Maximum likelihood estimation for severity and frequency distributions

Exam FAM – Fundamentals of Actuarial Mathematics · SOA (Society of Actuaries) · 7 min read

Welcome to Maximum Likelihood Estimation (MLE)!

Hello there! Today, we are diving into one of the most powerful tools in an actuary's toolkit: Maximum Likelihood Estimation, or MLE for short. If you have ever looked at a pile of insurance claims data and wondered, "Which mathematical model fits this best?", then MLE is your answer.

Think of MLE as a "Reverse Engineering" process. Usually, we are given a formula and asked to find the probability. In MLE, we have the results (the data) and we are trying to find the formula (the parameters) that was most likely to have produced those results. Don't worry if this seems a bit abstract right now—we'll break it down step-by-step!

1. The Core Idea: What is Likelihood?

Imagine you have a coin. You flip it 10 times and get 9 heads. You have two guesses for the probability of heads ($ p $): 0.5 (fair) or 0.9 (weighted). Which one is more likely? Obviously, 0.9! In actuarial science, we do the same thing with claim sizes (severity) and claim counts (frequency).

The Likelihood Function $ L(\theta) $:
The likelihood function is the probability (for discrete data) or the density (for continuous data) of observing our specific data set, given a parameter $ \theta $. If we have independent observations $ x_1, x_2, ..., x_n $, the likelihood is the product of their individual probabilities:

$ L(\theta) = f(x_1 | \theta) \times f(x_2 | \theta) \times ... \times f(x_n | \theta) = \prod_{i=1}^{n} f(x_i | \theta) $

The Goal: Find the value of $ \theta $ that makes $ L(\theta) $ as large as possible. This value is called the Maximum Likelihood Estimate, denoted as $ \hat{\theta} $.

Quick Tip: Why we love Logarithms

Multiplying a bunch of small decimals is messy. Calculus with products is even messier (Product Rule, yuck!). To make life easier, we take the Natural Logarithm of the likelihood. This turns multiplication into addition:

$ \ell(\theta) = \ln[L(\theta)] = \sum_{i=1}^{n} \ln[f(x_i | \theta)] $

Since the log function is "increasing," the value of $ \theta $ that maximizes the log-likelihood is the same value that maximizes the original likelihood!

Key Takeaway: The Likelihood Function is just the probability of your data occurring. We use logs to make the math manageable.

2. The Recipe for Finding the MLE

When you see an MLE problem on Exam FAM, follow these steps:

Step 1: Write down the density function $ f(x) $ or probability mass function $ p(x) $ for the distribution.
Step 2: Write the Likelihood Function $ L(\theta) $ (the product of the densities).
Step 3: Take the natural log to get the Log-Likelihood $ \ell(\theta) $.
Step 4: Take the derivative with respect to $ \theta $: $ \frac{d}{d\theta} \ell(\theta) $.
Step 5: Set the derivative to zero and solve for $ \theta $. This is your $ \hat{\theta} $!

Example: The Exponential Distribution

If we have claims $ x_1, ..., x_n $ from an Exponential distribution with mean $ \theta $, the density is $ f(x) = \frac{1}{\theta} e^{-x/\theta} $.
1. $ L(\theta) = \prod \frac{1}{\theta} e^{-x_i/\theta} = \theta^{-n} e^{-\sum x_i / \theta} $
2. $ \ell(\theta) = -n \ln(\theta) - \frac{\sum x_i}{\theta} $
3. $ \frac{d}{d\theta} \ell(\theta) = -\frac{n}{\theta} + \frac{\sum x_i}{\theta^2} = 0 $
4. Solving for $ \theta $ gives $ \hat{\theta} = \frac{\sum x_i}{n} = \bar{x} $ (the sample mean).

Did you know? For many common distributions, the MLE is just the sample mean! This includes the Exponential, Poisson, and Bernoulli distributions.

3. MLE for Severity: Dealing with Truncation and Censoring

In the real world, insurance data is rarely "clean." We often deal with deductibles (left-truncation) and policy limits (right-censoring). This is a favorite topic for Exam FAM!

A. Grouped Data

Sometimes we don't know the exact claim amount, only that it fell into a range (e.g., between \$500 and \$1,000). For an observation in the interval $ (c_{j-1}, c_j] $, the contribution to the likelihood is the probability of being in that range:

$ P(c_{j-1} < X \le c_j) = F(c_j) - F(c_{j-1}) $

B. Censored Data (Policy Limits)

If a claim is "censored" at a limit $ u $, it means the claim was actually at least $ u $, but we only recorded $ u $. For these data points, we don't use the density $ f(x) $; instead, we use the Survival Function $ S(u) $.

Logic: We don't know exactly where the claim was, just that it survived past $ u $.

C. Truncated Data (Deductibles)

If a policy has a deductible $ d $, we only see claims where $ X > d $. This is "left-truncated." We must use a Conditional Density:

$ f(x | X > d) = \frac{f(x)}{S(d)} $

Quick Review: The "Likelihood Contribution" Table

Individual, exact value $ x $: Use $ f(x) $
Censored at $ u $ (limit): Use $ S(u) $
Truncated at $ d $ (deductible): Divide the whole likelihood by $ S(d) $ for each such observation.

Common Mistake: Forgetting to divide by $ S(d) $ when a deductible is present. If the data is truncated, your probability space has shrunk, so you must "re-normalize" it!

4. MLE for Frequency Distributions

Frequency deals with the number of claims. The process is the same, but we use Probability Mass Functions (pmf) instead of densities.

The "Big Three" Frequency Distributions:

1. Poisson ($ \lambda $): The MLE $ \hat{\lambda} $ is simply the sample mean $ \bar{x} $.
2. Binomial ($ m, q $): If $ m $ is known, $ \hat{q} = \frac{\bar{x}}{m} $.
3. Negative Binomial ($ r, \beta $): If $ r $ is known, $ \hat{\beta} = \frac{\bar{x}}{r} $.

Wait, what if parameters are missing?

If you have a frequency table (e.g., 50 people had 0 claims, 30 had 1 claim, etc.), your likelihood is:
$ L = [P(X=0)]^{n_0} \times [P(X=1)]^{n_1} \times ... $

Summary: Frequency MLE is usually just finding the parameter that matches the observed average number of claims.

5. Properties of MLEs (Why we use them)

You might wonder why we go through all this calculus. MLEs have some "superpowers" when the sample size $ n $ gets very large:

1. Asymptotic Unbiasedness: As $ n $ grows, the bias goes to zero.
2. Consistency: As $ n $ grows, the estimate gets closer and closer to the true value.
3. Asymptotic Normality: For large $ n $, the distribution of $ \hat{\theta} $ looks like a Normal distribution! This allows us to build confidence intervals.
4. Invariance: This is a student favorite. If the MLE of $ \theta $ is $ \hat{\theta} $, then the MLE of any function $ g(\theta) $ is simply $ g(\hat{\theta}) $.
Example: If the MLE of the variance $ \sigma^2 $ is 100, the MLE of the standard deviation $ \sigma $ is $ \sqrt{100} = 10 $. No extra calculus needed!

Final Study Tips for Exam FAM

1. Practice the Logarithms: Most errors happen during the algebra of simplifying $ \ln(L) $. Remember that $ \ln(a^b) = b \ln(a) $ and $ \ln(ab) = \ln(a) + \ln(b) $.
2. Watch for the Shortcut: If the question asks for the MLE of a parameter for a standard distribution (like Exponential or Poisson) and there is no censoring or truncation, it is almost always the sample mean.
3. The "Deductible" Trap: Always check if the data is "per payment" or "per loss." If it's "per payment," the data is truncated at the deductible $ d $.
4. Don't Panic: If the derivative looks impossible to solve, look at the answer choices. Sometimes you can plug them in to see which one makes the derivative zero.

Key Takeaway: MLE is about finding the "best fit" by maximizing the probability of the data we actually saw. Master the log-likelihood steps and the handling of deductibles, and you'll be well on your way to passing Exam FAM!

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is the fundamental definition of the likelihood function $ L(\theta) $ for a set of independent observations?

The likelihood function is the product of the individual density functions (for continuous data) or probability mass functions (for discrete data) for all observations in the sample: $ L(\theta) = \prod_{i=1}^{n} f(x_i | \theta) $.

Why is the log-likelihood function $ \ell(\theta) $ preferred over the standard likelihood function in calculation?

Logarithms convert products into sums ($ \ell(\theta) = \sum \ln[f(x_i | \theta)] $), which simplifies the differentiation process required to find the maximum.

What are the five general steps to solve for a Maximum Likelihood Estimate (MLE)?

1. Write the density/mass function; 2. Form the Likelihood $ L(\theta) $; 3. Take the natural log to get $ \ell(\theta) $; 4. Differentiate with respect to $ \theta $; 5. Set to zero and solve.

What is the Maximum Likelihood Estimate for the parameter $ \theta $ in an Exponential distribution?

The MLE is the sample mean: $ \hat{\theta} = \bar{x} $.

How is the likelihood contribution calculated for grouped data within an interval $ (c_{j-1}, c_j] $?

The contribution for that observation is the probability of falling into the range: $ F(c_j) - F(c_{j-1}) $.

What term is contributed to the likelihood function for a right-censored observation at policy limit $ u $?

The contribution is the survival function evaluated at the limit: $ S(u) $.

How must the likelihood be adjusted for left-truncated data (e.g., claims above a deductible $ d $)?

Each observation's contribution must be divided by the survival function at the deductible: $ \frac{f(x)}{S(d)} $.

What is the MLE for the Poisson parameter $ \lambda $ given claim counts $ x_1, ..., x_n $?

The MLE is the sample mean of the counts: $ \hat{\lambda} = \bar{x} $.

What does the Invariance Property of MLEs state?

If $ \hat{\theta} $ is the MLE of $ \theta $, then the MLE of any function $ g(\theta) $ is simply $ g(\hat{\theta}) $.

What is the meaning of Asymptotic Normality for MLEs?

As the sample size $ n $ approaches infinity, the distribution of the estimator $ \hat{\theta} $ converges to a Normal distribution centered at the true parameter value.

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