Let $Y_1,...,Y_n$ be iid draws from a location family $\{f(\cdot - \theta) : \theta \in \mathbb{R}\}$. $f$ is a symmetric density w.r.t. the Lebesgue measure on $\mathbb{R}$ with finite variance. We want to test $H_0 : \theta = 0$ versus $H_1 : \theta > 0$.
Let $S_n$ denote the sign-test statistic which is given by $$S_n = \frac1n \sum_{i=1}^n I_{Y_i > 0} $$ and $T_n$ denote the $t$-test statistic which is given by $$T_n = \bar{Y}/S $$ where $S$ here is the sample standard deviation.
It can be deduced that the Pitman efficiency is $$\mathfrak{p}(\phi_{\cdot,S_n},\phi_{\cdot, T_n})(f) = 4f^2(0) \int y^2 f(y) \,dy. $$ We want to show that $$\inf_f \mathfrak{p}(\phi_{\cdot,S_n},\phi_{\cdot, T_n})(f) \geq \frac13. $$ My question: A priori, we can test various densities here and see that the uniform density gives us an efficiency of 1/3. From this knowledge, we can begin to deduce why $f$ must be uniform in order to show what the lower bound of the Pitman efficiency here is. I would like a different approach. Assume that we did not know a priori that the lower bound is 1/3. Suppose we simply want to minimize the integral $$\int y^2 f(y) \,dy. $$ How can we do this using variational calculus? Some constraints to keep in mind are that $0 \leq f \leq 1$, and that $f$ is symmetric. Further, since $f$ is a density, $\int f = 1$. I would like an approach which deduces that $f$ is the uniform density without knowing that the lower bound is 1/3 to begin with. Can we use some Euler-Lagrange equation here?
Edit: See comments for discussion on some assumptions.
