# Check if method of moments estimator is unbiased for $X_1...X_n$ being a random sample from $\mathcal{U}_{[-\theta,\theta]}$

I am not sure how to do this. To find the method of moments estimator I did:

$$E[X] = \frac{-\theta + \theta}{2} = 0$$

use 2nd moment:

$$E[X^2] = \frac{(-\theta)^2 + -(\theta^2) + \theta^2}{3} = \frac{\theta^2}{3} = \frac{1}{n} \sum_{i=1}^n X_i^2$$

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

And to find if it is unbiased, I tried this, but am not sure what to do with the integral

\begin{align}E[\hat{\theta}] &= \sqrt{\frac{3}{n}} \mathbb{E}\left[\sqrt{\sum_{i=1}^n X_i^2} \right] \\ &= \int_{-\theta}^\theta \sqrt{\frac{3}{n} \sum_{i=1}^n X_i^2} \frac{1}{2\theta} dx \\ &= \sqrt{\frac{3}{n}} * \frac{1}{2\theta} \int_{-\theta}^\theta \sqrt{\sum_{i=1}^n X_i^2} dx \end{align}

• Observe that $(3/n)\sum X_i^2$ is an unbiased estimator of $\theta^2$. What does Jensen's Inequality tell us about the relationship between the expected value of a square root and the square root of the expected value? Commented Mar 1 at 1:55

If we apply Jensen's Inequality as @jbowman suggested, which states that for a concave function $$f$$ (in this case, the square root function), and a random variable $$Y$$, we have: $$E[f(Y)] \leq f(E[Y]).$$ Therefore, since $$\hat{\theta}$$ involves taking a square root (a concave function), we have: $$E[\hat{\theta}] = E\left[\sqrt{\frac{3}{n} \sum_{i=1}^n X_i^2}\right] < \sqrt{E\left[\frac{3}{n} \sum_{i=1}^n X_i^2\right]} = \sqrt{3E[X^2]} = \sqrt{\theta^2} = \theta.$$
Since $$E[\hat{\theta}]$$ is less than $$\theta$$, we conclude that $$\hat{\theta}$$ is a biased estimator for $$\theta$$. Specifically, the bias is $$Bias(\hat{\theta}) = E[\hat{\theta}] - \theta < 0,$$ showing that the estimator tends to underestimate the true parameter.
Update: The transition from a weak inequality to a strict one in the application of Jensen's Inequality for the estimator $$\hat{\theta} = \sqrt{\frac{3}{n} \sum_{i=1}^n X_i^2}$$ occurs due to the strict concavity of the square root function. Jensen's Inequality tells us that for a concave function $$f$$ and a random variable $$Y$$, $$E[f(Y)] \leq f(E[Y])$$. This inequality is typically strict (i.e., $$E[f(Y)] < f(E[Y])$$) unless $$Y$$ is constant almost surely , which is not the case for sums of squared random variables unless they are all the same value almost surely.