Estimation, Evaluation, and Selection of Actuarial Models

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54 CHAPTER 3. SAMPLING PROPERTIES OF ESTIMATORS Exercise 63 (*) Observations can be censored, but there is no truncation. Let y i and y i+1 be consecutive death ages. A 95% linear confidence interval for H(y i ) using the Nelson-Åalen estimator is (0.07125,0.22875) while a similar interval for H(y i+1 ) is (0.15607,0.38635). Determine s i+1 . Exercise 64 (*) A mortality study is conducted on 50 lives, all observed from age 0. At age 15 there were 2 deaths; at age 17 there were 3 censored observations; at age 25 there were 4 deaths; at age 30 there were c censored observations; at age 32 there were 8 deaths; and at age 40 there were 2deaths.LetS be the product-limit estimate of S(35) and let V be the Greenwood estimate of this estimator’s variance. You are given V/S 2 =0.011467. Determine the value of c. Exercise 65 (*) Fifteen cancer patients were observed from the time of diagnosis until the earlier of death or 36 months from diagnosis. Deaths occurred as follows: At 15 months there were 2 deaths; at 20 months there were 3 deaths; at 24 months there were 2 deaths; at 30 months there were d deaths; at 34 months there were 2 deaths; and at 36 months there was 1 death. The Nelson-Åalen estimate of H(35) is 1.5641. Determine the variance of this estimator. Exercise 66 (*) You are given the following values: y j r j s j 1 100 15 8 65 20 17 40 13 25 31 31 Determine the standard deviation of the Nelson-Åalen estimator of the cumulative hazard function at time 20. 3.3.3 Information matrix and the delta method In general, it is not easy to determine the variance of complicated estimators such as the maximum likelihood estimator. However, it is possible to approximate the variance. The key is a theorem that can be found in most mathematical statistics books. The particular version stated here and its multi-parameter generalization is taken from Rohatgi (An Introduction to Probability Theory and Mathematical Statistics, Wiley, 1976) and stated without proof. Recall that L(θ) is the likelihood function and l(θ) its logarithm. All of the results assume that the population has a distribution that is a member of the chosen parametric family and that there is a single parameter of interest. Theorem 3.29 Assume that the pdf (pf in the discrete case) f(x; θ) satisfies the following (for θ in an interval containing the true value, and replace integrals by sums for discrete variables): (i) ln f(x; θ) is three times differentiable with respect to θ. (ii) R ∂ f(x; θ) dx =0. This allows the derivative to be taken outside the integral and so we are ∂θ just differentiating the constant 1. 5 (iii) R ∂ 2 2 f(x; θ) dx =0. This is the same concept for the second derivative. ∂θ 5 The integrals in (ii) and (iii) are to be evaluated over the range of x values for which f(x; θ) > 0.
3.3. VARIANCE AND CONFIDENCE INTERVALS 55 (iv) −∞ < R f(x; θ) ∂2 2 ln f(x; θ) dx < 0. This establishes that the indicated integral exists and ∂θ that the location where the derivative is zero is a maximum. (v) There exists a function H(x) such that R H(x)f(x; θ) dx < ∞ with ∂ 3 ¯ ∂θ 3 ln f(x; θ)¯¯¯¯
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