Let $X_{1},…,X_{n}$ be a sample from probability distribution with density $p_{\theta}(x)=\theta(1+x)^{-(1+\theta)}$ with $x>0$ and $0$ elsewhere, with $\theta>1$ unknown. Determine the method of moments estimator for $\theta$.
I let $\hat{\theta}_{MM}=E(p_{\theta}(x))$ Thus I need the find the expectation of the density function. Since it is continuous, I construct
$$E(p_{\theta}(x))=\int\limits_{0}^{\infty} x \theta(1+x)^{-(1+\theta)}dx,$$ but when I try to solve this by integration by parts I get stuck. I let $u=x$ and $dV=\theta(1+x)^{-(1+\theta)}$, which doesn't work.
Where do I go wrong? Any help, hints or suggestions would much be appreciated!
Best Answer
Integration by parts works. If $dv=\theta(1+x)^{-(1+\theta)}$, then $v=-(1+x)^{-\theta}$, and $$ \int_0^{\infty} x \theta(1+x)^{-(1+\theta)}dx=-\frac x{(1+x)^\theta}\,\Big|_0^\infty+\int_0^\infty(1+x)^{-\theta}\,dx\tag1 $$ Check that the first term on the RHS of (1) equals zero, while the second term equals $$ \frac{(1+x)^{-\theta+1}}{-\theta+1}\,\Big|_0^\infty=\frac1{\theta-1}. $$