial sums processes. Technical Report Series of the Laboratory for Research in Statistics and Probability, Carleton University,
Ottawa, 360.
Csörgő, M., Szyszkowicz, B., and Wang, Q. (2003). Donsker’s theorem for selfnormalized partial sums processes. The Annals of Probability, 31, 1228-1240.
Csörgő, M., Szyszkowicz, B., and Wang, Q. (2004). On weighted approximations and strong limit theorems for self-normalized partial sums processes. In L. Horváth and B. Szyszkowicz (Eds.), Asymptotic Methods in Stochastics (Vol. 44, p. 489-521).
AMS, Providence, Rhode Island: Fields Institute Communications.
Gine, E., Götze, F., and Mason, D. M. (1997). When is the Student t-statistic asymptotically normal? The Annals of Probability, 25, 1514-1531.
Gine, E., and Mason, D. M. (1998). On the LIL for self-normalized sums of IID random variables. Journal of Theory of Probability, 11, 351-370.
Gombay, E., and Horváth, L. (1994). An application of the maximum likelihood test to the change-point problem. Stochastic Processes and their Applications, 50, 161-171.
Gombay, E., and Horváth, L. (1996a). Applications for the time and change and the power function in change-point models. Journal of Statistical Planning and Inference, 52, 43-66.
Gombay, E., and Horváth, L. (1996b). On the rate of approximations for maximum likelihood test in chanage-point models. Journal of Multivariate Analysis, 56, 120-152.
Griffin, P. S. (2002). Tightness of the Student t-statistic. Electronic Communications in Probability, 7, 181-190.
Logan, B. F., Mallows, C. L., Rice, S. O., and Shepp, L. A. (1973). Limit distributions of self-normalized sums. The Annals of Probability, 1, 788–809.
Orasch, M., and Pouliot, W. (2004). Tabulating weighted sup-norm functionals used in change-point analysis. Journal of Statistical Computation and Simulation, 74, 249-276.
“Student”. (1908). The probable error of a mean. Biometrika, 6, 1-25.
Szyszkowicz, B. (1991). Weighted stochastic processes under contiguous alternatives. Comptes Rendus Mathématiques de l’Académie des Sciences, La Société Royale du Canada, 13, 211-216.
Szyszkowicz, B. (1996). Weighted approximations of partial sum processes in d[0;1). i. Studia Scientiarum Mathematicarum Hungarica, 31, 323-353.
Szyszkowicz, B. (1997). Weighted approximations of partial sum processes in d[0;1). ii. Studia Scientiarum Mathematicarum Hungarica, 33, 305-320.
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