Practice Problem Set 3 – maximum likelihood estimation

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

This problem set and the previous one present basic practice problems 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).

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Practice Problem 3-A

The following information is given about a sample of 5 observations.

  • The observations are drawn from a Weibull distribution with parameters \tau=2.5 and \theta.
  • Three of the observations are 13, 25 and 36.
  • The only thing known about the remaining two observations is that they exceed 40.

Determine the following:

  • The maximum likelihood estimate of the parameter \theta.
  • The median of the fitted distribution.

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Practice Problem 3-B

A random sample of 10 losses are given below:

    4, 60, 274, 95, 56, 121, 26, 228, 49, 56

The distribution that models the losses is known to be an exponential distribution with mean \theta.

  • Determine the maximum likelihood estimate of the parameter \theta.
  • A policy that covers these losses has a deductible of 50 and a maximum covered loss of 500 (the maximum payment per loss is 450). Determine the expected amount paid per loss under this policy.

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Practice Problem 3-C

In a study of patients with cardiovascular disease, 5 patients are observed for a period 5 years. Three of the patients die during the study with their times of death at 1, 1, 3. The remaining two patients survive to the end of the study. The time until death is modeled by a distribution with the following cumulative distribution function:

    \displaystyle F(x)=1-e^{-x^2/ \theta^2} \ \ \ \ \ \ x>0

  • Use the method of maximum likelihood estimation to estimate the parameter \theta.
  • Determine the probability of observing a patient surviving to the end of the study period.

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Practice Problem 3-D

An insurance coverage has a deductible of 50. A sample of 7 losses is given: 65, 100, 150, 200, 350, 505 and 600. No information is known about losses below 50.

The losses are known to follow an exponential distribution with mean \theta.

  • Estimate the parameter \theta using maximum likelihood estimation.
  • Determine the expected amount paid per loss under this insurance coverage.
  • Determine the expected amount paid per payment under this insurance coverage.

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Practice Problem 3-E

An insurance coverage has a deductible of 50 and a maximum covered loss of 500. A sample of 7 losses is given: 65, 100, 150, 200, 350, 500 and 500. No information is known about losses below 50. The two data points of 500 are the result of censoring at 500.

The losses are known to follow an exponential distribution with mean \theta.

  • Estimate the parameter \theta using maximum likelihood estimation.
  • Determine the expected amount paid per loss under this insurance coverage.
  • Determine the expected amount paid per payment under this insurance coverage.

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Practice Problem 3-F

An insurance coverage has a deductible of 50 and a maximum covered loss of 500. A sample of 7 payments is given: 15, 50, 100, 150, 300, 450 and 450. The two data points of 450 are the result of censoring at 500 and then subtracting the deductible.

The payments are known to follow an exponential distribution with mean \theta.

  • Estimate the parameter \theta using maximum likelihood estimation.
  • Determine the expected amount paid per payment under this insurance coverage.
  • Determine the expected amount paid per loss under this insurance coverage.

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Practice Problem 3-G

An insurance coverage has a deductible of 50 and a maximum covered loss of 500. A sample of 7 losses is given: 55, 60, 100, 150, 250, 500 and 500. No information is known about losses below 50. The two data points of 500 are the result of censoring at 500.

The losses (including the losses below the deductible and the losses exceeding the limit) are known to follow a Pareto distribution with parameters \alpha and \theta=150.

  • Estimate the parameter \alpha using maximum likelihood estimation.
  • Determine the expected amount paid per loss under this insurance coverage.
  • Determine the expected amount paid per payment under this insurance coverage.

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Practice Problem 3-H

An insurance coverage has a deductible of 5. The following claims are observed:

    7, 10, 12, 16, 22

The above sample is the result of a truncation below at 5. A Weibull distribution with parameters \tau=2 and \theta is fitted to the loss distribution (including losses below the deductible).

  • Determine the maximum likelihood estimate of \theta.
  • Determine the fitted median for losses.
  • Determine the fitted median for submitted claims.

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Practice Problem 3-I

Two groups of insureds are pooled for maximum likelihood estimation. Losses for Group 1 has a Pareto distribution with parameters \alpha and \theta=500. Losses for Group 2 has a Pareto distribution with parameters \alpha and \theta=1000. The following losses have been observed:

    Group 1: 500, 585, 900

    Group 2: 875, 980, 1500

  • Determine the maximum likelihood estimate for the parameter \alpha.

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Practice Problem 3-J

Suppose that the lifetimes of a certain type of washing machines have a Weibull distribution with parameters \tau=3 and \theta. Seven such machines are tested during a 5-year period. Two of the machines fail before the end of the testing period. Their times at failure are 2, 3. The other 5 machines are in working condition at the end of the testing period.

  • Determine the maximum likelihood estimate of the parameter \theta.
  • Using the fitted distribution, determine the median lifetime of such washing machines.

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Problem Answer
3-A
  • \hat{\theta}=40.7233
  • Median of fitted distribution: 35.17
3-B
  • \hat{\theta}=96.9
  • \displaystyle E[X \wedge 500]-E[X \wedge 50]=57.28378 where \displaystyle E[X \wedge 500]=\hat{\theta} (1-e^{-500/ \hat{\theta}}) and \displaystyle E[X \wedge 50]=\hat{\theta} (1-e^{-50/ \hat{\theta}})
3-C
  • \hat{\theta}=4.50925
  • \displaystyle P[X > 5]=0.292436
3-D
  • \hat{\theta}=231.4285714
  • \displaystyle E[X]-E[X \wedge 50]=186.46096 where \displaystyle E[X]=\hat{\theta} and \displaystyle E[X \wedge 50]=\hat{\theta} (1-e^{-50/ \hat{\theta}})
  • \hat{\theta}=231.4285714
3-E
  • \hat{\theta}=303
  • \displaystyle E[X \wedge 500]-E[X \wedge 50]=198.7260 where \displaystyle E[X \wedge x]=\hat{\theta} (1-e^{-x/ \hat{\theta}})
  • \displaystyle \frac{E[X \wedge 500]-E[X \wedge 50]}{1-F(50)}=\hat{\theta}=234.38
3-F
    Same answers as in 3-E because exponential distribution is memoryless.
3-G
  • \hat{\alpha}=1.332427776
  • \displaystyle E[X \wedge 500]-E[X \wedge 50]=132.9344 where \displaystyle E[X \wedge x]=\frac{150}{\hat{\alpha}-1} \biggl[1-\biggl(\frac{150}{x+150} \biggr)^{\hat{\alpha}-1} \biggr]
  • \displaystyle \frac{E[X \wedge 500]-E[X \wedge 50]}{1-F(50)}=195.0334
3-H
  • \displaystyle \hat{\theta}=13.4759
  • Fitted median for losses = 11.2194
  • Fitted median for submitted claims = 12.2831
3-I
  • \hat{\alpha}=1.2697
3-J
  • \hat{\alpha}=5.8964
  • median: 5.2183

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One thought on “Practice Problem Set 3 – maximum likelihood estimation

  1. […] More maximum likelihood practice problems are found in the next problem set. […]

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