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In this paper, a class of statistics based on high frequency observations ofoscillating and skew Brownian motions is considered. Their convergence ratetowards the local time of the underlying process is obtained in form of aCentral Limit Theorem. Oscillating and skew Brownian motions are solutions tostochastic differential equations with singular coefficients: piecewiseconstant diffusion coefficient or drift involving the local time. The result isapplied to provide estimators of the parameter of skew Brownian motion andstudy their asymptotic behavior. Moreover, in the case of the classicalstatistic given by the normalized number of crossings, the result is proved tohold for a larger class of It\^o-processes with singular coefficients.

Rates of convergence to the local time of Oscillating and Skew Brownian Motions