Practice Problem Set 2 – maximum likelihood estimation

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 2-A

The following losses are recorded for a group of insureds:

    19, 45, 12, 31, 32, 4, 1, 19, 30, 15

An exponential distribution with mean \theta is fitted to the loss data.

  • Determine the maximum likelihood estimate of the parameter \theta.
  • Suppose an insurance coverage with a deductible of 5 is to cover these losses. Determine the expected insurance payment per loss.
  • Suppose an insurance coverage with a deductible of 5 is to cover these losses. Determine the expected insurance payment per payment.

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Practice Problem 2-B

A sample of size 5 produced the values 332, 42, 94, 6, 9533. You fit a lognormal distribution with parameters \mu and \sigma using maximum likelihood estimation.

  • Determine the estimates of the lognormal parameters \mu and \sigma.
  • Use these estimates to determine the probability of observing a value exceeding 500.

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Practice Problem 2-C

The claim size follows a Weibull distribution with parameters \tau=2 and \theta. The following claim experience is recorded: 15, 5, 9, 10, 11, 20.

  • Use the method of maximum likelihood estimation to estimate the parameter \theta.
  • Determine the probability of observing a claim in excess of 15.

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Practice Problem 2-D

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

  • Estimate the parameter \theta using maximum likelihood estimation.
  • Determine the probability of observing a claim in excess of 10.2.

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Practice Problem 2-E

A claim size distribution is a Pareto distribution with parameters \alpha and \theta=100. 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.

  • Use maximum likelihood estimation to estimate the parameters \alpha.
  • According to the fitted Pareto distribution, determine the mean insurance payment per loss if the insurance coverage has no deductible.
  • According to the fitted Pareto distribution, determine the mean insurance payment per loss if the insurance coverage has a deductible of 20.

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Practice Problem 2-F

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

  • Estimate the parameters \theta using maximum likelihood.
  • Using the fitted distribution, determine the 85th percentile of the claim size.

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Practice Problem 2-G

A total of 40 claims have been observed for a loss distribution that is known to be an exponential distribution with mean \theta. The data is summarized in the table below.

Interval Frequency
(0, 40) 22
(40, 80) 7
(80, 120) 4
(120, \infty) 7
Total 40

Determine the maximum likelihood estimate of \theta.

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Practice Problem 2-H

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 \theta. Determine the maximum likelihood estimate of \theta.

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Practice Problem 2-I

An insurance coverage has a deductible of 20. The following losses are part of a data set that has been truncated at 20:

    25, 61, 110, 23, 27, 35.

The truncated claim data without modification is fitted to a Pareto distribution with parameter \alpha and \theta=100.

  • Determine the maximum likelihood estimate of \alpha.
  • Using the fitted distribution to determine the mean insurance payment per loss without a deductible.
  • Using the fitted distribution to determine the mean insurance payment per loss with respect to the deductible of 20.
  • Using the fitted distribution to determine the mean insurance payment over all losses exceeding the deductible of 20.

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Practice Problem 2-J
An insurance coverage has a deductible of 20. The following losses are part of a data set that has been truncated at 20:

    25, 61, 110, 23, 27, 35.

The truncated data is shifted by 20 and is fitted to a Pareto distribution with parameter \alpha and \theta=100.

  • Determine the maximum likelihood estimate of \alpha.
  • Using the fitted distribution to determine the mean insurance payment over all losses exceeding the deductible of 20. Compare this with the last part of Problem 2-I.

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Problem Answer
2-A
  • \hat{\theta}=20.8
  • \displaystyle E[X]-E[X \wedge 5]=\hat{\theta} \ e^{-5/\hat{\theta}}=16.3556
  • \displaystyle \frac{E[X]-E[X \wedge 5]}{1-F(5)}=\hat{\theta} =20.8
2-B
  • \hat{\mu}=5.008 and \hat{\sigma}=2.4523
  • \displaystyle P[X > 500]=0.3121
2-C
  • \hat{\theta}=12.5963
  • \displaystyle P[X > 15]=0.24218
2-D
  • \hat{\theta}=5.1
  • \displaystyle P[X > 10.2]=3 e^{-2}=0.4060
2-E
  • \hat{\alpha}=4.1546
  • \displaystyle E[X]-E[X \wedge 20]=31.70-13.8649=17.8351
2-F
  • \hat{\theta}=25.46775
  • 85th percdntile = 156.7064
2-G
  • \hat{\theta}=61.48288
2-H
  • \displaystyle \hat{\theta}=27.1429
2-I
  • \hat{\alpha}=5.48686
  • E[X]=22.287296
  • \displaystyle E[X]-E[X \wedge 20]=22.287296-12.45212=9.835176
  • \displaystyle \frac{E[X]-E[X \wedge 20]}{1-F(20)}=26.74475
2-J
  • \hat{\alpha}=4.7199
  • E[X]=26.8824

actuarial practice problems

Dan Ma actuarial

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3 thoughts on “Practice Problem Set 2 – maximum likelihood estimation

  1. […] This practice problem set has more exercises on maximum likelihood estimation, the continuation of Practice Problem Set 2. […]

  2. […] problems for maximum likelihood estimation for continuous distributions are found here and […]

  3. […] focus on continuous distributions – this one and this one. Two practice problem sets, Practice Problem Set 2 and Practice Problem Set 3, are to reinforce these two previous […]

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