Let X1 , X2 , … , Xn be uniformly distributed on the interval 0 to a. Recall that the maximum likelihood estimator of a is a X ˆ = max i ( ). (a) Argue intuitively why aˆ cannot be an unbiased estimator for a. (b) Suppose that E a na / n () ( ) ˆ = +1 . Is it reasonable that aˆ consistently underestimates a? Show that the bias in the estimator approaches zero as n gets large. (c) Propose an unbiased estimator for a. (d) Let Y X = max( )i . Use the fact that Y y ≤ if and only if each X y i ≤ to derive the cumulative distribution function of Y . Then show that the probability density function of Y is f y ny a , ya , n n ( ) = ≤ ≤ ⎧ ⎨ ⎪ ⎩ ⎪ −1 0 0 otherwise Use this result to show that the maximum likelihood estimator for a is biased. (e) We have two unbiased estimators for a: the moment estimator a X ˆ1 = 2 and an n X ˆ2 =+ 1 i [( ) / ] ( ) max , where max( ) Xi is the largest observation in a random sample of size n. It can be shown that V() ( ) a a/ n ˆ1 2 = 3 and that V ( ) [ ( )] a a / nn ˆ2 2 = + 2 . Show that if n > 1, aˆ2 is a better estimator than aˆ. In what sense is it a better estimator of a?
Let X1
, X2
, … , Xn
be uniformly distributed on the interval
0 to a. Recall that the maximum likelihood estimator of a is
a X ˆ = max i ( ).
(a) Argue intuitively why aˆ cannot be an unbiased estimator
for a.
(b) Suppose that E a na / n () ( ) ˆ = +1 . Is it reasonable that aˆ consistently underestimates a? Show that the bias in the estimator approaches zero as n gets large.
(c) Propose an unbiased estimator for a.
(d) Let Y X = max( )i . Use the fact that Y y ≤ if and only if
each X y i ≤ to derive the cumulative distribution
of Y . Then show that the probability density function of
Y is
f y
ny
a , ya
,
n
n ( ) = ≤ ≤
⎧
⎨
⎪
⎩
⎪
−1
0
0 otherwise
Use this result to show that the maximum likelihood estimator for a is biased.
(e) We have two unbiased estimators for a: the moment
estimator a X ˆ1 = 2 and an n X ˆ2 =+ 1 i [( ) / ] ( ) max , where
max( ) Xi is the largest observation in a random sample
of size n. It can be shown that V() ( ) a a/ n ˆ1 2 = 3 and that
V ( ) [ ( )] a a / nn ˆ2 2 = + 2 . Show that if n > 1, aˆ2 is a better
estimator than aˆ. In what sense is it a better estimator
of a?
Trending now
This is a popular solution!
Step by step
Solved in 4 steps with 3 images
