This question is concerned with the paper "A Lower Bound for a Probability Moment of any Absolutely Continuous Distribution with Finite Variance" by Sigeiti Moriguti appeared in Ann. Math. Statist. Volume 23, Number 2 (1952), 286-289.
In this paper the author has considered the following constrained optimization problem.
Consider all probability densities $f$ on $\mathbb{R}$ with respect the Lebesgue measure satisfying $$\int_{-\infty}^{\infty}x^2 f(x) dx = \sigma^2.$$ We need to find the probability density minimizing $$\int_{-\infty}^{\infty} f(x)^{\alpha} dx,$$ where $\alpha >1.$
Lagrangian associated with this problem is $$\int_{-\infty}^{\infty} f(x)^{\alpha} dx - \lambda \int_{-\infty}^{\infty}x^2 f(x) dx - \mu \int_{-\infty}^{\infty} f(x) dx$$ Then we obtain the characteristic equation $$\alpha \, f(x)^{\alpha-1} - \lambda x^2 - \mu = 0 .$$
From this we obtain \begin{equation} \label{} f(x) = \left[\frac{\lambda x^2 + \mu}{\alpha}\right]^{\frac{1}{\alpha-1}} \, \, \, \, ---\, \, (1) \end{equation}
Then he claims the following.
We should take $\lambda$ negative, and consquently $\mu$ is positive. Then the solution (1) is applicable in the interval $(-\sqrt{-\mu /\lambda}, \sqrt{-\mu /\lambda})$. Outside of this interval, $f(x)$ should be identically taken to be $0$.
My questions are the following.
- How is he able to claim that $\lambda$ should be negative?
- Outside of the mentioned interval $f(x)$ should be $0$ implies the following. $f(x) = 0$ implies $\lambda x^2 + \mu \le 0.$ Why should this be the case?
Can someone give a rigorous argument for these claims?
If $\lambda$ isn't negative, then for large $x$ we have $f(x)\sim x^r$ with $r=\frac{\alpha-1}{2}> 0$. So $f(x)$ doesn't vanish at infinity and as such the PDF cannot be normalized to have total probability $1$.
If $\lambda$ is negative, then $f(x)$ vanishes when $x^2=-\lambda/\mu$. Since probability densities must be nonnegative, the definition of $f(x)$ assumed initially can only be valid for $x$ such that $|x| \leq \sqrt{\lambda/\mu}$. The simplest solution is to just cut off the PDF here, which is the route they take.