Question

Let x1, x2 ….. xn be an independently, and identically distributed (iid) random sample drawn from a population that follows the Normal Distribution N(μ, σ2), where both the mean (μ) and variance (σ2) are unknown. Let $\bar{x}$ be the sample mean. The maximum likelihood estimator (MLE) of the variance ($\hat{\sigma}^2_{MLE}$) is/are then characterized by

• A. ${\hat{\sigma}}^2_{MLE}=\frac{1}{n}\sum^n_{i=1}(x_i-\bar{x})^2$ which is a biased estimator of σ2
• B. ${\hat{\sigma}}^2_{MLE}=\frac{1}{n}\sum^n_{i=1}(x_i^2-\bar{x})^2$ which is a consistent estimator of σ2
• C. ${\hat{\sigma}}^2_{MLE}=\frac{1}{n-1}\sum^n_{i=1}(x_i-\bar{x})^2$ which is an unbiased estimator of σ2
• D. ${\hat{\sigma}}^2_{MLE}=\frac{1}{n-1}\sum^{n-1}_{i=1}(x_i-\bar{x})^2$ which is an unbiased and consistent estimator of σ2

Answer 1

Answer: (A)

${\hat{\sigma}}^2_{MLE}=\frac{1}{n}\sum^n_{i=1}(x_i-\bar{x})^2$ which is a biased estimator of σ2

Answer 2

The correct option is (A) : ${\hat{\sigma}}^2_{MLE}=\frac{1}{n}\sum^n_{i=1}(x_i-\bar{x})^2$ which is a biased estimator of σ2.