Summary
Let (Z n )ℕ be a φ-mixing process which is valued in \(\mathbb{E} \subset \mathbb{R}^q \), g be a real measurable function defined on \(\mathbb{E}\), s and k be two positive integers. We suppose the existence of a function R satisfying R(.) = E(g(Z n+s )/[Z n−k+1 ,....Z n ]=.), ∀n∈ℕ, n≧k, and estimate the function R from a sequence {Z i, i=1, ..., n} by R n with
where K is a kernel of ℝkq and h n∈ℝ, h n>0. We give various conditions, on both the sequences (h n )ℕ and the sequence (φ n )ℕ which is associated with (Z n )ℕ, for the two following properties: uniform complete convergence to R for the functional estimator R n and complete convergence to 0 for the real random variable R n(Z n−k+1, ..., Z n)−R(Zn−k+1, ..., Z n). This last result concerns the predictor R n(Z n−k+1, ..., Z n) of g(Z n+s) from {Z i, i=1, ..., n} when the process (Z n )ℕ is stationary and markovian of order k.
These results are proved here for a more general problem: estimation of a regression E(Y/X) from {(X i, Y i), i=1, ..., n} when these couples are not independent. We also give a lemma which is an extension of the Bernstein inequality to the case of φ-mixing r.r.v.
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Collomb, G. Propriétés de convergence presque complète du prédicteur à noyau. Z. Wahrscheinlichkeitstheorie verw Gebiete 66, 441–460 (1984). https://doi.org/10.1007/BF00533708
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DOI: https://doi.org/10.1007/BF00533708