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Survival Models Assignment 2

Graduation and Mortality Projections

Your task is to write a report in two parts. Part 1 describes the graduation of selected data from the CMI data set while part 2 describes mortality projections based on this same data set.

Part 1: Graduation

The R-script Assignment2.r (available on Vision) selects data over the calendar year range 1950 to 2005, for a particular age MyAge from the CMI data set.  The age MyAge, which is in the range 50 to 70, is chosen randomly by running Assignment2.r with set.seed(xxxx) where xxxx is the last four digits of your student identification number.  You will need to have the file CMI read.r, the data files CMI Deaths.csv and CMI Exposures.csv, and the R-function file Test GoF.r in your working directory.  To change your working directory select ‘change dir...’  from the ‘file’ menu in R. Please do not include any line of the form setwd in a submitted R script.

Task 1:  For the age MyAge, fit a Gompertz generalized linear model with Binomial errors and logit link, with logit(E(gt )) = a + bt + c(t · )2  over the years 1950 to 2005 where t is calendar year and = 1977.5. Save the values of the fitted coefficients for a as Quantity1a, for b as Quantity1b and for c as Quantity1c (each worth 0.5 marks).

Task 2:  Apply the following statistical tests on the associated Pearson residuals to assess the suitability of the fitted model for actuarial use, including appropriate plots:

1. Chi-squared goodness of t. Save the test statistic as Quantity2a and the p-value as Quantity2b.

2. Standardised deviations with 4 equal areas. Save the test statistic as Quantity3a and the p-value as Quantity3b.

3. Sign Test. Save the test statistic as Quantity4a and the p-value as Quantity4b.

4. Change of Sign Test.   Save the test statistic as Quantity5a and the p-value as Quantity5b.

5. Grouping of Signs Test.  Save the test statistic as Quantity6a and the p-value as Quantity6b.

6. Serial Correlation Test. Save the test statistic as Quantity7.

For all of the above tests, use the exact calculation of the Pearson residuals, i.e. do not assume that 1 · gx  is negligible. All of the above quantities are worth 0.5 marks each.

Part 1 of your report should describe the fitted model, the statistical tests you have applied and your conclusions as to the suitability of the fitted model for actuarial use. Part 1 of your written report must state clearly the coefficients of the fitted GLM and all relevant details of the statistical tests.


Part 2: Mortality Projection

The second part of the R-script Assignment2.r selects data from the CMI data set over the age range 40 to 90, for calendar years 1950 to 1980.

Task 1: Fit a Lee-Carter model (again with binomial errors) to the data for calendar years 1950 to 1980. You should fit a model of the form

logit(gx,y ) = αx + βx Ky

where z is age and y is year. Please ensure the values of αx , βx and Ky satisfy the identifiability conditions

βx  = 1

x

and

K  = 0.

y

Task 2: Project the logit mortality rates forward for 25 years by modelling future values of y  as a time series (random walk with constant drift).

Part 2 of your report should describe: (a) the fitted model, the projection of future mortality at age MyAge, including a suitable measure of the uncertainty associated with the projected mortality; and (b) a comparison of the projected and the actual mortality in the 25 years since 1980 for age MyAge, including your conclusions concerning the suitability of the projected mortality from the point of view of an actuary in 1980.

Part 2 of your written report must state clearly the tted value of log(μMyAge+0.5,1980.5 ) and the projected value of log(μMyAge+0.5,2005.5 ), where you can assume that

gx,y

μx+0.5,y+0.5  =


You should save the vector of fitted values of αx  as Quantity8, the vector of fitted values of βx  as Quantity9, the vector of fitted values of Ky  as Quantity10 and the vector of projected values of Ky  as Quantity11.  Each of these vectors is worth 1 mark.  Please save the values of the tted value of log(μMyAge+0.5,1980.5 ) and the projected value of log(μMyAge+0.5,2005.5 ) as Quantity12a and Quantity12b, respectively. Each of these is worth 0.5 mark.


Your Report and R code

You must submit, using the links on Canvas:  (a) your written report, as a pdf file or as a Word document; and (b) an R script containing code that produces all your results. Please save your R script as Assignment2Code.R.

Twenty-five marks are available in total.  Twelve marks are for correctly carrying out all the mathematical and statistical tasks, and this will be assessed by running your R code and comparing the results with the verified correct output. You must ensure that you have defined the variables as above and ensure your code runs without generating errors. Failure to do so will result in 0 marks being awarded. You can check your code runs correctly by running it through the file TestSubmissionAssignment2.R.

Thirteen marks are for the written report. You should aim for your report to be at most 8 pages. The following qualities will be taken into account in awarding marks for the written report.

Structure and organization of the report: Your exposition should flow logically without jumps or gaps. You will have succeeded if the reader is always able to under- stand what you are doing and why, without having to refer backwards and forwards in your text.

Any tables or figures: In order to assist the readers understanding, tables and figures should be neat and clear, have an adequate caption, and be clearly referred to in the main text of your report.

Precise expression: Any statement relating to a mathematical or statistical quantity should be precise, unambiguous and based on well-defined quantities, whether the statement is expressed using words or mathematical notation. In particular, you should explain any assumptions you are making in your models.

Clarity of expression: At all times your writing, whether descriptive or mathemat- ical, should flow clearly.

Usefulness of your conclusions: The reader is reading your report in order to learn your conclusions, assess the strength of support for them, and in all probability to decide upon some subsequent action that may depend upon them. Can the reader make decisions with confidence based upon what you have written?