On sharp $L_2$-small ball asymptotics for a family of Durbin processes

In this paper, we calculate sharp asymptotics of small deviations in the $L_2$ norm for a family of Gaussian random processes that are special finite-dimensional perturbations of the Brownian bridge. These processes arise as limiting processes in statistics when constructing goodness-of-fit tests for testing a sample for the $p$-Gaussian (generalized Gaussian) distribution in the case where the shift and/or scale parameters are estimated from the sample. For $p=1$ (the Laplace distribution) and $p=2$ (the normal distribution), these results were previously obtained in the works of\break Yu. P. Petrova (2017) and A. I. Nazarov–Yu. P. Petrova (2015), respectively.