Estimation of the Mean of a Wiener Sheet

Abstract

A standard Wiener sheet shifted by an unknown parameter is observed above a decreasing curve Γ. By the help of a direct discrete approach and under weaker assumptions than in the paper of Arató, N. M. [3], an explicit formula is derived for the maximum likelihood estimator (MLE) of the shift parameter. The MLE is a weighted linear combination of the values at the endpoints of the curve Γ and weighted integrals of the observed process and its normal derivative along the curve Γ.Keywords. Wiener sheet, Radon-Nikodym derivatives, maximum likelihood estimator, stochastic integral along a curve, weighted integral of the normal derivative of a random process along a curve.

Parameter estimation of a shifted Wiener sheet

“…This definition is compatible with the notion of a weighted L 2 -Riemann integral of the normal derivative of an L 2 -process along a curve, see, for instance [12]. A weighted L 2 -Riemann integral of the normal derivative along a curve can be expressed by the help of weighted L 2 -Riemann integrals of the partial derivatives, namely…”

“…After forming a full square in the variable x i,j and integrating with respect to x i,j , S 1 can be replaced by (see [12])…”

“…In a similar way one can prove In order to determine Radon-Nikodym derivatives we have developed the following general method based on the results of [14, Section 2.3.2] (see e.g. [12,13]). Let X be a separable metric space.…”

“…Arató [11] used Rozanov's method to find the MLE of the shift parameter of a shifted Wiener sheet observed on a special domain. In [12] the authors considered the model of Arató [11], and applying an essentially simpler direct discrete approach they found the MLE of the shift parameter under much weaker conditions. In the present paper the discrete approach is applied for a more general model and more complicated domain of observations.…”