Exact L2-Small Ball Asymptotics for Some Durbin Processes

We find the exact L2-small ball asymptotics for some Durbin processes. These processes are finitedimensional perturbations of a Brownian bridge B(t) and naturally appear in statistics as limit ones when one constructs goodness-of-fit tests of ω2-type for testing a sample for some distribution with estimated parameters. Earlier, in the work of Nazarov and Petrova, Kac–Kiefer–Wolfowitz processes (which correspond for testing normality) were considered, where a technique for obtaining asymptotics of oscillating integrals with a slowly varying amplitude has been developed. Due to this, it is possible to calculate the asymptotics of small deviations for Durbin processes for certain distributions (Laplace, logistic, Gumbel, gamma).

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Correspondence to Yu. P. Petrova.

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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 466, 2017, pp. 211–233.

Translated by S. Yu. Pilyugin.

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Petrova, Y.P. Exact L2-Small Ball Asymptotics for Some Durbin Processes. J Math Sci 244, 842–857 (2020).

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