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Here are a set of exercises that guide the viewer through some of the theoretical foundations of Loss Data Analytics. Each tutorial is based on one or more questions from the professional actuarial examinations – typically the Society of Actuaries Exam C.
Tutorial Structure. Each guided tutorial has a strategy set that describes the context. When you hit the “Start quiz” button, you begin the tutorial that is comprised of a series of mini-questions designed to lead you to the target question. At each stage, hints are provided as well as feedback on the correct solution of each mini-question.
Your Assignment. In reviewing these exercises, ideally the viewer will:
- Work the problem posed referring only to basic theory
- Even if you get the answer correct, review the strategy for this type of problem by clicking (revealing) the Strategy for … header
- If you feel comfortable with the strategy and got the problem correct, then you may choose to move on. However, you might also decide to follow the step-by-step process for solving the problem by clicking on the “Start Quiz” button. It is not really a quiz — it is a guided tutorial.
Strategy for Mixture Problems
In some situations, we only observe a single outcome but can conceptualize an outcome as resulting from a two (or more) stage process. These are called two-stage, or “hierarchical,” type situations. Some special cases include:
- problems where the parameters of the distribution are random variables,
- mixture problems, where stage 1 represents the type of sub-population and stage 2 represents a random variable with a distribution that depends on population type
- an aggregate distribution, where stage 1 represents the number of events and stage two represents the amount per event.
To apply these rules,
- Identify the random variable that is being conditioned upon, typically a stage 1 outcome (that is not observed).
- Conditional on the stage 1 outcome, calculate summary measures such as a mean, variance, and the like.
- There are several results of the step (ii), one for each stage 1 outcome. Then, combine these results using the iterated expectations, law of total probability or the total variation rules.
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Two-Point Mixture 169
The distribution of a loss,
X, is a two-point mixture:
(i) With probability 0.8,
X has a two-parameter Pareto distribution with (alpha=2) and (theta=100).
(ii) With probability 0.2,
X has a two-parameter Pareto distribution with with (alpha=4) and (theta=3000).
Calculate Pr((Xleq200).)
[WpProQuiz 60]
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Strategy for Obtaining Maximum Likelihood Estimators for Severity Problems
In maximum likelihood estimation, the objective is to determine parameter estimates that maximize the (log-) likelihood of the observed values. To do this, you will need a general expression for the likelihood function. When observations are independent, the likelihood is the product of probabilities of individual observations. You can do the following:
- Using the problem information, identify the probability function (either the mass, density, or a hybrid) for a single observation.
- Take the logarithm of the probability function and sum over all observations to get the log-likelihood for the data set.
- Differentiate the log-likelihood.
- Determine the maximum likelihood estimates by setting the first derivative of the log-likelihood equal to zero and solving for the parameter estimates.
Pareto Likelihood 37
A random sample of three claims from a dental insurance plan is given below:
$$
{scriptsize
begin{matrix}
begin{array}{ccc}
225 & 525 & 950 \
end{array}
end{matrix}
}
$$Claims are assumed to follow a Pareto distribution with parameters (theta=150) and (alpha).
Calculate the maximum likelihood estimate of (alpha).
[WpProQuiz 62]
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Strategy for Obtaining Maximum Likelihood Estimators for Mixture Problems
In mixture problems, claims are observed from different sub-populations, usually the parameters of the probability distributions are conditional on the sub-population.For maximum likelihood estimation for mixture problems, the objective is to determine parameter estimates that maximize the (log-) likelihood of the observed values in the mixture population. To do this, you will need a general expression for the (log-) likelihood function, then do the following:
- Using the problem information, identify the probability function (either the mass, density, or a hybrid) for a single observation in each sub-population.
- Take the logarithm of the probability function for each sub-population and sum over all observations to get the log-likelihood for the sub-population. Sum the log-likelihood across all the sub-populations to get the log-likelihood for the mixture population
- Differentiate the log-likelihood.
- Determine the maximum likelihood estimates by setting the first derivative of the log-likelihood equal to zero and solving for the parameter estimates.
Mixture Distribution Likelihood Estimation 26
You are given:
(i) Low-hazard risks have an exponential claim size distribution with mean (theta).
(ii) Medium-hazard risks have an exponential claim size distribution with mean (2theta).
(iii) High-hazard risks have an exponential claim size distribution with mean (3theta).
(iv) No claims from low-hazard risks are observed.
(v) Three claims from medium-hazard risks are observed, of sizes 1, 2 and 3.
(vi) One claim from a high-hazard risk is observed, of size 15.
Calculate the maximum likelihood estimate of (theta)
[WpProQuiz 63]
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Strategy for Likelihood Ratio Test Problems
The Likelihood Ratio Test is used to compare two models where one model is a subset of the other. Generally, the model under the null hypothesis ((H_0)) is the subset of the model under the alternative hypothesis ((H_a)). Approach this problem using the following steps.
- Identify the model under the null and the alternative hypotheses. This is typically done by specifying parameter values.
- Compute the likelihood ratio test LRT statistic. To do that, determine (L(theta_0)), the log-likelihood under the null hypothesis and (L(theta_a)), the log-likelihood under the alternative. Use these to calculate
$$LRT=2[L(theta_a)-L(theta_0)] .$$
- The reference distribution is a chi-square (chi^2) distribution. The parameter, known as the “degrees of freedom” equals the number of restrictions on (theta). For some applications, this is the difference in the number of fitted parameters of the two models
- Draw a conclusion using the following decision rules.
- Decision rule A. Use significance level, typically denoted as (alpha), and reject ((H_0)) in favor of ((H_a)) if the test statistic exceeds the ( 1- alpha ) quantile of the reference distribution. That is, reject if
$$LRTgt chi^2_{1-alpha} .$$
- Decision rule B. Use significance level of (alpha), reject ((H_0)) in favor of ((H_a)) if :
$$p-valuelt alpha .$$ Here, the probability (p-value) is defined to be Pr((chi^2gt LRT)).
Pareto Likelihood 22
You fit a Pareto distribution to a sample of 200 claim amounts and use the likelihood ratio test to test the hypothesis that (alpha=1.5 ) and (theta=7.8 )
You are given:
(i) The maximum likelihood estimates are (widehat{alpha}=1.4) and (widehat{theta}=7.6)
(ii) The natural logarithm of the likelihood function evaluated at the maximum likelihood estimates is (-817.92)
(iii) (sumln(x_i +7.8)=607.64)
Determine the result of the test.
[WpProQuiz 61]
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Strategy for Loss Elimination Ratio Problems
Any deductible d imposed on an insurance policy reduces the insurer’s payment. The loss elimination ratio LER is the proportional decrease in the expected payment of the insurer as a result of imposing the deductible.For LER problems, you can do the following:
- Write down the general formulae for loss elimination ratio :
$$LER=frac{E(X)-E(Y^L)}{E(X)} .$$ Because ( E(Xwedge d)=Emin(X,d)=E(X)-E(Y^L)), a simpler formula for LER is
$$LER=frac{E(Xwedge d)}{E(X)}.$$
- From the problem, identify the distribution of losses and compute the mean (E(X)).
- Calculate the the limited expected value (E(Xwedge d)) using information from the question. Note that the actuarial Exam C tables contain expressions for (E(Xwedge d)) for some distributions. For distributions not provided in the tables, recall that
$$E(Xwedge d)=int_0^d x f_X(x) dx + int_d^{infty} d f_X(x) dx \
= int^d_0 (1-F(x)) dx . $$
Loss Elimination Ratio 89
You are given:
(i) Losses follow an exponential distribution with the same mean in all years.
(ii) The loss elimination ratio this year is 70%.
(iii) The ordinary deductible for the coming year is 4/3 of the current deductible.
Calculate the loss elimination ratio for the coming year.
[WpProQuiz 65]
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Strategy for Simulation using Inverse Transform Method Problems
To simulate a random variable X using the inverse transform method, you can do the following:
- From the question, identify the distribution of the random variable X.
- Usually the question will provide the random draws U from a uniform distribution on [0,1] . Set (F(x) = y) and solve for x; this yields the relationship (x = F^{-1}(y) ) where (F^{-1}(cdot) ) is the inverse of the distribution function.
- Create random draws X using the inverse distribution function, (X = F^{-1}(U)).
Note: Setting (F(x) =u) is the same mathematics as solving for a quantile, or percentile. The value at risk ({VaR}_p(X)) is also a quantile. It is tabulated for some distributions in, for example, the actuarial Exam C tables.
Simulation 202
Unlimited claim severities for a warranty product follow the lognormal distribution with parameters (mu=5.6) and (sigma =0.75). You use simulation to generate severities. The following are six uniform (0, 1) random numbers:
$$
{scriptsize
begin{matrix}
begin{array}{ccc}
0.6179 & 0.4602 & 0.9452 & 0.0808 & 0.7881 & 0.4207\
end{array}
end{matrix}
}
$$ Using these numbers and the inversion method, calculate the average payment per claim for a contract with a policy limit of 400.
[WpProQuiz 66]
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Strategy for Two-Point Mixture Simulation Problems
Consider a random variable X from a two-point mixture, where (F_1) and (F_2) represent the distribution functions from the two sub-populations and (alpha) is the probability of drawing from the first sub-population. To simulate a value of X, assume that you have a pair of independent random draws ([u_1,u_2]) from a uniform distribution on [0,1]. Then do the following
- For some problems, you need to identify the distribution function ( F(x)= alpha F_1(x) + (1-alpha) F_2(x)).
- Use the first random number (u_1) to determine the appropriate sub-population. Typically, you can compare (alpha ) to (u_1); if (u_1 leq alpha) then simulate from the first sub-population and otherwise simulate from the second.
- Determining the sub-population to draw from identifies the distribution function, either (F_1) or (F_2). Now use the inverse method. If you are working with the first sub-population, set (F_1(x) = u_2) and solve for the simulated variate X. For the second sub-population, set (F_2(x) = u_2) and solve for the simulated variate X
- Repeat this process for all pairs ([u_1,u_2]) provided in the question.
Note: Setting (F(x) =u) is the same mathematics as solving for a quantile, or percentile. The value at risk ({VaR}_p(X)) is also a quantile. It is tabulated for some distributions in, for example, the actuarial Exam C tables.
Mixture Simulation 290
A random variable
X has a two-point mixture distribution with pdf
$$ f(x) = frac{1}{8} e^{-x/2} + frac{1}{4} e^{-x/3}.$$You are to simulate one value,
x, from this distribution using uniform random numbers 0.2 and 0.6. Use the value 0.2 and the inversion method to simulate
J where
J=1 refers to the first random variable in the mixture and
J=2 refers to the second random variable. Then use 0.6 and the inversion method to simulate a value from
X.
Calculate the value of
x.
[WpProQuiz 67]
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Continuous Mixture 204
The length of time, in years, that a person will remember an actuarial statistic is modeled by an exponential distribution with mean 1/
Y. In a certain population,
Y has a gamma distribution with (alpha)=(theta)=2.
Calculate the probability that a person drawn at random from this population will remember an actuarial statistic less than 1/2 year.
[WpProQuiz 59]
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