Estimation, Evaluation, and Selection of Actuarial Models

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18 CHAPTER 2. MODEL ESTIMATION Exercise 6 Determine the Kaplan-Meier and Nelson-Åalen estimates of the distribution function of the amount of a workers compensation loss. First do this using the raw data from Data Set B. Then repeat the exercise, modifying the data by left truncation at 100 and right censoring at 1000. It is important to note that when the data in the above exercise are truncated, the resulting distribution function is the distribution function for payments given that they are above 100. Empirically, there is no information about observations below 100 and thus there can be no information for that range. Similarly, there is no information concerning losses above 1000, but we do have an estimate of the proportion of losses above that amount. It should be noted that all the notation and formulas in this Subsection are consistent with those in Subsection 2.2.2. If it turns out that there was no censoring or truncation, using the formulas in this Subsection will lead to the same results as when using the empirical formulas in Subsection 2.2.2. Exercise 7 (*) You are given the following times of first claim for five randomly selected auto insurance policies — 1, 2, 3, 4, 5. You are later told that one of the five times given is actually the time of policy lapse, but you are not told which one. The smallest product-limit estimate of S(4), the probability that the first claim occurs after time 4, would result if which of the given times arose from the lapsed policy? Exercise 8 (*) For a mortality study with right censored data, you are given: Time Number of deaths Number at risk t j s j r j 5 2 15 7 1 12 10 1 10 12 2 6 Calculate the estimate of the survival function at time 12 using the Nelson-Åalen estimate. Exercise 9 (*) 300 mice were observed at birth. An additional 20 mice were first observed at age 2 (days) and 30 more were first observed at age 4. There were 6 deaths at age 1, 10 at age 3, 10 at age 4, a at age 5, b at age 9, and 6 at age 12. In addition, 45 mice were lost to observation at age 7, 35 at age 10, and 15 at age 13. The following product-limit estimates were obtained: S 350 (7) = 0.892 and S 350 (13) = 0.856. Determine the values of a and b. Exercise 10 (*) n lives were observed from birth. None were censored and no two lives died at the same age. At the time of the ninth death, the Nelson-Åalen estimate of the cumulative hazard rate is 0.511 and at the time of the tenth death it is 0.588. Estimate the value of the survival function at the time of the third death. Exercise 11 (*) All members of a study joined at birth, however, some may leave the study by means other than death. At the time of the third death, there was 1 death (that is, s 3 =1); at the time of the fourth death there were 2 deaths; and at the time of the fifth death there was 1 death. The following product-limit estimates were obtained: S n (y 3 )=0.72, S n (y 4 )=0.60, andS n (y 5 )=0.50. Determine the number of censored observations between times y 4 and y 5 . Assume no observations were censored at the death times.
2.2. ESTIMATION USING DATA-DEPENDENT DISTRIBUTIONS 19 2.2.5 Kernel smoothing One problem with empirical distributions is that they are always discrete. If it is known that the true distribution is continuous, the empirical distribution may be viewed as a poor approximation. In this Subsection, a method of obtaining a smooth, empirical-like distribution, is introduced. Recall from Definition 2.4 on page 3 that the idea is to replace each discrete piece of probability by a continuous random variable. While not necessary, it is customary that the continuous variable have a mean equal to the value of the point it replaces. This ensures that the kernel estimate has the same mean as the empirical estimate. One way to think about such a model is that it produces the final observed value in two steps. The first step is to draw a value at random from the empirical distribution. The second step is to draw a value at random from a continuous distribution whose mean is equal to the value drawn at the first step. The selected continuous distribution is called the kernel. For notation, let p(y j ) be the probability assigned to the value y j (j =1,...,k) by the empirical distribution. Let K y (x) be a distribution function for a continuous distribution such that its mean is y. Letk y (x) be the corresponding density function. Definition 2.19 A kernel density estimator of a distribution function is ˆF (x) = and the estimator of the density function is ˆf(x) = kX p(y j )K yj (x) j=1 kX p(y j )k yj (x). The function k y (x) is called the kernel. In this Note, three kernels will be used. Definition 2.20 The uniform kernel is given by ⎧ ⎧ ⎨ 0, x < y − b ⎨ 0, x < y − b 1 k y (x) = ⎩ 2b , y − b ≤ x ≤ y + b x−y+b , K y (x) = ⎩ 2b , y − b ≤ x ≤ y + b 0, x > y + b 1, x > y + b. The triangular kernel is given by ⎧ 0, x < y − b ⎪⎨ x−y+b , y − b ≤ x ≤ y k y (x) = b y+b−x 2 , y ≤ x ≤ y + b ⎪⎩ b 2 0, x > y + b j=1 ⎧ ⎪⎨ , K y (x) = ⎪⎩ 0, x < y − b (x−y+b) 2 , 2b 2 y − b ≤ x ≤ y 1 − (y+b−x)2 2b , y ≤ x ≤ y + b 2 1, x > y + b. The gamma kernel is given by letting the kernel have a gamma distribution with shape parameter α and scale parameter y/α. That is, k y (x) = xα−1 e −xα/y (y/α) α Γ(α) . Note that the gamma distribution has a mean of α(y/α) =y andavarianceofα(y/α) 2 = y 2 /α.
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Estimation, Evaluation, and Selection of Actuarial Models

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