Statistics and Estimation for Actuaries

Fitting a distribution to data — a severity curve, a development pattern, a class relativity — always reduces to an estimation problem. This article reviews the two classical estimators actuaries use daily, contrasts frequentist and Bayesian inference, and clarifies a distinction that trips up even experienced analysts: a confidence interval is not a credible interval.

Method of moments

The method of moments (MoM) sets sample moments equal to theoretical moments and solves for the parameters. For a gamma distribution with mean $\alpha\theta$ and variance $\alpha\theta^2$, given sample mean $\bar x$ and sample variance $s^2$:

$\hat\alpha = \frac{\bar x^2}{s^2}, \qquad \hat\theta = \frac{s^2}{\bar x}.$

MoM is fast, always has a closed form for simple distributions, and is a good starting value for numerical MLE — but it is not efficient (its variance exceeds the MLE's asymptotically) and can produce nonsensical values (e.g., negative variance parameters) for small or unusual samples.

Maximum likelihood estimation

MLE chooses $\hat\theta$ to maximize the likelihood of the observed data, $L(\theta) = \prod_i f(x_i\mid\theta)$, equivalently the log-likelihood $\ell(\theta) = \sum_i \ln f(x_i\mid\theta)$. For an exponential severity with parameter $\theta$ (mean), $\ell(\theta) = -n\ln\theta - \sum x_i/\theta$, giving $\hat\theta_{MLE} = \bar x$ — the same as MoM in this case, but MLE generally differs from MoM and is asymptotically efficient (attains the Cramér–Rao lower bound) and asymptotically normal:

$\hat\theta \;\dot\sim\; N\!\left(\theta,\; I(\theta)^{-1}\right), \qquad I(\theta) = -E\!\left[\frac{\partial^2 \ell}{\partial \theta^2}\right],$

where $I(\theta)$ is the Fisher information. This asymptotic normality is what lets us build confidence intervals and standard errors around MLE parameter estimates — the same machinery underlying GLM output tables.

Censoring and truncation are handled naturally by MLE: for a policy with limit $L$, a claim that hits the limit contributes $1-F(L\mid\theta)$ (the survival probability) to the likelihood rather than the density $f$, correctly reflecting that we only know the loss exceeded $L$.

Bayesian estimation

Bayesian inference treats $\theta$ itself as random, with a prior $\pi(\theta)$ representing beliefs before seeing data. Bayes' theorem gives the posterior:

$\pi(\theta \mid x) = \frac{f(x\mid\theta)\pi(\theta)}{\int f(x\mid\theta)\pi(\theta)\,d\theta} \;\propto\; f(x\mid\theta)\,\pi(\theta).$

When the prior is conjugate to the likelihood (e.g., gamma prior with Poisson likelihood, or beta prior with binomial likelihood), the posterior has the same family as the prior with updated parameters — this conjugate-prior machinery is exactly the engine behind Bühlmann-type credibility formulas (see Credibility Theory). The posterior mean is often used as a point estimate; the full posterior distribution supports probabilistic statements that frequentist confidence intervals cannot make.

Confidence intervals vs. credible intervals — the key distinction

A 95% confidence interval is a random interval, constructed by a procedure that captures the true fixed parameter $\theta$ in 95% of repeated samples. It says nothing about the probability that $\theta$ lies in this particular interval — that probability is either 0 or 1 (the parameter is fixed) once the data are observed. A 95% credible interval is a fixed interval, computed from the posterior distribution, such that $P(\theta \in [L,U] \mid x) = 0.95$ — a genuine probability statement about the parameter given the observed data. In practice with diffuse priors and large samples the two often coincide numerically, but their interpretation differs fundamentally, and communicating a confidence interval as if it were a credible interval is a common but serious professional error (relevant under ASOP 41's requirement for clear disclosure).

Hypothesis testing

Actuaries use hypothesis tests to check whether a trend is statistically different from zero, whether two accident years' loss ratios differ, or whether a new rating variable is significant. The standard machinery: state $H_0$ (e.g., $\beta = 0$, no trend) and $H_1$ ($\beta \ne 0$); compute a test statistic (e.g., $t = \hat\beta/\widehat{\mathrm{se}}(\hat\beta)$); compare to a critical value or compute a p-value; reject $H_0$ if the p-value is below the significance level $\alpha$ (commonly 0.05).

Worked example. A trend model fit to 8 years of severity data gives $\hat\beta = 0.045$ (4.5% annual severity trend) with standard error $0.014$. Test $H_0: \beta = 0$ at $\alpha = 0.05$ using $n-2 = 6$ degrees of freedom.

$t = \frac{0.045 - 0}{0.014} = 3.214.$

The two-sided critical value $t_{0.975,6} \approx 2.447$. Since $3.214 > 2.447$, we reject $H_0$: the trend is statistically significant at the 5% level, supporting the actuary's use of a positive severity trend in the rate indication.

Pitfalls

• Reporting confidence intervals as if they were probability statements about $\theta$ — a persistent misstatement even in professional work.
• Ignoring model risk: standard errors from MLE assume the model form is correct; misspecification (wrong distribution family) is not captured by the Fisher information.
• Small-sample MLE bias — e.g., the MLE of a lognormal $\sigma^2$ is biased downward; a bias-correction or Bayesian shrinkage is often warranted with thin data.
• Multiple comparisons: testing many rating variables at $\alpha=0.05$ each inflates the family-wise false-positive rate; adjust (Bonferroni, false discovery rate) or use penalized/regularized selection instead.
• Priors chosen for convenience rather than genuine belief, which can materially bias thin-data credibility estimates.

Exam relevance

These estimation methods are directly examined on SOA Exam SRM/STAM, CAS Exam MAS-I, and underlie all model-fitting content on CAS Exams 5, 7, 8, and 9.