To check if $\hat{\theta}$ (the method of moments estimator) is unbiased, we need to verify if its expected value equals the true parameter $\theta$. That is, we want to check if $E(\hat{\theta}) = \theta$.

Given:

• $E(X) = \frac{2}{3} \theta$, the population mean.
• The method of moments estimator (MOM) for $\theta$ is given as:

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

Step 1: Find the expected value of $\hat{\theta}$

The estimator $\hat{\theta}$ involves the sample mean:

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

We can rewrite this as:

$\hat{\theta} = \frac{3}{2} - \overline{X}$ where $\overline{X}$ is the sample mean:

$\overline{X} = \frac{1}{n} \sum_{i=1}^{n} X_i$

Now, let's compute the expected value of $\hat{\theta}$:

$E(\hat{\theta}) = E\left(\frac{3}{2} - \overline{X}\right)$

Since the expectation of a constant is the constant itself, and the expectation of the sample mean $\overline{X}$ is the population mean $E(X)$, we have:

$E(\hat{\theta}) = \frac{3}{2} - E(\overline{X})$

$E(\hat{\theta}) = \frac{3}{2} - E(X)$

Substitute $E(X) = \frac{2}{3} \theta$:

$E(\hat{\theta}) = \frac{3}{2} - \frac{2}{3} \theta$

Step 2: Check if $\hat{\theta}$ is unbiased

For $\hat{\theta}$ to be unbiased, $E(\hat{\theta})$ should equal $\theta$. So we check if:

$E(\hat{\theta}) = \theta$

From the previous calculation, we have:

$E(\hat{\theta}) = \frac{3}{2} - \frac{2}{3} \theta$

Equating this to $\theta$:

$\theta = \frac{3}{2} - \frac{2}{3} \theta$

Multiply both sides by 3 to eliminate the fraction:

$3\theta = 9 - 2\theta$

Add $2\theta$ to both sides:

$5\theta = 9$

Solve for $\theta$:

$\theta = \frac{9}{5}$

Conclusion:

Since $\theta = \frac{9}{5}$, the estimator $\hat{\theta}$ is not unbiased unless $\theta = \frac{9}{5}$. Therefore, $\hat{\theta}$ is a biased estimator of $\theta$.

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Would you like further clarification or more details on this result?

Here are 5 related questions:

1. What conditions might make an estimator unbiased?
2. How does bias affect the accuracy of an estimator?
3. What is the difference between the method of moments (MOM) and maximum likelihood estimation (MLE)?
4. How would we adjust $\hat{\theta}$ to make it unbiased?
5. What is the variance of the estimator $\hat{\theta}$?

Tip: A biased estimator can still be useful if it has low variance—this creates a trade-off known as the bias-variance trade-off in estimation theory.