STAT 510
Homework 9
Due Date: 11:00 A.M., Wednesday, April 6
1. Suppose


 

y1
µ1
σ 2 σ 2 /2
0
 y2  ∼ N  µ1  ,  σ 2 /2 σ 2 σ 2 /2 
y3
µ2
0
σ 2 /2 σ 2

where µ1 ∈ R, µ2 ∈ R, and σ 2 > 0 are unknown parameters. Find the REML estimator of σ 2 .
2. Let yi1 denote the weight (in kg) gained by Holstein calf i from birth to one week of age. Let yi2
denote the weight (in kg) gained by Holstein calf i from one week of age to 12 weeks of age. Suppose
it is known that
Var(yi1 ) = Var(yi2 ) = 4 for all i
and that the correlation between yi1 and yi2 is 0.5 for all i. Suppose weight gained by any one calf is
independent of the weight gained by any other calf. Suppose the following information is available
for three randomly selected calves. (Note that the periods in the table denote data that are missing
completely at random.)
Calf Number (i)
1
2
3
yi1
51
48
52
yi2
54
.
.
(a) Determine the best linear unbiased estimator of the expected total weight of a Holstein calf at
age 12 weeks.
(b) Predict the weight gained by Holstein calf i = 2 from 1 week of age to 12 weeks; i.e., find the
BLUP of y22 .
3. For π ∈ [0, 1], a random variable has a Bernoulli distribution with success probability π if it takes the
value 1 with probability π and the value 0 with probability 1 − π. We use y ∼ Bernoulli(π) to indicate
that the random variable y has a Bernoulli distribution with success probability π.
If y ∼ Bernoulli(π), it follows that the probability mass function of y is
k
π (1 − π)(1−k) for k ∈ {0, 1}
.
P (y = k) =
0
otherwise
iid
Suppose y1 , . . . , yn ∼ Bernoulli(π) and answer all the following problems for this special case.
(a) Provide an expression for the likelihood function.
(b) Write down the score equation.
(c) Solve the score equation.
(d) Verify that the solution to the score equation maximizes the likelihood function; i.e., verify that
the solution to the score equation is an MLE for π.
(e) Find the Fisher information matrix. (This is just a 1 × 1 matrix in this case.)
(f) Find the inverse of the Fisher information matrix.
(g) Verify that the inverse of the Fisher information matrix gives the exact variance of the MLE in
this special case.
(h) Provide an expression for an estimator of the variance of the MLE by replacing the unknown
parameter π with the MLE of π in the inverse Fisher information matrix.
(i) Suppose researchers wish to estimate the probability that a certain type of plant will die when
infected with a virus. They randomly select 100 seeds from a large collection of seeds for the
plant type of interest. They grow the seeds into plants, infect the plants, and discover that 17 of
the plants die as a result of the infection. Find an approximate 95% confidence interval for the
proportion of all plants of this type that would die as a result of infection with the virus.
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