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Conditional least squares estimation in nonstationary nonlinear stochastic regression models

Abstract : Let {Z(n)} be a real nonstationary stochastic process such that E(Z(n)vertical bar F(n-1)) <(a.s.) infinity and E(Z(n)(2)vertical bar F(n-1)) <(a.s.) infinity, where {F(n)} is an increasing sequence of sigma-algebras. Assuming that E(Z(n)vertical bar F(n-1)) = gn(theta(0), nu(0)) = g(n)((1))(theta(0)) + g(n)((2))(theta(0), nu(0)), theta(0) is an element of R(p), p < infinity, nu(0) is an element of R(q) and q <= infinity, we study the symptotic properties of <(theta)over cap>(n) := arg min(theta) Sigma(n)(k=1) (Z(k) = g(k)(theta, (nu) over cap))(2)lambda(-1)(k), where lambda(k) is F(k-1)-measurable, (nu) over cap = {(nu) over cap (k)} is a sequence of estimations of nu(0), g(n)(theta, (nu) over cap) is Lipschits in theta and g(n)((2))(theta(0), (nu) over cap) - g(n)((2))(theta, (nu) over cap) is asymptotically negligible relative to g(n)((1))(theta(0)) - g(n)((1)) (theta). We first generalize to this nonlinear stochastic model the necessary and sufficient condition obtained for the strong consistency of {(theta) over cap (n)} in the linear model. For that, we prove a strong law of large numbers for a class of submartingales. Again using this strong law, we derive the general conditions leading to the asymptotic distribution of (theta) over cap (n). We illustrate the theoretical results with examples of branching processes, and extension to quasi-likelihood estimators is also considered.
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Christine Jacob. Conditional least squares estimation in nonstationary nonlinear stochastic regression models. Annals of Statistics, Institute of Mathematical Statistics, 2010, 38 (1), pp.566-597. ⟨10.1214/09-AOS733⟩. ⟨hal-02660576⟩



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