Let $X_1, X_2, \ldots, X_n$ independent and identically distributed random variables, $X_i \sim U(-\theta, \theta)$ where $\theta > 0$

Question

a) Determine method of moments estimator for $\theta$

b) Prove that the estimator found in a) is consistent.

Solution

a) $\hat{\theta} = \sqrt{\frac{3 \sum_{i=1}^{n} X_i^2}{n}}$

but I don't know how to prove b). I couldn't prove that the estimator is asymptotically unbiased. Any help? Thanks.

Answer 1

Note that unbiasedness is neither necessary nor sufficient for consistency. For instance, if $X_i$ are iid $ N(\theta,1)$, then $X_1$ is an unbiased but inconsistent estimator for $\theta$. Meanwhile, $\frac{1}{n}\left(1+\sum_{i=1}^n X_i\right)$ is a biased but consistent estimator for $\theta$.

Also, in terms of your updated post, asymptotic unbiasedness doesn't guarantee consistency by the same counterexample above. See this post for more general discussion.

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Coming to your question, WLLN and CMT immediately tell you that your estimator converges in probability to $\sqrt{3E[X_1^2]}=\theta.$

Answer 2

It is not necessarily to prove $\hat\theta_n$ is unbiased since we know the following theorems.

Thm 1 $\hat\theta_n \to \theta$ if $E \hat\theta_n \to \theta$ and $Var\hat\theta_n \to 0$.

Thm 2 $g(\hat\theta_n)\to g(\theta)$ if $\hat\theta_n\to\theta$ and $g(x)$ is continuous.

Then it suffices to prove $E\hat{\theta}^2_n \to \theta^2$ and $Var \hat{\theta}_n^2 \to 0$.

For $E\hat\theta^2_n$, $$ E\hat\theta^2_n = E \frac{3\sum X_i^2}{n} = 3EX_1^2 = \theta^2 $$

For $Var \hat\theta^2_n$, $$ Var \hat\theta^2_n = Var \frac{3\sum X_i^2}{n} = \frac9n Var X_1^2 \to 0 $$ as desired.