The post presents basic practice problems for the topic of parametric model selection, focusing on maximum likelihood estimation. The practice problems are to reinforce the concepts discussed in two posts – this one and this one. The first post shows how to obtain maximum likelihood estimates given complete data (individual data). The second post focuses on maximum likelihood estimation for other data scenarios (grouped data, censored data and truncated data).
More maximum likelihood practice problems are found in the next problem set.
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Practice Problem 2A 
The following losses are recorded for a group of insureds:
An exponential distribution with mean is fitted to the loss data.

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Practice Problem 2B 
A sample of size 5 produced the values 332, 42, 94, 6, 9533. You fit a lognormal distribution with parameters and using maximum likelihood estimation.

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Practice Problem 2C 
The claim size follows a Weibull distribution with parameters and . The following claim experience is recorded: 15, 5, 9, 10, 11, 20.

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Practice Problem 2D 
Five claims have been observed: 11, 13, 9, 8, 10. The claim distribution is known to be a gamma distribution with shape parameter and scale parameter .

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Practice Problem 2E 
A claim size distribution is a Pareto distribution with parameters and . A sample of 10 claims is observed: 20, 61, 110, 8, 23, 3, 27, 7, 35, 9. These observed claims are before the application of a deductible.

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Practice Problem 2F 
The distribution of claim size is an inverse exponential distribution with parameter . Eight claims are observed: 55, 8, 23, 22, 59, 64, 106, 25.

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Practice Problem 2G  
A total of 40 claims have been observed for a loss distribution that is known to be an exponential distribution with mean . The data is summarized in the table below.
Determine the maximum likelihood estimate of . 
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Practice Problem 2H 
An insurance coverage with a policy limit of 30 is purchased to cover a random loss. If the loss exceeds 30, the coverage will pay 30. Otherwise, the coverage pays for the loss in full. The reported losses are: 19, 30*, 12, 30*, 30*, 4, 1, 19, 30, 15. The loss distribution is known to be an exponential distribution with mean . Determine the maximum likelihood estimate of . 
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Practice Problem 2I 
An insurance coverage has a deductible of 20. The following losses are part of a data set that has been truncated at 20:
The truncated claim data without modification is fitted to a Pareto distribution with parameter and .

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Practice Problem 2J 
An insurance coverage has a deductible of 20. The following losses are part of a data set that has been truncated at 20:
The truncated data is shifted by 20 and is fitted to a Pareto distribution with parameter and .

Problem  Answer 

2A 

2B 

2C 

2D 

2E 

2F 

2G 

2H 

2I 

2J 

actuarial practice problems
Dan Ma actuarial
Daniel Ma actuarial
Daniel Ma Math
Daniel Ma Mathematics
Actuarial exam
2018 – Dan Ma
Tagged: Maximum Likelihood Estimation, Maximum Likelihood Estimators
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