-
8/11/2019 a2.w09
1/3
ACTSC 433/833 Winter 09
Assignment 2
Due: 10 March 2009 (hand in to the instructor at the beginning
of class)
1. You are given the following censored and truncated data:
I di xi ui
1 0 1 -
2 0 2 -
3 0 - 3
4 2.5 4 -
5 2 5 -
6 0 - 4.5
7 3 - 10
8 0 8 -
(a)Create a table of yj, rj, and sj, for j= 1, 2, , k, where k
is the total number ofdeath points.
(b)Calculate S8(2) and its variance.
(c)Calculate (4)H and its variance.
(d)Construct a 95% symmetric linear confidence interval for
S(2).(e)Construct a 95% log-transformed confidence interval for
S(2).
(f)
Repeat parts (d) and (e) forH(4).(g)On the basis of your answers
to parts (d) to (f), comment on the advantage of log-transformed
confidence intervals over symmetric linear confidence
intervals.
2. You are given the following random sample from the
distribution of X:
5, 8, 10, 12, 14, 20, 25, 35, 50, 50
(a)Calculate the empirical estimate of E(X) and E(X2)
(b)Calculate the smoothed empirical estimate of the 25thand
75thpercentiles.
(c)
Assuming that X follows a lognormal distribution with parameters
and ,estimate and using
(ii) the method of moments, and(iii) the method of percentile
matching based on the 25th and 75th
percentiles.
1
-
8/11/2019 a2.w09
2/3
3. You are given the following random sample from the
distribution of X:
Range Number of observations
(0 100] 50(100 200] 20
(200 400] 10(400 1000] 20
Total 100
(a)Plot an ogive on the basis of the random sample provided.
(b)Assuming thatXfollows a gamma distribution with parameters
and , estimate
and , using(i) the method of moments, and(ii) the method of
percentile matching based on the 25
thand 75
th
percentiles.Note: You may use GAMMADIST and Solver in Excel when
answering part (ii).
4. You are given the following information regarding a mortality
study:
(i) All (n) members of the study were observed from birth. Some
of themleft the study by means of other than death.
(ii) Between times 10 and 12, 6 observations were censored.(iii)
No observations were censored at times of death.
(a)Complete the values of a, b, c, d, e, andfin following
table:
j yj sj rj Sn(yj)1 1 a 20 0.902 2.5 b 18 c3 5.75 1 14 0.654 10 3
d 0.50
5 12 e f 0.25
(b)Using the results in (a), compute the Nelson-Aalen estimate
of S(11).
5. Assuming that there is no censoring or truncation, prove the
following:
(a) The Kaplan-Meier estimator of the survival function at death
pointyj,j=1, 2, ,k, is given by
1
1( )
k
n j
i j
S y sn = +
= i .
That is, in the absence of censoring and truncation, the
Kaplan-Meiers methodyields the same estimate of the survival
function as the empirical method.
2
-
8/11/2019 a2.w09
3/3
(b) The Greenwoods approximation of the estimated variance of
Sn(yj),j=1, 2, ,
k1, is given by
( ) ( )( ) 1 ( )
Var ( )n j n j
n j
S y S yS y
n
= .
That is, in the absence of censoring and truncation, the
Greenwoods method andthe empirical method yield the same variance
of Sn(t).
3
