Estimation, Evaluation, and Selection of Actuarial Models
Estimation, Evaluation, and Selection of Actuarial Models Estimation, Evaluation, and Selection of Actuarial Models
14 CHAPTER 2. MODEL ESTIMATION i d i x i u i i d i x i u i 1 0 - 0.1 16 0 4.8 - 2 0 - 0.5 17 0 - 4.8 3 0 - 0.8 18 0 - 4.8 4 0 0.8 - 19—30 0 - 5.0 5 0 - 1.8 31 0.3 - 5.0 6 0 - 1.8 32 0.7 - 5.0 7 0 - 2.1 33 1.0 4.1 - 8 0 - 2.5 34 1.8 3.1 - 9 0 - 2.8 35 2.1 - 3.9 10 0 2.9 - 36 2.9 - 5.0 11 0 2.9 - 37 2.9 - 4.8 12 0 - 3.9 38 3.2 4.0 - 13 0 4.0 - 39 3.4 - 5.0 14 0 - 4.0 40 3.9 - 5.0 15 0 - 4.1 j y j s j r j 1 0.8 1 32 − 0 − 2=30or 0+32− 0 − 2=30 2 2.9 2 35 − 1 − 8=26or 30 + 3 − 1 − 6=26 3 3.1 1 37 − 3 − 8=26or 26 + 2 − 2 − 0=26 4 4.0 2 40 − 4 − 10 = 26 or 26 + 3 − 1 − 2=26 5 4.1 1 40 − 6 − 11 = 23 or 26 + 0 − 2 − 1=23 6 4.8 1 40 − 7 − 12 = 21 or 23 + 0 − 1 − 1=21 ¤ Exercise 3 Repeat the above example, treating “surrender” as “death.” The easiest way to do this is to reverse the x and u labels and then use the above formula. In this case death produces censoring because those who die are lost to observation and thus their surrender time is never observed. Treat those who lasted the entire fiveyearsassurrendersatthattime. Despitealltheworkwehavedonetothispoint,wehaveyettoproduceanestimatorofthe survival function. The one most commonly used is called the Kaplan-Meier Product-Limit Estimator. Begin with S(0) = 1. Because no one died prior to y 1 , the survival function remains at 1 until this value. Thinking conditionally, just before y 1 , there were r 1 people available to die, of which s 1 did so. Thus, the probability of surviving past y 1 is (r 1 − s 1 )/r 1 .Thisbecomesthevalue of S(y 1 ) and the survival function remains at that value until y 2 . Again, thinking conditionally, the new survival value at y 2 is S(y 1 )(r 2 − s 2 )/r 2 . The general formula is ⎧ 1, 0 ≤ t
2.2. ESTIMATION USING DATA-DEPENDENT DISTRIBUTIONS 15 that at the age of the last death, there were still people alive, but all were censored prior to death. We know that survival past the last observed death age is possible, but there is no empirical data available to complete the survival function. One option (the first one used in the above formula) is to keep the function at its last value. This is clearly the largest reasonable choice. Another option is to declare the function to be zero past the last observed age, whether it is an observed death or a censored age. This is the smallest reasonable choice and makes it possible to calculate moments. An intermediate option is to use an exponential curve to reduce the value from its current level to zero. Let w =max{x 1 ,...,x n ,u 1 ,...,u n }. Then, for t ≥ w, S n (t) =e (t/w)lns∗ =(s ∗ ) t/w , where s ∗ = kY i=1 µ ri − s i . r i There is an alternative method of deriving the values of s j and r j that is more suitable for Excel spreadsheet work. 2 Thestepsareasfollows: 1. There should be one row for each data point. The points need not be in any particular order. 2. Each row should have three entries. The first entry should be d i , the second entry should be u i (the cell should be empty if this point was not censored), and the third entry should be x i (the cell should be empty if this point was censored). Assume, for example that the d i s occupy cells B6:B45, the u i s occupy C6:C45, and the x i s occupy D6:D45. Place the letter x on top of this column (in D5). 3. Highlight the column of x values (D5:D45) including the cell with the label. 4. Select Data | Filter | AdvancedFilterandtheninthedialogboxchecktheaction“Copy to another location.” The list range should already be D5:D45, the criteria range should be blank, the new location should be an unoccupied cell (say G15), and “Unique records only” should be checked. The result should be the set of y values in cells G15:Gxx where xx indicates the last cell needed. 5. Select the cells with y values (G16:Gxx). Sort the y values in ascending order by selecting Data | Sort from the menu. 6. Equation (2.1) is then used to calculate the r values. For example, in cell H16, it would be calculated as = COUNTIF(B$6:B$45,”
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2.2. ESTIMATION USING DATA-DEPENDENT DISTRIBUTIONS 15<br />
that at the age <strong>of</strong> the last death, there were still people alive, but all were censored prior to death.<br />
We know that survival past the last observed death age is possible, but there is no empirical data<br />
available to complete the survival function. One option (the first one used in the above formula) is<br />
to keep the function at its last value. This is clearly the largest reasonable choice. Another option<br />
is to declare the function to be zero past the last observed age, whether it is an observed death or<br />
a censored age. This is the smallest reasonable choice <strong>and</strong> makes it possible to calculate moments.<br />
An intermediate option is to use an exponential curve to reduce the value from its current level to<br />
zero. Let w =max{x 1 ,...,x n ,u 1 ,...,u n }. Then, for t ≥ w,<br />
S n (t) =e (t/w)lns∗ =(s ∗ ) t/w , where s ∗ =<br />
kY<br />
i=1<br />
µ <br />
ri − s i<br />
.<br />
r i<br />
There is an alternative method <strong>of</strong> deriving the values <strong>of</strong> s j <strong>and</strong> r j that is more suitable for Excel<br />
spreadsheet work. 2 Thestepsareasfollows:<br />
1. There should be one row for each data point. The points need not be in any particular order.<br />
2. Each row should have three entries. The first entry should be d i , the second entry should be<br />
u i (the cell should be empty if this point was not censored), <strong>and</strong> the third entry should be<br />
x i (the cell should be empty if this point was censored). Assume, for example that the d i s<br />
occupy cells B6:B45, the u i s occupy C6:C45, <strong>and</strong> the x i s occupy D6:D45. Place the letter x<br />
on top <strong>of</strong> this column (in D5).<br />
3. Highlight the column <strong>of</strong> x values (D5:D45) including the cell with the label.<br />
4. Select Data | Filter | AdvancedFilter<strong>and</strong>theninthedialogboxchecktheaction“Copy<br />
to another location.” The list range should already be D5:D45, the criteria range should<br />
be blank, the new location should be an unoccupied cell (say G15), <strong>and</strong> “Unique records<br />
only” should be checked. The result should be the set <strong>of</strong> y values in cells G15:Gxx where xx<br />
indicates the last cell needed.<br />
5. Select the cells with y values (G16:Gxx). Sort the y values in ascending order by selecting<br />
Data | Sort from the menu.<br />
6. Equation (2.1) is then used to calculate the r values. For example, in cell H16, it would be<br />
calculated as<br />
= COUNTIF(B$6:B$45,”
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