2. (Cox model) In the Bonus-Malus system, car-insurance premiums are adjusted if the customer has ever made a claim in the past. The increase in the premium may affect the cancellation of the insurance contract. The data below records the lifespan of insurance contracts until their cancellation is observed. Claim 1 if there is a claim made; Cancellation = 1 if the contract is cancelled. Lifespan Cancellation ID Claim 1 1 112 1 2 99 0 3 108 0 4 1 100 1 95 1 0 111 0 Table 1: Data for Question 2 Consider the Cox proportional model λ(t|x) = λo(t)ex for given covariate x. We want to study the effect of a claim on the cancellation of the insurance contract. 1 (a) Explain what are the consequences of the model. (b) Write down the partial likelihood based on the dataset given in Table 1. (c) Derive the score function and observed Fisher information for ẞ. (d) Calculate the maximum partial likelihood estimator ẞ of the effect of the claim to the cancellation of the insurance contract. (e) Determine the estimated variance of the estimator B. (f) State appropriate null and alternative hypotheses, Ho and H₁, to test the significance of the effect of a claim on the cancellation of the insurance contract. (g) State your conclusions about the null hypothesis at a significance level of a = 5%, using both the likelihood ratio statistic and the Z-score.
2. (Cox model) In the Bonus-Malus system, car-insurance premiums are adjusted if the customer has ever made a claim in the past. The increase in the premium may affect the cancellation of the insurance contract. The data below records the lifespan of insurance contracts until their cancellation is observed. Claim 1 if there is a claim made; Cancellation = 1 if the contract is cancelled. Lifespan Cancellation ID Claim 1 1 112 1 2 99 0 3 108 0 4 1 100 1 95 1 0 111 0 Table 1: Data for Question 2 Consider the Cox proportional model λ(t|x) = λo(t)ex for given covariate x. We want to study the effect of a claim on the cancellation of the insurance contract. 1 (a) Explain what are the consequences of the model. (b) Write down the partial likelihood based on the dataset given in Table 1. (c) Derive the score function and observed Fisher information for ẞ. (d) Calculate the maximum partial likelihood estimator ẞ of the effect of the claim to the cancellation of the insurance contract. (e) Determine the estimated variance of the estimator B. (f) State appropriate null and alternative hypotheses, Ho and H₁, to test the significance of the effect of a claim on the cancellation of the insurance contract. (g) State your conclusions about the null hypothesis at a significance level of a = 5%, using both the likelihood ratio statistic and the Z-score.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.3: Geometric Sequences
Problem 80E
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Please assist with the following questions with the answers.
Transcribed Image Text:2. (Cox model)
In the Bonus-Malus system, car-insurance premiums are adjusted if the customer has ever
made a claim in the past. The increase in the premium may affect the cancellation of the
insurance contract. The data below records the lifespan of insurance contracts until their
cancellation is observed.
Claim 1 if there is a claim made; Cancellation = 1 if the contract is cancelled.
Lifespan Cancellation
ID
Claim
1
1
112
1
2
99
0
3
108
0
4
1
100
1
95
1
0
111
0
Table 1: Data for Question 2
Consider the Cox proportional model λ(t|x) = λo(t)ex for given covariate x.
We want to study the effect of a claim on the cancellation of the insurance contract.
1
(a) Explain what are the consequences of the model.
(b) Write down the partial likelihood based on the dataset given in Table 1.
(c) Derive the score function and observed Fisher information for ẞ.
(d) Calculate the maximum partial likelihood estimator ẞ of the effect of the claim to
the cancellation of the insurance contract.
(e) Determine the estimated variance of the estimator B.
(f) State appropriate null and alternative hypotheses, Ho and H₁, to test the significance
of the effect of a claim on the cancellation of the insurance contract.
(g) State your conclusions about the null hypothesis at a significance level of a = 5%,
using both the likelihood ratio statistic and the Z-score.
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