We are tasked with determining if $S_n^2$, as given in the problem, is a consistent estimator of the population variance $\sigma^2$.

Problem Overview:

Let $X_1, X_2, \dots, X_n$ be independent and identically distributed (i.i.d.) random variables with:

• $\mathbb{E}[X_i] = \mu$ (the population mean),
• $\text{Var}(X_i) = \sigma^2$ (the population variance).

The estimator $S_n^2$ is given as: $S_n^2 = \frac{1}{n} \sum_{i=1}^{n} X_i^2 - \overline{X}^2,$ where $\overline{X} = \frac{1}{n} \sum_{i=1}^{n} X_i$ is the sample mean.

Consistency of an Estimator:

An estimator $T_n$ of a parameter $\theta$ is said to be consistent if:

1. Unbiasedness: $\mathbb{E}[T_n] \to \theta$ as $n \to \infty$,
2. Convergence in Probability: $T_n \overset{p}{\to} \theta$ as $n \to \infty$, meaning for any $\epsilon > 0$, $P(|T_n - \theta| \geq \epsilon) \to 0$ as $n \to \infty$.

Steps to Determine Consistency:

1. Expectation of $S_n^2$:

We need to find $\mathbb{E}[S_n^2]$. Break it down into two parts: $S_n^2 = \frac{1}{n} \sum_{i=1}^{n} X_i^2 - \overline{X}^2.$

• $\mathbb{E}\left[ \frac{1}{n} \sum_{i=1}^{n} X_i^2 \right] = \frac{1}{n} \sum_{i=1}^{n} \mathbb{E}[X_i^2] = \mathbb{E}[X_1^2]$. Using $\text{Var}(X_i) = \sigma^2$ and $\mathbb{E}[X_i] = \mu$, we know $\mathbb{E}[X_i^2] = \sigma^2 + \mu^2$. Thus: $\mathbb{E}\left[ \frac{1}{n} \sum_{i=1}^{n} X_i^2 \right] = \sigma^2 + \mu^2.$

• $\mathbb{E}[\overline{X}^2] = \mathbb{E}\left[\left(\frac{1}{n} \sum_{i=1}^{n} X_i\right)^2 \right] = \frac{\sigma^2}{n} + \mu^2$.

Thus, the expectation of $S_n^2$ becomes: $\mathbb{E}[S_n^2] = (\sigma^2 + \mu^2) - \left(\frac{\sigma^2}{n} + \mu^2\right) = \sigma^2 \left(1 - \frac{1}{n}\right).$

As $n \to \infty$, $\mathbb{E}[S_n^2] \to \sigma^2$, indicating that the estimator is asymptotically unbiased.

2. Variance of $S_n^2$:

To establish consistency, we also need to examine the variance of $S_n^2$. Without diving into the full calculation, we know that the variance of $S_n^2$ tends to 0 as $n \to \infty$, meaning that $S_n^2$ converges to $\sigma^2$ in probability.

Conclusion:

Since $\mathbb{E}[S_n^2] \to \sigma^2$ and $\text{Var}(S_n^2) \to 0$, we can conclude that $S_n^2$ is a consistent estimator of $\sigma^2$.

Would you like more details on any part of this solution or have any questions?

Here are 5 related questions for further exploration:

1. What is the difference between a biased and an unbiased estimator?
2. How does the law of large numbers relate to consistency of estimators?
3. Can you derive the variance of $S_n^2$ explicitly to confirm consistency?
4. What conditions ensure an estimator is efficient, in addition to being consistent?
5. How does $S_n^2$ compare to the sample variance $\hat{\sigma}^2 = \frac{1}{n-1} \sum_{i=1}^{n} (X_i - \overline{X})^2$?

Tip: Consistency ensures that as you collect more data, your estimator converges to the true parameter value. It's often the first criterion for a good estimator.