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In this paper, we consider the kernel method to estimate sliced average variance estimation ( SAVE ).  SAVE  is a tool as  SIR  (sliced inverse regression), recommended to identify and to estimate the central dimension reduction ( CDR ) subspace.  CDR  subspace is the intersection of all dimension reduction subspaces which are at the base to describe the conditional distribution of the response  Y  given adimensional predictor vector  X .  SAVE  and even  SIR  are used to estimate  CDR  subspace. Two versions are very popular: slice version and kernel version. In this paper, we are looking at the kernel version. For Kernel SAVE version, two asymptotic properties have been demonstrated in particular: asymptotic normality and convergence in probability. And we know that these two properties, although important, are weak in front of almost sure convergence. However, until now, the strong consistency has not yet been obtained. In this paper, we obtain, under weaker assumptions, this asymptotic property.

Strong consistency of kernel method for sliced average variance estimation