An insurer is modelling time to death of lives insured at age [math]x[/math] using the Kaplan-Meier estimator. You are given the following information.

(i) There were 100 policies in force at time 0

(ii) There were no new policies entering the study

(iii) At time 10.0, immediately after a death, there were 50 policies remaining in force

(iv) The Kaplan-Meier estimate of the survival function for death at time 10 is [math]\hat{S}(10.0)=0.92[/math]

(v) The next death after time 10.0 occurred when there was one death at time 10.8

(vi) During the period from time 10.0 to time 10.8 , a total of 10 policies terminated for reasons other than death

Calculate [math]\hat{S}(10.8)[/math], the Kaplan-Meier estimate of the survival function [math]S(10.8)[/math].

• 0.897
• 0.903
• 0.909
• 0.910
• 0.920

In a study of 1,000 people with a particular illness, 200 died within one year of diagnosis. Calculate a 95% (linear) confidence interval for the one-year empirical survival function.

• (0.745,0.855)
• (0.755,0.845)
• (0.765,0.835)
• (0.775,0.825)
• (0.785,0.815)

A cohort of 100 newborns is observed from birth. During the first year, 10 drop out of the study and one dies at time 1. Eight more drop out during the next six months, then, at time 1.5, three deaths occur.

Calculate [math]\hat{S}(1.5)[/math], the Nelson-Aalen estimator of the survival function, [math]S(1.5)[/math].

• 0.950
• 0.951
• 0.952
• 0.953
• 0.954

You are given the following data based on 60 lives at time 0 :

Calculate the upper limit of the 80% linear confidence interval for [math]S(21.0)[/math] using the Kaplan Meier estimate and Greenwood's approximation.

• 0.872
• 0.893
• 0.915
• 0.944
• 0.968

In a mortality study, the following grouped death data were collected from 100 lives, all studied beginning at age 40 .

There were no terminations other than death.

Calculate [math]\hat{S}_{40}(32)[/math] using the ogive empirical distribution function.

• 0.44
• 0.48
• 0.52
• 0.56
• 0.60

You are doing a mortality study of insureds between ages 70 and 90 . Two specific lives contributed this data to the study:

You assume mortality follows Gompertz law [math]\mu_{x}=B \times c^{x}[/math] and plan to use maximum likelihood estimation.

[math]L[/math] is the likelihood function associated with these two lives.

[math]L^{*}[/math] denotes the value of [math]L[/math] if the Gompertz parameters are [math]B=0.000003[/math] and [math]c=1.1[/math].

Calculate [math]L^{*}[/math].

• 0.0115
• 0.0131
• 0.0147
• 0.0163
• 0.0179

You are doing a mortality study of insureds between ages 60 and 90 . Two specific lives contributed this data to the study:

You assume mortality follows Gompertz law [math]\mu_{x}=B \times c^{x}[/math] and plan to use maximum likelihood estimation.

[math]L[/math] is the log-likelihood function (using natural logs) associated with these two lives.

[math]L^{*}[/math] denotes the value of [math]L[/math] if the Gompertz parameters are [math]B=0.000004[/math] and [math]c=1.12[/math].

Calculate [math]L^{*}[/math].

• -4,67
• -4.53
• -4.39
• -4.25
• -4.11

You are given the following seriatim data on survival times for a group of 12 lives. The superscript + indicates a right-censored value.

[math]25,32^{+}, 35^{+}, 36,40^{+}, 44,48,60,62^{+}, 65,67,70^{+}[/math]

Calculate the standard deviation of the estimate of [math]S(50)[/math] using the Nelson-Aalen estimator.

• 0.1455
• 0.1519
• 0.1547
• 0.1621
• 0.1650

Initially, 80 lives are included in an observation of survival times following a specific medical treatment. You are given excerpted information from the study data in the table below.

Calculate the Kaplan-Meier estimate of [math]S(2)[/math].

• 0.931
• 0.952
• 0.960
• 0.969
• 0.972

Initially, 50 lives are included in an observation of survival times following a specific medical treatment. You are given excerpted information from the study data in the table below.

Calculate the Nelson-Aalen estimate of [math]S(2)[/math].

• 0.910
• 0.916
• 0.922
• 0.928
• 0.934