Estimation of a regression function on a Poisson process
Let \(f_ 0\) be an \(R_+\times R\)-valued point process, let \(f_ 0(A)\) be the total number of points of \(f_ 0\) falling in A, and let \(\ell =f_ 0(R_+\times R)\) be a.s. finite. The first projection of \(f_ 0\) is assumed to be a Poisson process. For \(\ell \geq 1\), let \((X_ 1,Y_ 1),(X_ 2,Y_ 2), ... ,(X_{\ell},Y_{\ell})\) be the points of the process \(f_ 0\). It is assumed that \[ \psi (x):=E(Y_ j| \quad X_ 1=x_ 1,...,X_{j-1}=x_{j-1},X_ j=x,X_{j+1}=x_{j+1},...,X_{\ell}=x\quad_{\ell}) \] depends neither on \(\ell\) nor on j. On the basis of n i.i.d. copies of \(f_ 0\), a simple estimator \(\psi_ n(x)\) of the regression function \(\psi\) (x) is constructed, and it is shown that \((\forall x\in R_+)\) \(E(\psi_ n(x)-\psi (x))^ 2\to 0\) as \(n \to\infty.\)
sufficient conditions of weak consistency
scientific article; zbMATH DE number 4050769