Identification of a hereditary system with distributed delay

Although extensive research has been done on the distributed system parameter identification problem (see [l] for a recent survey), relatively little attention has been paid, in the stochastic framework, to the problem of estimating unknown parameters that are infinite dimensional. This is not surprising, as Rozanov pointed out in his classic paper [2] that directly estimating infinite-dimensional parameters by the method of maximum likelihood either may not be possible or may turn out to be not consistent. To circumvent these difficulties, Grenander [3] suggested the method of sieves to estimate parameters in abstract spaces. We apply his idea to a canonical example where one wants to identify the distributed delay function of a linear hereditary system. Recently, Nguyen and Pham [4] applied the method of sieves to the problem of identifying a nonstationary diffusion model.