Difference-based estimation for error variances in repeated measurement regression models

Consider a repeated measurement regression model yij=g(xi)+[epsilon]ij where i=1,...,n, j=1,...,m, yij's are responses, g(·) is an unknown function, xi's are design points, [epsilon]ij's are random errors with a one-way error component structure, i.e. [epsilon]ij=[mu]i+[nu]ij, [mu]i and [nu]ij's are i.i.d random variables with mean zero, variance and , respectively. This paper focuses on estimating and . It is well known that although the residual-based estimator of works very well the residual-based estimator of works poorly, especially when the sample size is small. We here propose a difference-based estimation and show the resulted estimator of performs much better than the residual-based one. In addition, we show the difference-based estimator of is equal to the residual-based one. This explains why the residual-based estimator of works very well even when the sample size is small. Another advantage of the difference-based estimation is that it does not need to estimate the unknown function g(·). The asymptotic normalities of the difference-based estimators are established.