Estimation, Evaluation, and Selection of Actuarial Models
Estimation, Evaluation, and Selection of Actuarial Models Estimation, Evaluation, and Selection of Actuarial Models
16 CHAPTER 2. MODEL ESTIMATION 8. Calculate the first S(y) value. For this example, cell J16 becomes = (H16 − I16)/H16. 9. The remaining values of the survival function can be calculated in column J. For cell J17, the formula is = J16*(H17 − I17)/H17 which can be copied into the remaining cells. Example 2.17 Determine the Kaplan-Meier estimate for Data Set D2. Based on the previous example, we have ⎧ 1, 0 ≤ t
2.2. ESTIMATION USING DATA-DEPENDENT DISTRIBUTIONS 17 Integrating both sides yields Z t Z ds(u) t 0 r(u) = h(t)dt = H(t). 0 Now replace the true expected count s(t) by ŝ(t), the observed number of deaths by time t. Itisa step function, increasing by s i at each death time. Therefore, the left-hand side becomes X s i r i which defines the estimator, Ĥ(t). The Nelson-Åalen estimator is ⎧ ⎪⎨ 0, 0 ≤ t
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16 CHAPTER 2. MODEL ESTIMATION<br />
8. Calculate the first S(y) value. For this example, cell J16 becomes<br />
= (H16 − I16)/H16.<br />
9. The remaining values <strong>of</strong> the survival function can be calculated in column J. For cell J17, the<br />
formula is<br />
= J16*(H17 − I17)/H17<br />
which can be copied into the remaining cells.<br />
Example 2.17 Determine the Kaplan-Meier estimate for Data Set D2.<br />
Based on the previous example, we have<br />
⎧<br />
1, 0 ≤ t
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