3 Assignment ACTSC 433/833, Winter 2012
(Due at the beginning of the class on March 1)
1. You are given the following ages at time of death for 10 individuals: 38, 45, 52, 52, 56,
59, 68, 70, 74, and 82. Determine the kernel density estimate for the probability that an
individual will survive at least to age 60 using
(a) the uniform kernel with bandwidth b = 5;
[5 marks]
(b) and the uniform kernel with bandwidth b = 15.
[5 marks]
2. You are given the following loss amounts for 5 insurance portfolios: 120, 190, 241, 442,
and 578. Determine the kernel density estimate for the probability that the loss in a
portfolio will exceed 220 using
(a) the triangular kernel with bandwidth b = 75;
[5 marks]
(b) and the triangular kernel with bandwidth b = 100.
[5 marks]
3. Let X1 , ..., Xn (n > 2) be a random sample from the exponential density function f (x) =
ex , x > 0, > 0.
(a) Determine the MLE for .
(b) Show that MLE the is consistent for .
[3 marks]
[7 marks]
4. Let X1 , ..., Xn be a random sample from the normal distribution N (, 2 ), where the
parameters < < and > 0.
(a) Suppose that the parameter is known.
i. Determine the MLE 2 for 2 .
[3 marks]
ii. Show that the MLE 2 is consistent for 2 .
[5 marks]
(b) Suppose that both the parameters and 2 are unknown.
i. Determine the MLE for and the MLE 2 for 2 .
[6 marks]
ii. Verify if the MLE is consistent for .
[4 marks]
2
2
iii. Verify if the MLE is consistent for .
[7 marks]
5. Let X1 , ..., Xn be a random sample from the uniform distribution U ( 1, + 1), where
the parameter > 1.
(a) Determine the MLE of .
(b) Verify if
X(1) +X(n)
2
[3 marks]
is an MLE for , where X(1) = min{X1 , ..., Xn } and X(n) =
max{X1 , ..., Xn }.
(c) Verify if
X(1) +X(n)
2
[2 marks]
is a consistent estimator of .
[10 marks]
[Total: 70 marks]
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