Test 2 Flashcards
Briefly explain the principle of correspondence
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.
Briefly explain the concept of a rate interval
A rate interval is a period of one year during which a life’s recorded age remains the same.
Briefly explain the weaknesses of the chi-square test.
- 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,
Explain the theoretical rationale for the derivation of the degrees of freedom used for the calculation of the critical value.
- 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.
State the common null hypotheses when testing graduated rates:
H0: The graduated rates reflect the true underlying rate of ___ of this population at each curate duration.
State the alternative hypothesis when testing graduated rates:
H1: For at least one duration, the graduated rates to not reflect the true underlying ___ rate of this population.
List points to comment on when using the standard deviations test.
- Overall Shape
- Symmetry
- Absolute deviations
- Outliers
- Conclusion
3 Methods of graduation
- Parametric formula
- Reference to a standard table
- Graphical
What must graduated rates be tested for?
- Smoothness
- Adherence to data
Chi-Square Tests: Purpose
Overall goodness of fit.
Chi-Square Test Assumptions
- 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.
Standard deviations test: Purpose
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.
Standard Deviations Test: Assumptions
Assumes the normal approx. provides a good approximation at all ages.
Standard Deviations Test: Rationale
Under the hypothesis, the Zx’s comprise m independent samples from a N(0,1) distribution. It just tests for that normality.
Signs Test: Purpose
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.
Signs Test: Rationale
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.
Cumulative Deviations test: Purpose
Detects overall bias or long runs of deviations of the same sign.
Grouping of Signs test: Purpose
To test for overgraduation.
Tests “clumping” of deviations of the same sign.
Serial Correlations test: Purpose
To test for overgraduation.
Detects grouping of signs of deviations.
List the 3 assumptions applicable to the census process (using different definitions of age)
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.
Explain why data is divided into homogenous groups when doing a mortality investigation
- 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.
Name 8 types of censoring
- Right censoring
- Left censoring
- interval censoring
- random censoring
- informative censoring
- non-informative censoring
- Type I censoring
- Type II censoring
Right Censoring
Cuts short observations in progress.
When:
- Policyholders surrender
- Active lives retire (pension scheme)
- Endowment Assurance policies mature
Left Censoring
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.
Interval Censoring
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.
Random Censoring
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
Non-informative censoring
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
Examples of informative censoring
- Withdrawals in life insurance policy (likely to be in better health than the rest)
- Ill-health retirements in pension schemes.
Type I censoring
If the censoring times {Ci} are known in advance, then the mechanism is called “Type I censoring”.
Type II censoring
If observation is continued until a predetermined number of deaths has occurred.
Explain why a mortality experience would need to be graduated.
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.
Explain the difference between the central and the initial exposed to risk
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.
Describe three shortcomings of the χ2 test for comparing crude estimates of mortality with a standard table and why they may occur.
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.
Name a test that would detect the shortcoming of:
Outliers
not being detected in the Chi-Square Test
Individual Standardised Deviations Test.
Name a test that would detect the shortcoming of:
Small bias
not being detected in the Chi-Square Test
Signs Test
Cumulative Deviations Test.
Name a test that would detect the shortcoming of:
Clumps or runs
not being detected in the Chi-Square Test
Grouping of Signs Test
Serial Correlations Test.
