I observe a set of $n$ observations $(X_i,Y_i)$, $X_i, Y_i$ are both scalars. I wish to construct an optimization criterion which
- for low values of the tuning parameter $\lambda$ produces a cubic fit to the data and
- for high values of $\lambda$ produces an exact linear fit to the data
My proposed answer is $$\operatorname{argmin}_\beta \sum_i(Y_i-\beta_1 X_i-\beta_2X_i^3)^2+\frac{1}{\lambda}||\beta_1||+\lambda||\beta_2||$$ where $||\bullet||$ is $L_1$ norm.
and I am not sure if it is legit. Also the question asks to draw (on a single plot) three example curves ($\lambda=0,\lambda=\infty, 0<\lambda<\infty$ that might be produced by the criterion and I am not sure which curve should I draw.
Any help would be appreciated. Thanks!