Statistical Inference for Stochastic Processes

, Volume 14, Issue 3, pp 273–305 | Cite as

On estimation of delay location

  • Alexander A. Gushchin
  • Uwe Küchler


Assume that we observe a stationary Gaussian process X(t), \({t \in [-r, T]}\) , which satisfies the affine stochastic delay differential equation
$$d X(t) = \int\limits_{[-r,0]}X(t+u)\, a_\vartheta (du)\,dt +dW(t), \quad t\ge 0,$$
where W(t), t ≥ 0, is a standard Wiener process independent of X(t), \({t\in [-r, 0]}\) , and \({a_\vartheta}\) is a finite signed measure on [−r, 0], \({\vartheta\in\Theta}\) . The parameter \({\vartheta}\) is unknown and has to be estimated based on the observation. In this paper we consider the case where \({\Theta=(\vartheta_0,\vartheta_1)}\) , \({-\infty\,<\,\vartheta_0 <0 \,<\,\vartheta_1\,<\,\infty}\) , and the measures \({a_\vartheta}\) are of the form
$$a_\vartheta = a+b_\vartheta-b,$$
where a and b are finite signed measure on [−r, 0] and \({b_\vartheta}\) is the translate of b by \({\vartheta}\) . We study the limit behaviour of the normalized likelihoods
$$Z_{T,\vartheta}(u) = \frac{dP_T^{\vartheta+\delta_T u}}{dP_T^\vartheta}$$
as T→ ∞, where \({P_T^\vartheta}\) is the distribution of the observation if the true value of the parameter is \({\vartheta}\) . A necessary and sufficient condition for the existence of a rescaling function δ T such that \({Z_{T,\vartheta}(u)}\) converges in distribution to an appropriate nondegenerate limiting function \({Z_{\vartheta}(u)}\) is found. It turns out that then the limiting function \({Z_{\vartheta}(u)}\) is of the form
$$Z_\vartheta(u)=\exp\left(B^H(u) - E[B^H(u)]^2/2\right),$$
where \({H\in[1/2,1]}\) and B H (u), \({u\in\mathbb{R}}\) , is a fractional Brownian motion with index H, and δ T  = T −1/(2H) (T) with a slowly varying function . Every \({H\in[1/2,1]}\) may occur in this framework. As a consequence, the asymptotic behaviour of maximum likelihood and Bayes estimators is found.


Affine stochastic delay differential equation Bayes estimator Fractional Brownian motion Local asymptotic normality Maximum likelihood estimator Regular variation Stationary Gaussian process 

Mathematics Subject Classification (2000)

62M09 34K50 60G10 60G30 62F12 


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© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteMoscowRussia
  2. 2.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany

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