On Penalized Least-Squares: Its Mean Squared Error and a Quasi-Optimal Weight Ratio

It is well known in a Random Effects Model, that the Best inhomogeneously LInear Prediction (inhomBLIP) of the random effects vector is equivalently generated by the standard Least-Squares (LS) approach. This LS solution is based on an objective function that consists of two parts, the first related to the observations and the second to the prior information on the random effects; for more details, we refer to the book by Rao, Toutenburg, Shalabh and Heumann (2008). We emphasize that, in this context, the second part cannot be interpreted as “penalization term”.

A very similar objective function, however, could be applied in the Gauss-Markov model where no prior information is available for the unknown parameters. In this case, the additional term would serve as “penalization” indeed as it forces the Penalized Least-Squares (PLS) solution into a chosen neighborhood, not specialized through the model. This idea goes, at least, back to Tykhonov (1963) and Phillips (1962) and has since become known as (a special case of) “Tykhonov regularization” for which the weight ratio between the first and the second term in the objective function determines the degree of smoothing to which the estimated parameters are subjected to. This weight ratio is widely known as “Tykhonov regularization parameter”; for more details, we refer to Grafarend and Schaffrin (1993) or Engl et al. (1996), for instance.