Let x 1 , x 2 ….. x n be an independently, and identically distributed (iid) random sample drawn from a population that follows the Normal Distribution N(μ, σ 2 ), where both the mean (μ) and variance (σ 2 ) 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

Question

Let x 1 , x 2 ….. x n be an independently, and identically distributed (iid) random sample drawn from a population that follows the Normal Distribution N(μ, σ 2 ), where both the mean (μ) and variance (σ 2 ) 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

cdquestions Admin · Accepted Answer

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 σ 2 .

Answer

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

Answer

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

Answer

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

Answer

${\hat{\sigma}}^2_{MLE}=\frac{1}{n-1}\sum^{n-1}_{i=1}(x_i-\bar{x})^2$ which is an unbiased and consistent estimator of σ²

X	-2	-1	0	1	2
P(X)	0.2	0.1	0.3	0.2	0.2

Variable	Coefficient	Standard Error	Z-Statistic	Probability
Constant	-0.064	0.399	-0.161	0.871
X	0.029	0.010	2.916	0.003

The Correct Option is A

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