The truth about the effective dimension

The effective dimension of a model is a useful measure of its complexity. In linear models, the trace of the so-called hat matrix is a convenient choice. It has strong links to variance estimation in mixed models, and it suggests straightforward partial effective dimensions for additive model components. Efron (2004) casts doubt on the trace of the hat matrix and advocates an alternative definition of the effective dimension, based on a covariance formula. Unfortunately, he uses the robust lowess smoother, which is strongly nonlinear. This blurs the issue and invalidates his conclusions. I show that the problems disappear if a linear smoother is being used. The computation of the trace of the hat matrix is much more efficient than using the covariance formula, which needs bootstrapping.

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