Question 2: Suppose we have n sample pairs {xi, yi}, with a < x₁ <...<n<b. The smoothing cubic spline estimate f is defined as n f= * = arg min Σ (9. − f(x;))² + x [" (ƒ" (t))³²dt. A i=1 In the class, we mentioned that it happens that the minimizer of the above problem is unique and is a natural cubic spline with knots at the input point x₁,...,n. Here we will prove it. First, we assume that f is any twice differentiable function on [a, b]. 1. Show that there exists a natural cubic spline f with knots at ₁,..., n (in the form of linear combination of those basis functions) such that f(x) = f(xi), i = 1,..., n. Define h(X) = f(X)-f(X). ["* ƒ"(x)h"(x)dx = 0. 2. Prove the following claim Hint: you may need to use integration by parts. 3. Now, we can show ["* ƒ" (x)²³dx ≤ ["°*ƒ"(x)²dx, with equality if and only if h"(x) = 0 for all x [a, b]. Note that h"(x) = 0 implies that h must be linear, and since we already know that h(x) = 0 for all x₁, this is equivalent to h(x) = 0. And we finish the proof.

Fig: 1