We are given that $X \sim \text{Binom}(n, p)$ and that an estimator for $p$ is

$\hat{p} = \frac{X}{n}$

where $X$ is a single sample from the binomial distribution.

The variance of $X$ is known to be

$V(X) = np(1 - p).$

The given estimator for $V(X)$ is:

$\hat{V} = n \hat{p} (1 - \hat{p}) = n \left(\frac{X}{n}\right) \left(1 - \frac{X}{n}\right) = \frac{X}{n} (n - X).$

Step 1: Compute the Expected Value of $\hat{V}$

To determine whether $\hat{V}$ is biased, we compute its expectation:

[ E[\hat{V}] = E \left[ \frac{X}{n} (n - X) \right]. ]

Expanding,

[ E[\hat{V}] = \frac{1}{n} E[X(n - X)]. ]

Using the properties of expectation,

[ E[X(n - X)] = nE[X] - E[X^2]. ]

Since $X \sim \text{Binom}(n, p)$, we know:

$E[X] = np.$

For $E[X^2]$, we use the second moment formula for a binomial variable:

[ E[X^2] = V(X) + (E[X])^2 = np(1 - p) + (np)^2. ]

Thus,

[ E[X(n - X)] = n(np) - \left[ np(1 - p) + (np)^2 \right]. ]

Simplifying,

$E[X(n - X)] = n^2p - np - np^2n + np.$

$= n^2p(1 - p).$

Dividing by $n$, we get:

$E[\hat{V}] = \frac{n^2p(1 - p)}{n} = (n - 1)p(1 - p).$

Step 2: Bias of $\hat{V}$

Since the true variance is $np(1 - p)$, we compute the bias:

$\text{Bias}(\hat{V}) = E[\hat{V}] - V(X) = (n - 1)p(1 - p) - np(1 - p).$

$= np(1 - p) - p(1 - p) = -p(1 - p).$

Thus, $\hat{V}$ is biased.

Step 3: Asymptotic Unbiasedness

Since

$\lim_{n \to \infty} \frac{E[\hat{V}]}{V(X)} = \lim_{n \to \infty} \frac{(n - 1)p(1 - p)}{np(1 - p)} = \lim_{n \to \infty} \frac{n - 1}{n} = 1,$

we conclude that $\hat{V}$ is asymptotically unbiased.

Step 4: Unbiased Estimator

To construct an unbiased estimator, we use the fact that the expectation of $\hat{V}$ is $(n - 1)p(1 - p)$, whereas the true variance is $np(1 - p)$. To correct this bias, we scale $\hat{V}$ by $\frac{n}{n - 1}$:

$\hat{V}_{\text{unbiased}} = \frac{n}{n - 1} \hat{V}.$

Thus, the final unbiased estimator for $V(X)$ is:

$\hat{V}_{\text{unbiased}} = \frac{n}{n - 1} \cdot \frac{X}{n} (n - X) = \frac{X(n - X)}{n - 1}.$

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Summary:

1. The given estimator $\hat{V} = \frac{X}{n} (n - X)$ is biased.

2. The bias is $-p(1 - p)$, meaning it systematically underestimates the variance.

3. The estimator is asymptotically unbiased as $n \to \infty$.

4. An unbiased estimator can be obtained by scaling $\hat{V}$ by $\frac{n}{n - 1}$, resulting in:

   $\hat{V}_{\text{unbiased}} = \frac{X(n - X)}{n - 1}.$

Would you like further clarification on any step?

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Related Questions:

1. How do you derive the expectation of $X^2$ for a binomial variable?
2. Why does the estimator become unbiased as $n \to \infty$?
3. How does this method compare to variance estimation in other distributions?
4. What other methods exist to estimate the variance of a binomial distribution?
5. How does bias affect statistical inference in practical applications?

Tip: In statistical estimation, a common trick to remove bias is multiplying the biased estimator by a correction factor derived from expectation calculations.