Define and briefly summarize each concept both statistically, and, in informal terms.
• Maximum likelihood estimator
Let {𝑥ଵ, . . . , 𝑥} ∼iid 𝑁(𝜇, 𝜎
ଶ
) where 𝑓(𝑥|𝜇, 𝜎
ଶ
) = ଵ
√ଶగఙమ 𝑒
ି
భ
మమ
(௫ିఓ)
మ
.
a) Write the likelihood function 𝐿(𝜇, 𝜎
ଶ
|𝑋).
b) Write the log-likelihood function ℓ(𝜇, 𝜎
ଶ
|𝑋).
c) Compute 𝜇̂MLE.
d) Verify that your answer is a maximum by checking that the second derivative is negative.
Sample Solution