Solution to a certain moment problem

Question

I'm looking for a function $f$ that satisfies

1. $f(x)\geq0$
2. $\int f(x) \mathrm{d}x=1$
3. $\int xf(x) \mathrm{d}x=0$
4. $\int x^2f(x)\mathrm{d}x=1$
5. $\int x^4f(x)\mathrm{d}x=\delta$
6. $\int x^5f(x)\mathrm{d}x=\infty$
7. It has a "nice" antiderivative, not like the density of the Student's $t$-distribution with 5 degrees of freedom and not like a series.

So basically I'm looking for a density $f$ that solves this truncated moment problem.

My approach so far was to start with the right side of the real line:$$f_+(x)=\frac{1}{(x+\rho_1)(x+\rho_2)^2(x+\rho_3)^3}, x\geq0 $$ Then I would solve for $\rho_1,\rho_2,\rho_2$ such that $\frac{1}{2}$ of the specified moments are matched. Later I would get my $f$ by replacing $(\cdot)$ by $|\cdot|$. However, I run into ugly lengthy partial fraction decompositions and ugly integrals. Are there more accessible approaches than these rational functions?

Answer 1

If you set $f(x)=1/x^6$ for $x\ge m>0$ then you have taken care of the tail. The remaining conditions can be met by a polynomial of degree 4 on $-m\le x\le m$ with $f(x):=0$ for $x<-m$. However, the condition $f(x)\ge 0$ discards many solutions. In general, $m$ will depend on $\delta$. Another way of obtaining a solution is to define $f$ constant on certain intervals. For example, if $\delta=3$ you can set $m=1$ and $$ f(x):=\begin{cases}0&\text{if }\qquad\quad\ x<-3\\ \frac{2}{75}&\text{if }\ -3\le x<-2\\ \frac{3}{25}&\text{if }\ -2\le x<-1\\ \frac{33}{100}&\text{if }\ -1\le x<0\\ \frac{97}{300}&\text{if }\ \quad\ 0\le x<1\\ \frac{1}{x^6}&\text{if }\ \ \quad 1\le x \end{cases} $$

Moreover, you can construct solutions that are $C^{\infty}$, approximating the locally constant solutions by such functions.

${\bf{EDIT:}}$ Playing around with the limits and locally constant functions one can set for example $$ f(x):=\begin{cases}0&\text{if }\qquad\quad\ x<-2\delta\\ \frac{k}{6\delta^6}&\text{if }\ -2\delta\le x<-\delta\\ a&\text{if }\ 0\le |x|<1\\ b&\text{if }\ \quad\ 1\le |x|<(\delta+1)/2\\ c&\text{if }\ \quad\ (\delta+1)/2\le |x|<\delta\\ \frac{k}{x^6}&\text{if }\ \ \quad \delta\le x \end{cases} $$ Then $a,b,c$ can be determined by the conditions 2., 4. and 5. If one set $k:=\delta^2/800$, then $a,b,c$ are positive for $\delta\in[1.0026,6.9]$.