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

Consider a repeated measurement regression model yij=g(xi)+εij where i=1,…,n, j=1,…,m, yij's are responses, g(·) is an unknown function, xi's are design points, εij's are random errors with a one-way error component structure, i.e. εij=μi+νij, μi and νij's are i.i.d random variables with mean zero, variance σμ2 and σν2, respectively. This paper focuses on estimating σμ2 and σν2. It is well known that although the residual-based estimator of σν2 works very well the residual-based estimator of σμ2 works poorly, especially when the sample size is small. We here propose a difference-based estimation and show the resulted estimator of σμ2 performs much better than the residual-based one. In addition, we show the difference-based estimator of σν2 is equal to the residual-based one. This explains why the residual-based estimator of σν2 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.