2 Assignment ACTSC 433/833, Winter 2012
(Due at the beginning of the class on February 9)
1. Let X1 , ..., Xn be a random sample from the distribution function F (x) = 1 S (x) and
Fn (x) be the empirical estimator for F (x).
(a) Show that for any > 0, Pr |Fn (x) F (x)| >
2n
1
.
2
[6 marks]
(b) Let = Pr{a < X b}, where a < b. Write out a consistent empirical estimator n
for and explain why your estimator is consistent.
[6 marks]
(c) Calculate Cov (Fn (x), Fn (y )) for x < y .
[6 marks]
(x) be the Product-Limit estimator for the survival function S (x). Calculate
(d) Let S
Cov (S (yj ), S (yj +1 )) for j = 1, ..., k 1.
[6 marks]
2. Let X be a non-negative discrete random variable with the probability function pn =
Pr{X = n} and the discrete hazard rate function hn = Pr{X = n|X n} for n =
0, 1, 2, ....
n
1
(a) Assume that X has a geometric distribution with pn = 1+ 1+ for n = 0, 1, 2, ...,
where > 0. Calculate hn .
[2 marks]
(b) Assume hn = for n = 0, 1, 2, ..., where 0 < < 1. Calculate the probability
function pn .
[4 marks]
3. Consider the portfolio of an auto insurance that has 100 policyholders. The numbers of
accidents for the policyholders in one year were recorded as follows:
Number of accidents Number of policyholders
0
86
9
1
2
3
1
3
4
1
(a) Use the sample mean to estimate the expectation of the number of accidents for one
policyholder in one year.
[2 marks]
(b) Use the sample variance to estimate the variance of the number of accidents for one
policyholder in one year.
[2 marks]
(c) Determine the empirical distribution function F100 (x) for the number of accidents
for one policyholder in one year for all x.
[3 marks]
(d) Determine the empirical estimate for the discrete hazard rate function h(k ) of the
number of accidents for one policyholder in one year for k = 0, 1, 2, 3, 4. [3 marks]
(e) Let be the probability that one policyholder will have more than one accident
in one year. Use a consistent estimator for and the normal approximation for
E ()
to determine the interval estimate for at the condence level of 90%.
V ar()
[5 marks]
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4. Ten cancer patients were observed from the time of diagnosis until the earlier of death or
24 months from diagnosis. Deaths occurred as follows: At 6 months there were 2 deaths;
at 10 months there were d deaths; at 15 months there was one death; at 20 months there
was one death; and at 24 months there was one death.
(a) The Nelson-Aalen estimate for the cumulative hazard function H (t) of the death
time of one cancer patient at the 18 month is 0.775. Determine the Klein estimate
for V ar(H (18)).
[6 marks]
(b) Determine the Nelson-Aalen estimate for the probability that a cancer patient will
survive at least 22 months.
[3 marks]
5. You are given the following data set for the (in times years) of deaths or censored times
for 25 individuals from a group:
2, 3, 3, 3+ , 4, 4, 4, 4, 4+ , 5+ , 6, 6, 7, 7, 7, 7+ , 7+ , 8, 9, 10, 12+ , 13, 13, 14, 16+ .
(a) Table the values of yj , sj and rj for all possible values of j .
[4 marks]
(b) Determine the Kaplan-Meier estimate for the probability that an individual in the
group will survive at least 11 years.
[3 marks]
(c) Determine the Greenwood estimate for the variance of the Product-Limit estimate
S (11).
[2 marks]
(d) Use the Brown, Hollander, and Kowar method to estimate the probability that an
individual in the group will survive at least 20 years.
[3 marks]
6. The record of the claim amount with deductibles and policy limits for 15 policyholders
in an insurance portfolio was given as follows:
Policy
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Deductible
0
0
0
0
0
0
0
0
0
0
1
2
2
3
5
Recorded claim amount
1
2
3
10+
2
6
10
13+
13
6
10
13
18+
16
10
Policy limit
10
10
10
10
12
12
14
13
13
15
15
15
18
18
18
(a) Table the values of yj , sj and rj for all possible values of j .
2
[5 marks]
(b) Determine the Nelson-Aalen estimate H (x) for the cumulative hazard function H (x)
of the claim amount of one policyholder for all x [0, x ), where x is the largest
observed value.
[4 marks]
(c) Determine the Aalen estimate for V ar(H (10)).
[2 marks]
(d) Determine a 90% linear condence interval for H (10).
[2 marks]
(e) Determine a 90% log-transformed condence interval for H (10).
[4 marks]
7. In a 5-year term mortality study for a group of 15 members aged 90. One member may
join the study at the beginning or later while one member may terminate the study by
death or by surrender or at the expiration of the ve-year period. The following table
presents the results for the 15 members. The column headed Starting time gives the
time at which one member joined the study, the column headed Termination time gives
the time at which one member terminated the study, and the column headed Event
indicates the reasons of terminations.
Member
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Starting time
0
0
0
0
0
0
0
0
0.3
0.5
1.4
1.5
2.0
2.8
3.5
Termination time
0.2
0.5
0.6
3.9
5.0
4.2
2.8
2.8
2.3
5.0
3.2
3.9
2.8
3.2
5.0
Event
s
s
d
d
e
d
d
s
d
e
s
d
s
d
e
Here s means surrender, d means death, and e means expiration of the 5-year
period.
(a) Table the values of yj , sj and rj for all possible values of j .
[5 marks]
(t) for the survival function S (t) of the
(b) Determine the Product-Limit estimate S
death time of one member for all t [0, t ), where t is the largest observed value.
[4 marks]
(c) Calculate Greenwoods approximation for the variance V ar(S (2.3)).
[2 marks]
(d) Determine a 95% linear condence interval for S (2.3).
[2 marks]
(e) Determine a 95% log-transformed condence interval for S (2.3).
[4 marks]
[Total: 100 marks]
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