Maximum likelihood estimation for frequency and severity distributions

Exam ASTAM – Advanced Short-Term Actuarial Mathematics · SOA (Society of Actuaries) · 6 min read

Welcome to the World of Likelihood!

Hello there, future Actuary! Welcome to one of the most important chapters in your ASTAM journey: Maximum Likelihood Estimation (MLE). If you’ve ever wondered how insurance companies decide on the specific "numbers" (parameters) for their models—like the average claim size or how often accidents happen—you are in the right place.

Think of MLE as being a detective. You have the clues (the data), and you want to find the most likely culprit (the parameter) that caused those clues to appear. It is the "Gold Standard" for parameter estimation because it has some very powerful mathematical properties that make our models reliable. Don't worry if this seems a bit heavy at first; we will break it down step-by-step!

1. The Big Idea: What is Likelihood?

In everyday language, "likelihood" and "probability" mean the same thing. But in Actuarial Science, they have a subtle difference:
• Probability is used when we know the parameters and want to predict the data.
• Likelihood is used when we have the data and want to guess the parameters.

The Likelihood Function, denoted as $L(\theta)$, represents the probability of seeing the data we actually observed, given a specific parameter $\theta$. Our goal is to find the value of $\theta$ that makes our observed data the most probable outcome. We call this estimate $\hat{\theta}$ (pronounced "theta-hat").

Why do we use the Natural Log?

In most problems, the Likelihood function is a giant product of probabilities. Multiplying dozens of tiny decimals is a nightmare! To make life easier, we take the natural logarithm to get the Log-likelihood, denoted as $l(\theta)$ or $\ln L(\theta)$.
Did you know? Because the logarithm is an "increasing function," the value of $\theta$ that maximizes the log-likelihood is exactly the same value that maximizes the original likelihood. It turns multiplication into addition, making the calculus much friendlier!

2. MLE for Individual Severity Data

Let's start with Severity (the size of a claim). Suppose we have a set of independent claim observations $x_1, x_2, ..., x_n$.

The Basic Case:
If we have a complete set of data with no deductibles or limits, the likelihood is simply the product of the probability density functions (PDF) for each observation:
$L(\theta) = \prod_{i=1}^{n} f(x_i; \theta)$

The Actuarial Reality: Truncation and Censoring
In the real world of ASTAM, we rarely see "pure" data. We deal with deductibles (d) and policy limits (u). This changes how we write our likelihood!

• Left Truncation (Deductibles): If a claim is only recorded if it exceeds a deductible $d$, we are looking at a conditional probability. The contribution of an observation $x$ to the likelihood is: $\frac{f(x)}{S(d)}$.
• Right Censoring (Policy Limits): If a claim is capped at a limit $u$, we don't know the exact value—only that it was "at least $u$." The contribution to the likelihood for a capped claim is: $S(u)$.

Step-by-Step Process for Severity MLE:

1. Construct the Likelihood: For each data point, determine if it's a specific value $f(x)$ or a censored value $S(u)$, and adjust for deductibles $/ S(d)$ if necessary.
2. Take the Log: Apply $\ln$ to the whole thing.
3. Differentiate: Take the derivative with respect to $\theta$.
4. Solve for Zero: Set the derivative equal to zero and solve for $\hat{\theta}$.
5. Verify: (Technically) check the second derivative to ensure it's a maximum, though on the exam, the first solution is almost always the one you need.

Key Takeaway: For severity, remember: "PDF for exact values, Survival for capped values, and divide by Survival of the deductible for truncated data."

3. MLE for Frequency Data

Frequency refers to how many claims occur. Here, our data points $n_i$ are counts (0, 1, 2...).

If we observe $k$ claims $n_1, n_2, ..., n_k$, the likelihood is the product of the probability mass functions (PMF):
$L(\theta) = \prod_{i=1}^{k} P(N = n_i; \theta)$

Often, frequency data is given in a table (e.g., "There were 50 policies with 0 claims, 30 policies with 1 claim..."). In this case, use the counts as exponents:
$L(\theta) = [P(N=0)]^{50} \times [P(N=1)]^{30} ...$

Common Frequency Shortcuts:

For some distributions, you don't even need calculus! The MLEs are intuitive:
• Poisson: $\hat{\lambda}$ is simply the sample mean ($\bar{x}$).
• Binomial: $\hat{q}$ is the sample mean divided by $m$.
• Negative Binomial: If $r$ is fixed, $\hat{\beta} = \bar{x} / r$.

Quick Review Box:
• Severity = Continuous (use $f(x)$ and $S(x)$)
• Frequency = Discrete (use $p_k$)
• Logarithm is your best friend!

4. Properties of Maximum Likelihood Estimators

Why do actuaries love MLE so much? Because when the sample size $n$ gets large, MLEs become "the perfect estimators." Here are the three "A's" you should know:

1. Asymptotic Unbiasedness: As you get more data, the average of your estimate $\hat{\theta}$ hits the true value perfectly.
2. Asymptotic Normality: For large samples, the distribution of $\hat{\theta}$ looks like a Bell Curve (Normal Distribution). This is great for building confidence intervals!
3. Asymptotic Efficiency: Among all "good" estimators, MLE has the smallest possible variance. It's the most "precise" tool in our kit.

Calculating the Variance of an MLE

To find out how much our estimate might "wiggle," we use the Information Matrix (or Fisher Information).
The variance of $\hat{\theta}$ is approximately:
$Var(\hat{\theta}) \approx \frac{1}{-E[l''(\theta)]}$ or more commonly on exams: $\frac{1}{-l''(\hat{\theta})}$

Memory Aid: Think of the second derivative ($l''$) as the "curvature." If the log-likelihood curve is very pointy (high curvature), we are very certain about our estimate, so the variance is small!

5. Grouped Data (The "Bucket" Problem)

Sometimes, we don't know the exact claim amounts. We just know how many claims fell into certain "buckets" (e.g., 10 claims were between \$0 and \$1,000).
For a group between boundaries $c_{j-1}$ and $c_j$ with $n_j$ observations, the contribution to the likelihood is:
$[F(c_j) - F(c_{j-1})]^{n_j}$

Common Mistake to Avoid: Don't use the PDF $f(x)$ for grouped data! You don't have a specific $x$. You must use the difference in the Cumulative Distribution Function (CDF).

6. Summary and Final Tips

• Stay Organized: MLE problems can be long. Write out your PDF or PMF clearly before starting the log-transformation.
• Watch the Deductible: If the data is "per payment," you usually divide by $S(d)$. If it's "per loss" and you are only told about losses above $d$, the same logic applies.
• Practice the Derivatives: Many ASTAM students lose points not on the actuarial concepts, but on basic calculus errors. Brush up on $\frac{d}{dx} \ln(u) = \frac{1}{u} u'$.
• Check your Distribution: Is it a Gamma? A Pareto? An Exponential? Each has its own shortcut or specific PDF. Keep your ASTAM tables handy!

Key Takeaway for the Exam: MLE is about finding the parameter that makes the observed reality the most likely outcome. Whether it's frequency or severity, the pattern is: Likelihood $\rightarrow$ Log $\rightarrow$ Derivative $\rightarrow$ Zero. You've got this!

Quick check

Can you answer these now?

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

What is the conceptual difference between probability and likelihood in actuarial estimation?

Probability is used to predict data when parameters are known, whereas likelihood is used to estimate unknown parameters based on observed data.

What is the definition of the Likelihood Function $L(\theta)$?

The Likelihood Function represents the probability of observing the specific sample of data given a specific parameter value $\theta$.

Why is the log-likelihood $l(\theta)$ preferred over the likelihood function $L(\theta)$ for calculation?

The natural log transforms the product of probabilities into a sum, which simplifies differentiation, while the maximum point remains the same.

How is the likelihood function constructed for a complete set of severity observations $x_1, x_2, ..., x_n$?

It is the product of the probability density functions (PDF) for each individual observation: $L(\theta) = \prod_{i=1}^{n} f(x_i; \theta)$.

How does a deductible $d$ (left truncation) change the likelihood contribution of a claim $x$?

The contribution becomes a conditional probability: $\frac{f(x)}{S(d)}$.

What is the likelihood contribution for a claim that is right-censored at a policy limit $u$?

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

What are the standard calculus steps for finding the Maximum Likelihood Estimate (MLE)?

1. Construct Likelihood $L(\theta)$; 2. Take the Log $l(\theta)$; 3. Differentiate $l'(\theta)$; 4. Set to zero and solve for $\hat{\theta}$.

How is the likelihood function constructed for frequency data provided in a table of counts?

Use the probability mass function (PMF) for each outcome, using the observed counts as exponents: $L(\theta) = \prod [P(N=k)]^{n_k}$.

What are the shortcut MLEs for Poisson and Binomial distributions?

For Poisson, $\hat{\lambda} = \bar{x}$. For Binomial, $\hat{q} = \bar{x} / m$.

What are the three "Asymptotic" properties of MLEs as the sample size $n$ increases?

MLEs are asymptotically unbiased, asymptotically normal, and asymptotically efficient.

How is the variance of an MLE $\hat{\theta}$ calculated?

The variance is the reciprocal of the negative second derivative of the log-likelihood: $Var(\hat{\theta}) \approx \frac{1}{-l''(\hat{\theta})}$.

What is the likelihood contribution for grouped data (bucketed data)?

The contribution for a group between $c_{j-1}$ and $c_j$ is the difference in the CDF: $F(c_j) - F(c_{j-1})$.

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