Let's say we have a log-logistic random variable $X$ with probability density function:

$$f(x)=\frac{(\beta/\alpha)(x/\alpha)^\beta}{(1+(x/\alpha)^{\beta})^{2}}$$

where $\alpha>0$, $\beta>0$ and support on $x\in[0,\infty)$.

This random variable has mean and variance defined as follows:

$$\mathbb{E}[X]=\frac{\alpha(\pi/\beta)}{\text{sin}(\pi/\beta)}$$ $$\text{Var}[X]=\alpha^{2}\bigg[\frac{2(\pi/\beta)}{\text{sin}(2(\pi/\beta))}-\frac{(\pi/\beta)^{2}}{\text{sin}^{2}(\pi/\beta)}\bigg]$$

or, more generally, with $k$th raw moment defined for $k<\beta$: $$\mathbb{E}[X^{k}]=\alpha^{k}\frac{k(\pi/\beta)}{\text{sin}(k\pi/\beta)}$$

Now, is there an clever way to estimate the parameters $(\alpha,\beta)$ such that it provides a desired mean and variance?

As far as I can tell, we can arrive at an equation in terms of either parameter, but solving requires some form of optimization. I was curious to know if there are any methods to avoid such optimization?

Let's say we have a log-logistic random variable $X$ with probability density function:

$$f(x)=\frac{(\beta/\alpha)(x/\alpha)^{\beta-1}}{(1+(x/\alpha)^{\beta})^{2}}$$

where $\alpha>0$, $\beta>0$ and support on $x\in[0,\infty)$.

This random variable has mean and variance defined as follows:

$$\mathbb{E}[X]=\frac{\alpha(\pi/\beta)}{\text{sin}(\pi/\beta)}$$ $$\text{Var}[X]=\alpha^{2}\bigg[\frac{2(\pi/\beta)}{\text{sin}(2(\pi/\beta))}-\frac{(\pi/\beta)^{2}}{\text{sin}^{2}(\pi/\beta)}\bigg]$$

or, more generally, with $k$th raw moment defined for $k<\beta$: $$\mathbb{E}[X^{k}]=\alpha^{k}\frac{k(\pi/\beta)}{\text{sin}(k\pi/\beta)}$$

Now, is there an clever way to estimate the parameters $(\alpha,\beta)$ such that it provides a desired mean and variance?

As far as I can tell, we can arrive at an equation in terms of either parameter, but solving requires some form of optimization. I was curious to know if there are any methods to avoid such optimization?