Estimation, Evaluation, and Selection of Actuarial Models
Estimation, Evaluation, and Selection of Actuarial Models Estimation, Evaluation, and Selection of Actuarial Models
92 CHAPTER 5. MODELS WITH COVARIATES Summing such items and doing the same for the other partial derivatives yields the information matrix, and its inverse, the covariance matrix. · ¸ · ¸ 1171.054 5.976519 I = , Var= 5.976519 1.322283 d 0.000874 −0.003950 . −0.003950 0.774125 Therelativeriskistheratioofthec-values for the two cases. For a house of age x, therelative risk of wood versus brick is e z 1β 1 +β 2/e z 1 β 1 = e β 2. A 95% confidence interval for β2 is −0.91994 ± 1.96 √ 0.774125 or (−2.6444, 0.80455). Exponentiating the endpoints gives the confidence interval for the relative risk, (0.07105, 2.2357). ¤ Exercise 93 (*) A Cox proportional hazards model has z 1 =1for males and 0 for females and z 2 =1for adults and 0 for children. The maximum likelihood estimates of the coefficients are ˆβ 1 =0.25 and ˆβ 2 = −0.45. The covariance matrix of the estimators is · 0.36 0.10 0.10 0.20 Determine a 95% confidence interval for the relative risk of a male child subject compared to a female adult subject. Exercise 94 (*) Four insureds were observed from birth to death. The two from Class A died at times 1 and 9 while the two from Class B died at times 2 and 4. A proportional hazards model uses z 1 =1for Class B and 0 for Class A. Let b = ˆβ 1 . Estimate the cumulative hazard rate at time 3 for a member of Class A. Exercise 95 (*) A Cox proportional hazards model has three covariates. The life that died first has values 1,0,0 for z 1 ,z 2 ,z 3 . The second to die has values 0,1,0 and the third to die had values 0,0,1. Determine the partial likelihood function (as a function of β 1 , β 2 , and β 3 ). 5.3 The generalized linear model This model, while not a direct generalization of the previous two, does provide more flexibility. A comprehensive reference is Generalized Linear Models, 2nd ed. by P. McCullagh and J. Nelder (Chapman and Hall, 1989). Actuarial papers using this model include S. Mildenhall (“A Systematic relationship between minimum bias and generalized linear models,” PCAS, 1999, 393—487), F. Guiahi (“Fitting to loss distributions with emphasis on rating variables,” CAS Forum, Winter 2001, 133—174), K. Murphy, M. Brockman, and P. Lee (“Using generalized linear models to build dynamic pricing systems for personal lines insurance,” CAS Forum, Winter 2000, 107—140), and K. Holler, D. Sommer, and G. Trahair (“Something old, something new in classification ratemaking with a novel use of generalized linear models for credit insurance, CAS Forum, Winter 1999, 31—84). The definition of this model given below is even slightly more general than the usual one. Definition 5.10 Suppose a parametric distribution has parameters µ and θ where µ is the mean and θ is a vector of additional parameters. The mean must not depend on the additional parameters ¸ .
5.3. THE GENERALIZED LINEAR MODEL 93 and the additional parameters must not depend on the mean. Let z be a vector of covariates for an individual, let β be a vector of coefficients, and let η(µ) and c(y) be functions. The generalized linear model then states that the random variable, X, has as its distribution function where µ is such that η(µ) =c(β T z). F (x|θ, β) =F (x|µ, θ) The model indicates that the mean of an individual observation is related to the covariates through a particular set of functions. Normally, these functions do not involve parameters, but instead are used to provide a good fit or to ensure that only legitimate values of µ are encountered. Example 5.11 Demonstrate that the ordinary linear regression model is a special case of the generalized linear model. For ordinary linear regression, X has a normal distribution with µ = µ and θ = σ. Bothη and c are the identity function, resulting in µ = β T z. ¤ The model presented here is more general than the one usually used. Usually, only a few distributions are allowed to be used for X. The reason is that for these distributions, people have been able to develop the full set of regression tools, such as residual analysis. Because such analyses are covered in a different part of Part 4, that restriction is not needed here. For many of the distributions we have been using, the mean is not a parameter. However, it could be. For example, we could parameterize the Pareto distribution by setting µ = θ/(α − 1) or equivalently, replacing θ with µ(α − 1). The distribution function is now · µ(α − 1) ¸α F (x|µ, α) =1− , µ > 0, α > 1. µ(α − 1) + x Note the restriction on α in the parameter space.
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92 CHAPTER 5. MODELS WITH COVARIATES<br />
Summing such items <strong>and</strong> doing the same for the other partial derivatives yields the information<br />
matrix, <strong>and</strong> its inverse, the covariance matrix.<br />
· ¸ · ¸<br />
1171.054 5.976519<br />
I =<br />
, Var=<br />
5.976519 1.322283<br />
d 0.000874 −0.003950<br />
.<br />
−0.003950 0.774125<br />
Therelativeriskistheratio<strong>of</strong>thec-values for the two cases. For a house <strong>of</strong> age x, therelative<br />
risk <strong>of</strong> wood versus brick is e z 1β 1 +β 2/e z 1 β 1 = e<br />
β 2. A 95% confidence interval for β2 is −0.91994 ±<br />
1.96 √ 0.774125 or (−2.6444, 0.80455). Exponentiating the endpoints gives the confidence interval<br />
for the relative risk, (0.07105, 2.2357). ¤<br />
Exercise 93 (*) A Cox proportional hazards model has z 1 =1for males <strong>and</strong> 0 for females <strong>and</strong><br />
z 2 =1for adults <strong>and</strong> 0 for children. The maximum likelihood estimates <strong>of</strong> the coefficients are<br />
ˆβ 1 =0.25 <strong>and</strong> ˆβ 2 = −0.45. The covariance matrix <strong>of</strong> the estimators is<br />
· 0.36 0.10<br />
0.10 0.20<br />
Determine a 95% confidence interval for the relative risk <strong>of</strong> a male child subject compared to a<br />
female adult subject.<br />
Exercise 94 (*) Four insureds were observed from birth to death. The two from Class A died at<br />
times 1 <strong>and</strong> 9 while the two from Class B died at times 2 <strong>and</strong> 4. A proportional hazards model uses<br />
z 1 =1for Class B <strong>and</strong> 0 for Class A. Let b = ˆβ 1 . Estimate the cumulative hazard rate at time 3<br />
for a member <strong>of</strong> Class A.<br />
Exercise 95 (*) A Cox proportional hazards model has three covariates. The life that died first<br />
has values 1,0,0 for z 1 ,z 2 ,z 3 . The second to die has values 0,1,0 <strong>and</strong> the third to die had values<br />
0,0,1. Determine the partial likelihood function (as a function <strong>of</strong> β 1 , β 2 , <strong>and</strong> β 3 ).<br />
5.3 The generalized linear model<br />
This model, while not a direct generalization <strong>of</strong> the previous two, does provide more flexibility.<br />
A comprehensive reference is Generalized Linear <strong>Models</strong>, 2nd ed. by P. McCullagh <strong>and</strong> J. Nelder<br />
(Chapman <strong>and</strong> Hall, 1989). <strong>Actuarial</strong> papers using this model include S. Mildenhall (“A Systematic<br />
relationship between minimum bias <strong>and</strong> generalized linear models,” PCAS, 1999, 393—487), F.<br />
Guiahi (“Fitting to loss distributions with emphasis on rating variables,” CAS Forum, Winter<br />
2001, 133—174), K. Murphy, M. Brockman, <strong>and</strong> P. Lee (“Using generalized linear models to build<br />
dynamic pricing systems for personal lines insurance,” CAS Forum, Winter 2000, 107—140), <strong>and</strong> K.<br />
Holler, D. Sommer, <strong>and</strong> G. Trahair (“Something old, something new in classification ratemaking<br />
with a novel use <strong>of</strong> generalized linear models for credit insurance, CAS Forum, Winter 1999, 31—84).<br />
The definition <strong>of</strong> this model given below is even slightly more general than the usual one.<br />
Definition 5.10 Suppose a parametric distribution has parameters µ <strong>and</strong> θ where µ is the mean<br />
<strong>and</strong> θ is a vector <strong>of</strong> additional parameters. The mean must not depend on the additional parameters<br />
¸<br />
.
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