I'm working on a problem from my mathematical statistics book and the following is asked from me. Let $X_{1},…,X_{n}$ be a sample from a probability density function with density $p_{\theta}(x)= \frac{2x}{\theta^{2}}1_{0 \leq x \leq \theta}.$
1) Determine method of moments estimator for $\theta$.
2) Determine method of moments estimator for $\theta^{2}$.
I think I've found the solution for 1, but I have no idea where to start with the solution for 2. Any hints would be much appreciated. So far my solution for 1)
Because we are determining a method of moments estimator for $\theta$, we set $E(X_{i}^{j})=\bar{X^{j}}$. In this case we let $j=1$, since that solution exists as we shall see.
We define a moments estimator $T$ for $\theta$ as follows,
\begin{equation}
\begin{split}
E(p_{\theta}) &= \int_{-\infty}^{\infty} x p_{\theta}(x) dx\\
&= \frac{2}{\theta^2} \int_{0}^{\theta} x^{2} dx \text{ (since $1_{0 \leq x \leq \theta}$ we let $0$ and $\theta$ be the boundaries for $x$)} \\
&= \frac{2}{\theta^2} \left[ \frac{1}{3}x^{3} \right]_{x=0}^{x=\theta}\\
&= \frac{2\theta}{3} = \bar{X}.
\end{split}
\end{equation}
Which implies $\hat{\theta} = \frac{3\bar{X}}{2}$ is an estimator for $\theta$.
But now, how to determine an estimator for $\theta^{2}$? Do I just substitute $\theta^{2}$ for $\theta$ so $p_{\theta^{2}}=\frac{2x}{\theta^{4}}$ and then iterate my previous method?
Best Answer
Just as you did with Q1, in the same manner, you could say \begin{equation} E(\theta^2) = \frac{2}{\theta^2} \int_0^{\theta} x^3 \ dx = \frac{2}{\theta^2} \frac{1}{4} \left[ x^4\right]_{x=0}^{x=\theta} = \frac{\theta^2}{2} = \bar{X}_2 \end{equation} where $\bar{X}_2 = \frac{1}{n}\sum_{i=1}^n x_i^2$ is the sample estimate of the second moment. So you could estimate $\theta$ as \begin{equation} \hat{\theta} = \sqrt{2\bar{X}_2} \end{equation}