Estimation for the function of a time deformation in the model of the stationary reduction (Q2740081)

This article deals with the problem of reduction of the non-stationary process \(Z(t)\), \(t\in T,\) to the stationary process. The non-stationary random process \(Z(t)\) with correlation function \(r(t,s)=EZ(t)Z(s)\) is modelled by \(Z(t)=\delta(\Phi(t))\), \(t\in T\), where \(\Phi:T\to R^1\) is a time deformation, \(\delta(s), s\in R^1\), is a stationary random process with zero mean and the correlation function \(R(s-t)\). The representation \(Z(t)=\delta(\Phi(t))\) is possible if and only if \(r(t,s)=R(\Phi(s)-\Phi(t))\), \(s,t\in T\). The author constructs the non-parametric consistent estimate of the time deformation \(\Phi(t)\) by the Baxter sums of the random process \(Z(t)\), \(t\in [0,1]\).