MATH 557 - ASSIGNMENT 2
To be handed in not later than 5pm, 14th February 2008.
Please hand in during lectures, to Burnside 1235, or to the Mathematics Ofce Burnside 1005
1 Suppose that X1 , . . . , Xn are a random sample from the pdf
fX| (x|) =
1
expcfw_
MATH 557 - EXERCISES 2 These exercises are not for assessment
1 Suppose that X Binomial(n, ) for 0 < < 1. (a) Verify that the estimator T (X) = X/n is unbiased for . (b) Consider () = 1/. Find an unbiased estimator of (). 2 Suppose that X1 , . . . , Xn U
MATH 557 - MID-TERM 2008 - SOLUTIONS
1.
(a) Note rst that by standard expansion into a quartic polynomial
x
k
4
k
wj (, )xj = w0 (, ) +
= w0 (, ) +
j=1
wj (, )tj (x)
j=1
say, where wj (, ) are constant functions of and . Thus
k
fX|, (x|, ) = h(x)c(, ) ex
M ATH 557 - ASSIGNMENT 3
S OLUTIONS
1
(a) To nd the UMP test, consider
H0 : = 1
H1 : = 1
for 1 > 1. By Neyman-Pearson, the rejection region is constructed by looking at
n
fX | (x|1 )
fX | (x|1)
where T (x) =
n
1 (1 xi )1 1
=
i=1
= n cfw_T (x)1 1
1
1
(1 xi
M ATH 557 - ASSIGNMENT 3
To be handed in not later than 5pm, 20th March 2008.
Please hand in during lectures, to Burnside 1235, or to the Mathematics Ofce Burnside 1005
1 Let X1 , . . . , Xn be a random sample from the Beta(1, ) probability model, for par