Consider a sequence of independent and iden1cally distributed random variables !! , !" , … , !#
drawn from a normal distribu1on $(&, ( " ), where both parameters are unknown.
Define the following two es1mators for the popula1on variance ( " ,
*! " = ∑#*+!
(%! &'()"
, and *" " = ∑#*+!
#&!
(%! &'( )"
#
.
(1) Show that *! " is an unbiased es1mator of ( " while *" " is biased. Calculate the bias of
*" " . (3)
(2) Find the variance of both es1mators. (4)
(3) Compare the mean squared error (MSE) of both es1mators. (4)
(4) Can *! " be improved by applying the Rao-Blackwell theorem? Why or why not. (4)