STA131C, Spring 2007
Prof. W. Polonik
HW # 4
due: May 9, 2007 in class
1. An ecologist takes data (
Y
i
, x
i
)
, i
= 1
, . . . , n
where
x
i
is the size of an area and
Y
i
is
the number of moss plants in that area. We model the data by assuming the
Y
i
to be
independent with
Y
i
∼
Poisson(
θ
·
x
i
)
, i
= 1
, . . . , n.
(a) (10 pts) Show that the LSE of
θ
is given by
hatwide
θ
LSE
=
P
n
i
=1
x
i
Y
i
P
n
i
=1
x
2
i
.
(b) (10 pts) Show that
hatwide
θ
LSE
is unbiased with Var(
hatwide
θ
LSE
) =
θ
P
n
i
=1
x
3
i
(
P
n
i
=1
x
2
i
)
2
.
(c) (10 pts) Show that the MLE of
θ
is
hatwide
θ
MLE
=
P
n
i
=1
Y
i
P
n
i
=1
x
i
.
(d) (10 pts) Show that the
hatwide
θ
MLE
is also unbiased and find its variance.
(e) (10 pts) Find the UMVUE of
θ
and show that its variance attains the Cram´
erRao
bound. Is this UMVUE also BLUE?
2. Consider the oneway ANOVA model
Y
ij
=
μ
i
+
ǫ
ij
,
j
= 1
, . . . , n
i
, i
= 1
, . . . , I,
with
ǫ
ij
iid random variables with mean 0 and variance
σ
2
.
Let
Y
i
squaresmallsolid
=
1
n
i
∑
n
i
j
=1
Y
i
,
and let
n
=
∑
I
i
=1
n
i
denote the total sample size.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '07
 Polonik
 Statistics, Normal Distribution, Standard Deviation, Variance, pts, Yi

Click to edit the document details