Question

Two brothers, Liam and Noel, play in a rock and roll band. The brothers each have 3 sons. Liam and Noel want to predict the mean drum roll speed (in hits per ten seconds) at age 5 years that would be expected for a new son. To do this, they'll use estimators based on the six children's drum roll speeds, all measured at age 5. Due to a rare genetic variant that Liam has (and that Noel does not have), it is expected that the variance of drum roll speeds at age 5 years for Liam's children is twice that for Noel's children i.e. ??? ? = ??? and ??? ? = ??. However, the mean drum roll speed at age 5 is assumed to be the same for both Liam's children and Noel's children i.e. ?(?) = ? ? = ?. What is more, the distributions of drum roll speeds at age 5 are assumed to be normal with mean ?. Liam would like to make a best guess about the expected drum roll speed (at age 5 years) should he have a fourth child. He first considers the estimator

? = (L1 + L2 + L3 + N1 + N2 + N3)/6 .
where Li denotes the random variable that is the drum roll speed (at age 5 years) for the ?!!son of

Liam, and Ni denotes the random variable that is the drum roll speed (at age 5 years) for the ?!!son of Noel. We denote by ? and ?! the population mean and population variance of drum roll speed (at age 5 years) for Noel's children, and we assume that all children's drum speeds are independent.

a) What is the bias of ? as an estimator of ??

b) What is the variance of ??

c) What is the Mean Square Error (MSE) of ??

Noel remembers from his introductory statistics course that sometimes unbiased estimators can have a smaller variance if one uses appropriate weights. He wants to consider a weighted average of the six random variables:

?~ = ! [ N1 + N2 + N3 + 0.5 L1 + L2 + L3 ] . d) It turns out that ?~ is unbiased. What is the MSE of ?~?

e) Which estimator, ? or ?~, is preferable for estimating ?? Which criterion are you using to determine which estimator is preferred? Explain your answer in complete sentences.

Answer 1

(a) What is the bias of as an estimator of ?

definition of bias of an estimator for the population mean,

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