Question

Problem 1. Define the unbiased or corrected sample variance as follows: $S^2 = \frac{1}{n-1} \sum_{i=1}^{n} (Y_i - \bar{Y})^2$, while the biased sample variance is given by $\tilde{S}^2 = \frac{1}{n} \sum_{i=1}^{n} (Y_i - \bar{Y})^2$. (i) Given $Y_i \stackrel{iid}{\sim} N(\mu, \sigma^2)$, show that $S^2$ and $\tilde{S}^2$ are both consistent estimators of $\sigma^2$, whereas only $S^2$ is an unbiased estimator of the unknown parameter $\sigma^2$ (the reason why it's called an unbiased sample variance). (ii) Given $Y_i \stackrel{iid}{\sim} N(\mu, \sigma^2)$, show that $S$ is a biased estimator of $\sigma$. (iii) Given $Y_i \sim \mu, \sigma^2$ and independent (a sample which is independent but not necessarily identically distributed), show that $S^2$ is an unbiased and consistent estimator of $\sigma^2$. (iv) Given $Y_i \sim \mu, \sigma^2$ and independent (a sample which is independent but not necessarily identically distributed), show that $\tilde{S}^2$ is a biased and consistent estimator of $\sigma^2$.

          Problem 1. Define the unbiased or corrected sample variance as follows:
$S^2 = \frac{1}{n-1} \sum_{i=1}^{n} (Y_i - \bar{Y})^2$,
while the biased sample variance is given by
$\tilde{S}^2 = \frac{1}{n} \sum_{i=1}^{n} (Y_i - \bar{Y})^2$.
(i) Given $Y_i \stackrel{iid}{\sim} N(\mu, \sigma^2)$, show that $S^2$ and $\tilde{S}^2$ are both consistent estimators of $\sigma^2$, whereas only $S^2$ is an
unbiased estimator of the unknown parameter $\sigma^2$ (the reason why it's called an unbiased sample variance).
(ii) Given $Y_i \stackrel{iid}{\sim} N(\mu, \sigma^2)$, show that $S$ is a biased estimator of $\sigma$.
(iii) Given $Y_i \sim \mu, \sigma^2$ and independent (a sample which is independent but not necessarily identically
distributed), show that $S^2$ is an unbiased and consistent estimator of $\sigma^2$.
(iv) Given $Y_i \sim \mu, \sigma^2$ and independent (a sample which is independent but not necessarily identically
distributed), show that $\tilde{S}^2$ is a biased and consistent estimator of $\sigma^2$.

Problem 1. Define the unbiased or corrected sample variance as follows:
S^2 = (1)/(n-1)∑i=1^n (Yi - Y̅)^2,
while the biased sample variance is given by
S̃^2 = (1)/(n)∑i=1^n (Yi - Y̅)^2.
(i) Given Yi iid∼ N(μ, σ^2), show that S^2 and S̃^2 are both consistent estimators of σ^2, whereas only S^2 is an
unbiased estimator of the unknown parameter σ^2 (the reason why it's called an unbiased sample variance).
(ii) Given Yi iid∼ N(μ, σ^2), show that S is a biased estimator of σ.
(iii) Given Yi ∼μ, σ^2 and independent (a sample which is independent but not necessarily identically
distributed), show that S^2 is an unbiased and consistent estimator of σ^2.
(iv) Given Yi ∼μ, σ^2 and independent (a sample which is independent but not necessarily identically
distributed), show that S̃^2 is a biased and consistent estimator of σ^2.

Added by Keith V.

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