Problem Set 7
Robert Kohn
UNSW School of Business
University of New South Wales
ECON 3203 2020
November 10, 2020
This problem set is to be handed in by 5 pm on November 20.
Q1; 13 marks Suppose that Y has a Poisson distribution with λ. This means that
Pr(Y = y;λ) = exp(−λ)λ
y
y!
for y = 0, 1, 2, . . . ,. We can show that E(Y ) = λ and σ2 = Var(Y ) = λ.
The following 10 observations are from a Poisson distribution with
mean λ.
5, 6, 5, 8, 8, 4, 2, 4, 5, 4
(a) (2 marks) Find the maximum likelihood estimate λ̂ of λ.
(b) (5 marks) Use the parametric bootstrap to estimate the bias, stan-
dard error and rmse of λ̂ as an estimator of λ.
(c) (5 marks) One estimate of σ2 is σ̂2 = λ̂. A second way is to use
the sample variance
s2 =
∑10
i=1(yi − y)2
9
.
Use the parametric bootstrap to estimate the bias, standard error
and rmse of s2 as an estimator of σ2.
1
(d) (1 mark) Which of λ̂ and s2 is better, as judged by rmse?
(i) λ̂
(ii) s2
Q2. 11 marks Consider exactly the same setup as in Q1, but now we are interested
in estimating the ratio R = E(Y )/sd(Y ) =
√
λ. We can estimate R by
r =
√
λ̂ or by s =
√
s2.
(a) (5 marks) Estimate the bias, standard error and rmse of r as an
estimator of
√
λ.
(b) (5 marks) Estimate the bias, standard error and rmse of s as an
estimator of
√
λ.
(c) (1 mark) Which of λ̂ and s2 is better, as judged by rmse?
(i) r
(ii) s
Q3.6 marks Suppose that in Q1 and Q2 the true value of λ is 5.
i. (3 marks) What is the true bias of λ̂ as an estimator of λ?
ii. (3 marks) What is the true bias of r as an estimator of R?
2

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