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I am asked to show that the solution to a smoothing splines problem of the form $$ \text{PRSS}(f,\lambda) = \sum_{i=1}^N\left[y_i-f(x_i)\right]^2 + \lambda \int f''(t)^2 dt, $$ with $$ f(x) = \sum_{j=1}^{N+4} \gamma_j B_j(x), $$ where $B_j(x)$ is a natural spline, reduces to the regular least-squares fit as $\lambda \rightarrow \infty$.

I understand that the first term disappears in the limit and that we are left with minimizing $$ \int f''(t)^2 dt $$ but I am not sure how to transform this into something that resembles the least squares. Could anyone please provide me a hint as to how to get started?

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  • $\begingroup$ Since PRSS is a differentiable function of the $\gamma_j,$ it is natural to find critical values by equating the gradient of PRSS to zero. $\endgroup$ – whuber Sep 19 '18 at 17:29
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When $\lambda \rightarrow \infty$, $\text{PRSS}(f,\lambda)$ is finite if and only if $$ \int f''(t)^2 dt = 0 \implies f''(t) = 0, $$ which is only true if $f(t)$ is a linear function of $t$. The objective is then reduced into $$ \sum_{i=1}^N\left[y_i-\beta_0 - \beta_1x_i\right]^2, $$ which is the objective for classical linear least-squares regression.

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    $\begingroup$ This is good intuition but it lacks rigor: it is not necessarily the case that a sequence of solutions to a sequence of minimization problems will converge to a solution of the limit of the minimization problems. The objective function you write down here does not agree with the objective you wrote in your question, either. $\endgroup$ – whuber Sep 25 '18 at 17:25

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