Regularizingstratifiedmodels. We want to choose the two model parameters (1) and (2) to minimize

A(1)(1)y(1)2+A(2)(2)y(2)2+(1)(2)2

Where 0 is a parameter. The first term is the least squares residual for the first model on the first data set (say, smokers); the second term is the least squares residual for the second model on the second data set (say, non-smokers); the third term is a regularization term that encourages the two model parameters to be close to each other.

Note that when =0, we simply fit each model separately, and when is very large, we are essentially fitting one model to all the data. Of course the choice of an appropriate is obtained using out-of-sample validation (or cross-validation).

(a) Giveaformulafortheoptimal ((1)^,(2)^). If your formula requires one or more matrices to have linearly independent columns, give the condition(s) for which this holds true.

(b) Stratifyingacrossagegroups. Suppose we fit a model with each data point representing a person, and we stratify over the person's agegroup, which is a range of consecutive ages such as 18-24, 24-32, 33-45, and so on. Our goal is to fit a model for each age of k groups, with the parameters for adjacent age groups similar, or not too far, from each other. Suggestamethodfordoingthis.