Approximations and Designs for Estimating Regression Coefficients of IndependentStochastic Processes

摘要

The problem of estimating the regression coefficients of a linear model for Nindependent stochastic processes Y(sub i), i = 1,...,N, is considered. The time integrated least square estimators depend on a random integral and, thus are hard to compute in practice. Optimal approximations of these estimators are given, which depend on the inverse of the covariance matrix generated by the observations of the process at a finite number of points of a regular sampling design and, thus are unstable and not robust. Simple nonparametric approximations based on an adjusted trapezoidal rule using regular sampling designs are derived. The estimators obtained from the two approximations are asymptotically unbiased, consistent, and asymptotically jointly normal. For processes Y(sub i), i = 1,...,N, having K quadratic mean derivatives, K = 0,1,2,..., exact rates of convergence of the performances in terms of variance are derived along with asymptotically optical designs.