On estimation of delay location

Assume that we observe a stationary Gaussian process X(t), t ∈ [-r,T], which satisfies the affine stochastic delay differential equation, where W(t), t ≥ 0, is a standard Wiener process independent of X(t), t ∈ [-r,0], and av is a finite signed measure on [-r, 0], v ∈ Θ. The parameter v is unknown and has to be estimated based on the observation. In this paper we consider the case where Θ = (v0,v1, -∞ < v0 0 < v1 < ∞, and the measures av are of the form av = a + bv - b, where a and b are finite signed measure on [-r, 0] and bv is the translate of b by v. We study the limit behaviour of the normalized likelihoods, as T→ ∞, where PvT is the distribution of the observation if the true value of the parameter is v. A necessary and sufficient condition for the existence of a rescaling function δT such that ZT,v(u) converges in distribution to an appropriate nondegenerate limiting function Zv(u) is found. It turns out that then the limiting function Zv(u) is of the form, where H ∈ [1/2, 1] and BH(u), u ∈ ℝ, is a fractional Brownian motion with index H, and δT = T-1/(2H)ℓ(T) with a slowly varying function ℓ. Every H ∈ [1/2, 1] may occur in this framework. As a consequence, the asymptotic behaviour of maximum likelihood and Bayes estimators is found. © 2011 Springer Science+Business Media B.V.