Test 2

Question 1

Briefly explain the principle of correspondence

Answer 1

A life alive at 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 dx at age x.

Question 2

Briefly explain the concept of a rate interval

Answer 2

A rate interval is a period of one year during which a life’s recorded age remains the same.

Question 3

Briefly explain the weaknesses of the chi-square test.

Answer 3

• A few large deviations are offset by a lot of very small deviations.
• Deviations may be biased (constantly above or below the data) over the whole range or over sections
• The squared deviations also do not give an indication of the direction of the bias, if any,

Question 4

Explain the theoretical rationale for the derivation of the degrees of freedom used for the calculation of the critical value.

Answer 4

• It is firstly assumed that the number of withdrawals at each duration has an approximate normal distribution and therefore the standardized deviations have approximate normal(0,1) and z_x^2 ~ X^2_1
• The withdrawals at durations are assumed to be independent and therefore the sum of the squared standardized deviations will have chi-square distribution with degrees of freedom equal to the number of individual standardized deviations.
• An adjustment is required because a formula has been fitted to the data. Therefore there is dependence between the graduated rates and the actual data used to assess the fit of the graduated rates. The rule of thumb is that one degree of freedom is removed for every parameter being estimated.

Question 5

State the common null hypotheses when testing graduated rates:

Answer 5

H0: The graduated rates reflect the true underlying rate of ___ of this population at each curate duration.

Question 6

State the alternative hypothesis when testing graduated rates:

Answer 6

H1: For at least one duration, the graduated rates to not reflect the true underlying ___ rate of this population.

Question 7

List points to comment on when using the standard deviations test.

Answer 7

• Overall Shape
• Symmetry
• Absolute deviations
• Outliers
• Conclusion

Question 8

3 Methods of graduation

Answer 8

• Parametric formula
• Reference to a standard table
• Graphical

Question 9

What must graduated rates be tested for?

Answer 9

• Smoothness

- Adherence to data

Question 10

Chi-Square Tests: Purpose

Answer 10

Overall goodness of fit.

Question 11

Chi-Square Test Assumptions

Answer 11

• No heterogeneity of mortality within each group and lives are independent.
• The expected numbers of deaths are high enough for the approximation to be valid.

Question 12

Standard deviations test: Purpose

Answer 12

Use to look for the first defect of the chi-square test.
Tests overall goodness of fit.
Reveals problems due to under/overgraduation or heterogeneity.

Question 13

Standard Deviations Test: Assumptions

Answer 13

Assumes the normal approx. provides a good approximation at all ages.

Question 14

Standard Deviations Test: Rationale

Answer 14

Under the hypothesis, the Zx’s comprise m independent samples from a N(0,1) distribution. It just tests for that normality.

Question 15

Signs Test: Purpose

Answer 15

Simple test for overall bias (i.e. whether the graduated rates are too high or too low)
It will identify the second deficiency of the chi square test, ie failure to detect where there is an imbalance between positive and negative deviations.

Question 16

Signs Test: Rationale

Answer 16

We would expect roughly half the graduated values to be above the crude rates and half below.
Normally a two-tailed test - looking for both positive and negative bias.

Question 17

Cumulative Deviations test: Purpose

Answer 17

Detects overall bias or long runs of deviations of the same sign.

Question 18

Grouping of Signs test: Purpose

Answer 18

To test for overgraduation.

Tests “clumping” of deviations of the same sign.

Question 19

Serial Correlations test: Purpose

Answer 19

To test for overgraduation.

Detects grouping of signs of deviations.

Question 20

List the 3 assumptions applicable to the census process (using different definitions of age)

Answer 20

Converting from nearest to last birthday, ex:
- Assume birthdays are uniformly spread over the calendar year.

Calculating central Exposure (converting integral to summation):
- Assume exposure varies linearly over the calendar year.

Calculating the estimate for µ, using d/Ec:
- Assume the force of mortality is constant over the years in the period and over the life interval.

Question 21

Explain why data is divided into homogenous groups when doing a mortality investigation

Answer 21

• The estimated mortality rate is an average rate based on the lives included in the group under investigation.
• If it is expected that the lives included in this group are homogenous wrt their true underlying mortality experience, this will improve the estimate of that underlying mortality rate.

Question 22

Name 8 types of censoring

Answer 22

• Right censoring
• Left censoring
• interval censoring
• random censoring
• informative censoring
• non-informative censoring
• Type I censoring
• Type II censoring

Question 23

Right Censoring

Answer 23

Cuts short observations in progress.

When:

• Policyholders surrender
• Active lives retire (pension scheme)
• Endowment Assurance policies mature

Question 24

Left Censoring

Answer 24

Censoring mechanism prevents us from knowing when entry into the state observed took place.

Example:
- Medical studies, we know only when diagnosed, not when onset fell.

Question 25

Interval Censoring

Answer 25

If the observational plan only allows us to say that an event fell within some interval of time.

Example:
- Actuarial: only know calendar year of withdrawal/death.

Question 26

Random Censoring

Answer 26

The time Ci at which observation of the i’th lifetime is censored is a random variable.

Example:

• When individuals may leave the observation by means OTHER THAN DEATH of which the time is not know in advance.
• Withdrawals

Question 27

Non-informative censoring

Answer 27

Gives no information about the lifetimes {Ti}

Just means the mortality of the remaining lives in the at-risk group is the same as the mortality of the lives that have been censored.

Example
- End of investigation period

Question 28

Examples of informative censoring

Answer 28

• Withdrawals in life insurance policy (likely to be in better health than the rest)
• Ill-health retirements in pension schemes.

Question 29

Type I censoring

Answer 29

If the censoring times {Ci} are known in advance, then the mechanism is called “Type I censoring”.

Question 30

Type II censoring

Answer 30

If observation is continued until a predetermined number of deaths has occurred.

Question 31

Explain why a mortality experience would need to be graduated.

Answer 31

We believe that mortality varies smoothly with age.
Therefore the crude estimate of mortality at any age carries information about mortality at adjacent ages.

By smoothing the experience, we can make use of data at adjacent ages to improve the estimates at each age.
This reduces sampling (or random) errors.
The mortality experience may be used in financial calculations.

Irregularities, jumps and anomalies in financial quantities (such as premiums for life insurance contracts) are hard to justify to customers.

Question 32

Explain the difference between the central and the initial exposed to risk

Answer 32

The central exposed to risk at age x, Ec , is the waiting time in a multiple-state or Poisson model.

The initial exposed to risk is equal to the central exposed to risk plus the time elapsing between the date of death and the end of the rate interval for those who are observed to die during the rate interval.

Question 33

Describe three shortcomings of the χ2 test for comparing crude estimates of mortality with a standard table and why they may occur.

Answer 33

Outliers. Since all the information is summarised in one number, a few large deviations may be offset or hidden by a large number of small deviations.

Small bias. Since the squares of the differences are used, the sign of the differences are lost, hence small but consistent bias above or below may not be noticed.

Clumps or runs. Again, because the squares of the differences are used, the sign of the differences are lost, so significant groups of (clumps or runs) of bias over ranges of the data may not be detected.

Question 34

Name a test that would detect the shortcoming of:
Outliers
not being detected in the Chi-Square Test

Answer 34

Individual Standardised Deviations Test.

Question 35

Name a test that would detect the shortcoming of:
Small bias
not being detected in the Chi-Square Test

Answer 35

Signs Test

Cumulative Deviations Test.

Question 36

Name a test that would detect the shortcoming of:
Clumps or runs
not being detected in the Chi-Square Test

Answer 36

Grouping of Signs Test

Serial Correlations Test.