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