We find the exact L_{2}-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 ω^{2}-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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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 *L*_{2}-Small Ball Asymptotics for Some Durbin Processes.
*J Math Sci* **244, **842–857 (2020). https://doi.org/10.1007/s10958-020-04657-9

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