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C. R. Acad. Sci. Paris Ser. I Math. 325 83 86 CSÖRGÖ, M. and HORVÁTH, L., Invariance principles for

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Summary Page’s CUSUM test for detecting change in a sequence of independent observations is extended to the general parametric model involving nuisance parameters. The test statistic is the standardized efficient score vector. The model of nested random effects is analyzed in detail.

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Abstract  

Modified least squares processes (MLSP’s) and self-randomized MLSP’s are introduced in D[0, 1] for the slope in linear structural and functional error-in-variables models (EIVM’s). Sup-norm approximations in probability and, as a consequence, functional central limit theorems (CLT’s) are established for the data-based self-normalized versions of these MLSP’s and self-randomized MLSP’s. The MLSP’s are believed to be new types of objects of study, and the invariance principles for them constitute new asymptotics, in EIVM’s. Moreover, the obtained data-based functional CLT’s for the MLSP’s open up new possibilities for constructing various asymptotic confidence intervals (CI’s) for the slope that are named functional asymptotic CI’s here. Three special examples of such CI’s are given.

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BERKES, I. and HORVÁTH, L., almost sure invariance principles for logarithmic averages, Studia Sci. Math. Hungar. 33 (1997), 1-24. MR 98f :60054 Almost sure invariance principles for

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, H. , Philipp , W. 1982 Almost sure invariance principles for weakly dependent vector-valued random variables Ann. Probab. 10 689 – 701 10.1214/aop/1176993777 . [4] Einmahl , U. , Li

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Peligrad, M. , Invariance principles for mixing sequences of random variables, Ann. Probab. , 10 (1982), 968–981. MR 84c :60054 Peligrad M. Invariance principles for

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-Raikov Probab. Theory Relat. Fields 1992 93 407 438 Philipp, W. , Almost sure invariance principles

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