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Chapter 14, Question 7 of Van Der Vaart's Asymptotic Statistics is to find the probability density $f$ minimizing $\int y^2f(y)dy$ over all densities $f$ bounded by $1$. Question 8 is to find the pdf minimizing $\int f^2(y)dy$ over all $f$ with mean $0$ and variance $1$. I have an idea how to solve these problems with a lot of calculus but my question is how to solve them using the hint he suggests. He says to use the following fact. "Suppose $\phi:\mathcal{F}\mapsto\mathbb{R}$ and $\psi:\mathcal{F}\mapsto\mathbb{R}^m$ are arbitrary maps on an arbitrary set $\mathcal{F}$ and we wish to find the minimum value of $\phi$ over $\{f\in\mathcal{F}:\psi(f)=0\}$. If the map $f\mapsto\phi(f)+a^T\psi(f)$ attains its min over $\mathcal{F}$ at $f_a$ for each fixed $a$ in an arbitrary set $A$, and there exists $a_0\in A$ such that $\psi(f_{a_0})=0$, then the desired min is $\phi(f_{a_0})$." I can see that the constraints like mean $0$ and variance $1$ correspond to $\psi$, and $\phi$ is the objective like $\int f^2(y)dy$, but I don't see how the hint helps. Finding the minimizing $f_a$ looks as hard as the original problem.

kimla.2
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  • Finding $f_a$ is easier than finding $f$, because the constraint $\psi(f) = 0$ has been lifted. – SolubleFish Mar 30 '22 at 06:55
  • @SolubleFish But at the expense of making the objective much messier. If this were optimizing in a space where I could just differentiate, that wouldn't make much of a difference. But domain here are functions. – kimla.2 Mar 30 '22 at 07:23
  • Isn't this issue to be placed into "calculus of variations" ? – Jean Marie Mar 30 '22 at 07:39
  • This is an optimization problem over a space of functions (which, as @JeanMarie pointed out, probably falls into the calculus of variations). It is still more manageable to solve a class of such problems without constraints, than the problem with contraints. – SolubleFish Mar 30 '22 at 14:17
  • Minimizing $\int y^2f(y),dy$ subject to $\int f(y),dy=1$ is equivalent to minimizing $\int(y^2-c^2)f(y),dy$ subject to $\int f(y),dy=1$. This suggest the optimal $f$:

    $$f_0(y)=\begin{cases}1&,\text{ if }y^2<c^2 \ 0 &,\text{ if }y^2\ge c^2\end{cases}$$ Solving for $c$, you get a density for uniform distribution. The other question is related to a derivation of the Epanechnikov kernel. Not quite the same, but something similar: https://math.stackexchange.com/q/46590/321264.

     – StubbornAtom Jun 18 '22 at 07:39
  • @StubbornAtom how about making this comment into an answer ? – Stratos supports the strike Jun 20 '22 at 02:57
  • @StratosFair OP is looking for a specific proof. – StubbornAtom Jun 20 '22 at 18:13

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