Practice Problem Set 1 – method of moments estimation

The post presents basic practice problems for the topic of parametric model selection, in particular estimating distributional parameters using the method of moments.

The task at hand is that of estimating an unknown parameter (or parameters) of a distribution. A relatively easy and conceptually clear approach is the method of moments. The approach is to set the sample moments equal to distributional moments and solve for the parameters in the resulting equations. For example, if the distribution has a single parameter \theta, then set the sample mean equal to the expression in terms of \theta (the mean of the distribution). If there are two parameters, then set the sample mean and sample second moment equal to the appropriate expressions for the first and second moments of the distribution. The method of moments estimates are the solutions of the two equations.

The method of moments estimates are not unique. The same sample data do not always produce the same estimates. For a 2-parameter estimation, we can obtain the estimates by setting the first two sample moments equal to the first two distributional moments. We can equate the third and fourth moments. But by doing so, we will obtain different numerical estimates. We take the natural approach of equating the first k moments if there are k parameters. For some distributions whose mean and variance do not exist, we may have to equate the -1 or -2 moments (e.g. inverse exponential distribution).

The topic of method of moments was covered in the old Exam C but is no longer part of the syllabus of the short-term actuarial math (STAM). However, the method of maximum likelihood estimation (MLE) is still in the STAM syllabus. The method of moments is a great contrast to MLE. For certain distributions (e.g. binomial, Poisson, negative binomial, exponential, gamma and normal), the MLE estimates are identical to the method of moments estimates. It will be advantageous to know the method of moments for these distributions since the method of moments is, in many cases, easier to compute. The method of moments requires the use of moments of various probability distributions. Thus it is also a great review of basic information – moments of various distributions and other distributional quantities in additional to being background information for MLE.

The method of moments is also discussed here in a companion site.

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Here’s basic information on lognormal distribution.

Here’s basic information on Pareto distribution.

Here’s basic information on gamma distribution.

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Practice Problem 1-A
An examination of 1,000 insurance claims produces the following summary.

    \displaystyle \sum \limits_{i=1}^{1000} x_i=5476.51

    \displaystyle \sum \limits_{i=1}^{1000} x_i^2=126450.53

A Pareto distribution is fitted to the data using the method of moments.

  • Determine the estimates of the Pareto parameters \alpha and \theta.
  • Calculate the probability that a claim exceeds 5 using the fitted distribution.
  • .

    Practice Problem 1-B

    A sample of size 5 produced the values 1.76, 39.37, 5.81, 7.49 and 0.92. You fit a lognormal distribution using the method of moments.

  • Determine the estimates of the lognoemal parameters \mu and \sigma.
  • Use these estimates to estimate the probability of observing a value exceeding 2.5.
  • .

    Practice Problem 1-C

    The claim size follows a gamma distribution with shape parameters \alpha=2 and scale parameter \theta. A random sample of 8 claims are obtained and their values are as follows:

      5.72, 12.75, 14.51, 8.65, 7.41, 12.55, 9.44, 4.86
  • Use the method of moments to estimate the scale parameter \theta.
  • Determine the mean and variance of the fitted gamma distribution.
  • .

    Practice Problem 1-D

    In the current year, there are 500 claims with a total amount of 475,000 from a large pool of policyholders. Claim size is subject to an increase of 10% from the previous year due to inflation.

    A gamma distribution with shape parameter \alpha=2 and unknown scale parameter \theta is used to model the claim size distribution.

  • Use the method of moments to estimate the scale parameter \theta for the next year.
  • Determine the probability of observing a claim next year in excess of 1045.
  • .

    Practice Problem 1-E

    A claim size distribution is a mixture of two exponential distributions, one with mean \theta (weight 75%) and one with mean \tau (weight 25%).

    A sample of 20 claims is observed and is summarized below.

      \displaystyle \sum \limits_{i=1}^{20} x_i=550

      \displaystyle \sum \limits_{i=1}^{20} x_i^2=99370

  • Use the method of moments to estimate the parameters \theta and \tau.
  • Determine the probability of observing a claim in excess of 10.
  • .

    Practice Problem 1-F

    The distribution of claim size is a gamma distribution with parameters \alpha and \theta (scale). The following is a random sample of 6 claims.

      33, 29, 21, 54, 12, 3

    Use the method of moments to estimate the parameters \alpha and \theta.

    .

    Practice Problem 1-G

    The following is a random sample of losses.

      2, 4, 3, 6, 50, 4, 7, 1

    The losses X are assumed to follow a Pareto distribution with parameters \alpha and \theta.

  • Use the method of moments to estimate the parameters \alpha and \theta.
  • Determine E[X \wedge 50], the limited expected value at 50.
  • .

    Practice Problem 1-H

    Observations x_1,x_2,\cdots,x_n are made about a certain distribution. The following is the summary.

      \displaystyle \frac{1}{n} \sum \limits_{i=1}^{n} x_i=0.825

      \displaystyle \frac{1}{n} \sum \limits_{i=1}^{n} x_i^2=0.720

    The beta distribution with the following density function

      \displaystyle f(x)=\frac{\Gamma(a+b)}{\Gamma(a) \ \Gamma(b)} \ x^{a-1} \ (1-x)^{b-1} \ \ \ \ \ \ 0<x<1

    is fitted to the observed data.

    Use the method of moments to estimate the parameters a and b.

    .

    Practice Problem 1-I

    You are given the following claim data for a large group of insurance policyholders.

    Year # of Claims Total Claim Amount
    1 200 35,000
    2 250 48,000

    Sizes of claims from this group of policyholders are subject to an annual inflation of 10%. Claim size is to be modeled by a Pareto distribution with parameters \alpha=3.5 and \theta.

    Use the method of moments to estimate the parameter \theta in year 3.

    .

    Practice Problem 1-J

    The following 5 claims are sampled from a lognormal distribution with parameters \mu and \sigma.

      5, 43, 8, 11, 3

  • Use the method of moments to estimate the parameters \mu and \sigma.
  • Determine the 80th percentile of the fitted lognormal distribution.
  • \text{ }

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    Problem Answer
    1-A
    • \hat{\alpha}=2.9025 and \hat{\theta}=10.4189
    • \displaystyle P[X > 5]=0.321
    1-B
    • \hat{\mu}=1.9108 and \hat{\sigma}=0.9934
    • \displaystyle P[X > 2.5]=0.8413
    1-C
    • \hat{\theta}=4.743125
    • E[X]=9.48625 and Var[X]=44.99446953
    1-D
    • \hat{\theta}=522.5
    • \displaystyle P[X > 1045]=3 e^{-2}=0.4060
    1-E
    • \hat{\theta}=3.5 and \hat{\tau}=99.5
    • \displaystyle P[X > 10]=0.26917
    1-F
    • \hat{\alpha}=2.4228 and \hat{\theta}=10.4561
    1-G
    • \hat{\alpha}=3.2903 and \hat{\theta}=22.0443
    • E[X \wedge 50]=8.9861
    1-H
    • \displaystyle \hat{a}=\frac{11}{5}=2.2 and \displaystyle \hat{b}=\frac{7}{15}=0.4667
    1-I
    • \displaystyle \hat{\theta}=528.6111
    1-J
    • \displaystyle \hat{\mu}=2.2657 and \displaystyle \hat{\sigma}=0.8642
    • 80th percentile = e^{2.991628}=19.918

    actuarial practice problems

    Dan Ma actuarial

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    \copyright 2018 – Dan Ma

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    2 thoughts on “Practice Problem Set 1 – method of moments estimation

    1. […] the method of moments, which is relative easy to use (for the most parts). This is the focus of the practice problem set. In this post, we discuss the method of maximum likelihood […]

    2. […] problems involving the method of moments are found here in a companion […]

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