Statistics Question

Question

Suppose that Y₁, Y₂,...,Y_n constitutes a random sample from the normal distribution with a mean of zero and variance σ², such that σ²>0. Further, it has been shown that σthe MLE for σ² is σ²-hat = (∑y_i²)/n. Is this estimator unbiased?

*Please provide a motivation of your response.

Answer 1

Hello Naeema,

Compute the expected value of the estimator σ²-hat.

`E[σ²-hat] = E[∑y_i²/n]

=(1/n)*E[∑y_i²]

= (1/n)*∑E[y_i²] (Equation 1)

Now recall that by definition, the variance of a random variable X is given by

Var = E[(X - μ)²]

where μ = E[X] is the expected value (or mean) of X. This may be rewritten as

Var = E[X²] - μ².

Since for each i, y_i has mean 0 and variance σ², we have

σ² = E[y_i²]

Using the above result in Equation 1, we now have

E[σ²-hat] = (1/n)*∑σ²

E[σ²-hat] = (1/n)*nσ²

E[σ²-hat] = σ².

Since the expected value of σ²-hat equals σ², we conclude that σ²-hat is an unbiased estimator of σ².

Hope that helps! Let me know if you have any questions.

William