Question

An operator using a gauge measure collection of n randomly selected parts twice. Let Xi and Yi; denote the measured values for the i^th part.

Assume that these two random variables are independent and normally distributed and that both have true mean μi and variance ².

a) Derive the maximum likelihood estimator for ².

b) Compute the bias of the maximum likelihood estimator that you derived in part a).

For part a, please explain how you obtained the likelihood function. For part b, please explain the computations for bias. Thank you!

Answer 1

(a)

Given that Let denote the two observed weights for the i^{th specimen. Suppose} are independent of one another, each normally distributed with

Then the likelihood estimator of is given by

  

  

Take log on both sides we have

Differentiate the above with respect to we have

Equating to zero we have

Put this value in the log likrlihood equation we have

  

  

Differentiate with respect to we have

  

Equating to zero we have

(b)

  

So the MLE is definitely not unbiased, the expected value of the estimator is oly half the value of what is being estimated