Chapter 3

Question 1

What is N in the binomial model

Answer 1

The individuals subject to death within a given time frame

Question 2

What is q in the binomial model

Answer 2

Probability of event of death occuring in each case

Question 3

What is N called and denoted by

Answer 3

N is the initial exposed to risk and is denoted E

Question 4

What is the ratio Q hat called?

Answer 4

Crude occurance rate or crude mortality rate

Question 5

What are the qualities of the crude occurance rate estimator q hat

Answer 5

It is unbiased
It is the MLE of q
It is a consistent estimator of q
It is an efficient estimator of q meaning its variances matches the cramer rao bound

Question 6

What can the sampling distribution be written as if N is very large when we have a number of deaths ~ binomial model

Answer 6

We can use Normal distribution and can find confidence intervals of Q hat

Question 7

Explain the central exposed to risk between x and x+1

Answer 7

Measure the length of time in years each individual is exposed to this risk and sum over all individuals to get the total.

Question 8

What si the max length of time a person can be exposed to risk of death between x and x+1

Answer 8

1 year

Question 9

Explain person years of exposure?

Answer 9

Same as central exposed to risk just usually called this in medical stsatistics

Question 10

Explain the assumptions in the posson model

Answer 10

Assuming the rate of occurrence is the same for everyone and it acts on each life independently. Then the total number of deaths at age x is distributed by a Poisson distribution, by summing the independent Poisson for each life and its length of exposure

Question 11

What are the properties of the estimator mew(x+0.5) hat

Answer 11

Unbiased,
Maximum likelihood estimator
Consistent estimator
Efficient estimator

Question 12

Explain Consistent estimator

Answer 12

If at the limit n → ∞ the estimator tend to be always right (or at least arbitrarily close to the target), it is said to be consistent

Question 13

Explain efficient estimator

Answer 13

The most efficient estimator among a group of unbiased estimators is the one with the smallest variance

Question 14

Comparing central exposed to risk and inital exposed to risk

Answer 14

Central exposed to risk is easily observable - just record the time each life was observed , carries through unchanged to multiple state models
Initial exposed to risk requires adjustments in respect of lives who die

Question 15

Which is easier to calculate central exposed to risk and inital exposed to risk

Answer 15

central exposed to risk

Question 16

What is mx

Answer 16

Probability a life between x and x+1 dies

Question 17

Explain what the homogeneity assumption means

Answer 17

Usually we assume we observe groups with the same mortality characteristics or homogeneous groups with identifical lives and death rates in mortality studies. In reality this is near impossible

Question 18

What balance needs to be struck when organising groups for mortality studies

Answer 18

Balance between subdividing groups to achieve increased homogeneity and the resultant group being too small that they lack volume for statistically significant results. Sample error would be too big

Question 19

Give some common factors used to subdivide data for mortality studeis

Answer 19

Sex, age, type of policy, smoker, level of underwriting, duration of policy, occupation, BMI, impairments

Question 20

What is another factor that limits subdivision abilities

Answer 20

Actually collecting the data to allow for subdivision is difficult

Question 21

State the principle of correspondence

Answer 21

A life alive at time t should be included in the exposure at age x at time t f and only if, were that life to die immediately, they would be counted in the death data dx age x

Question 22

Explain the idea of the principle of correspondence

Answer 22

Data deaths and exposure should be defined consistently or the ratio d/n is meaningless

Question 23

What are the assumptions in calculation of central exposed to risk with ideal data if we had it all

Answer 23

Independent and identically distributed with respect to mortality lives!
Non informative censoring
Model specific assumptions that may apply

Question 24

Why can we not calculate the central exposure to risk exactly -

Answer 24

in reality we do not have all the information we would need. We would need birth, death dates of everyone and when they entered and left the study

Question 25

Define dx in context of calculating approximation of the central exposed to risk

Answer 25

Number of deaths age x last birthday during calendar years K, K+1…..

Question 26

Define Px,t in context of calculating approximation of the central exposed to risk

Answer 26

Number of lives under observation aged x last birthday at time t where t=1 January in calendar years K , k+1, …, K+N, K+N+1

Question 27

What approximation do we used to find central exposed to risk and why?

Answer 27

Trapezium rule - we only have avalue of Px,t at certain points so we cannot find the area under the plotted points by integrals. Instead we assume linearity between census dates and use trapezium rule

Question 28

What data determines the correct rate interval to use

Answer 28

Death data carries the most information so it determines wich rate interval to use

Question 29

What is a rate interval

Answer 29

A different year of age studied as defined by age nearest birthday, age next birthday, age last birthday.

Question 30

Define d^2 x

Answer 30

Total number of deaths age x nearest birthday during calendar years K, K+1,…, K+n

Question 31

Define d^3 x

Answer 31

Total number of deaths age x next birthday during calendar years K, K+1,…, K+n

Question 32

What is the rate interval when we define x as age last birthday

Answer 32

[x,x+1)

Question 33

What is the rate interval when we define x as age nearest birthday

Answer 33

[x-1/2,x+1/2)

Question 34

What is the rate interval when we define x as age next birthday

Answer 34

[x-1,x)

Question 35

What is the census corresponding to the interval [x-1,x)

Answer 35

P^3 x,t is Number of lives under observation age x next birthday at time t where t=1 is january in calendar year K, K+1,…, K+n, K+n+1

Question 36

What is the census corresponding to the interval [x-1/2,x+1/2)

Answer 36

P^2 x,t is Number of lives under observation age x nearest birthday at time t where t=1 is january in calendar year K, K+1,…, K+n, K+n+1

Question 37

Why do we adjust the age definition in the census data to
correspond with that of the deaths data

Answer 37

To ensure that they follow the principle of correspondence,
which states that a life alive time t should be included in the exposure at age x at time t if and only if, were that life to die immediately, he or she would be counted in the death data at age x.
The deaths data “carry most information” when mortality rates are small, so we adjust the census data not the deaths data.

Question 38

What assumptions are made using the trapezium estimate to calculate central exposed to risk

Answer 38

We need to assume that births are uniformly distributed across the calendar year.
To use the trapezium rule we must assume that the population varies linearly between census dates.
We assume that the population enumerated in the census of 1 January 2015 can be taken to be the population at the end of the calendar year 2014

Question 39

How could you estimate death rates at age x nearest birthday for future years

Answer 39

To compute the population aged x last birthday some kind of forecasting/modelling will be required
Extrapolation of the linear change at each age x between
dates you know is one option
Can adjust census data to match the death data

Question 40

Two reasons why those with a life assurance policy tend to have lower mortality than
those without such a policy:

Answer 40

Because of self-selection: proposers for assurance policies are generally
wealthier and /higher educated than the general population.
* Because of temporary initial selection: because policyholders have passed the
underwriting criteria of the life assurance company when they originally took out
the policy, so are healthier on average than the rest of the population.

Question 41

What statistical problem can arise when data is sub divided

Answer 41

We can subdivide until the data is too small for the mortality investigation to have
statistical significance. So there is a tension in sub-dividing to reduce heterogeneity in
mortality rates and in achieving groups with sufficient data (number of deaths) to produce
statistically significant estimates of the mortality rates.