Perform point estimation of statistical parameters using maximum likelihood estimation (MLE) applying criteria such as consistency, unbiasedness, sufficiency, efficiency, minimum variance, and mean square error (including censoring and truncation).

A reserving actuary fits a severity distribution to a stack of closed claims, but half the policies were still open at year-end and three policy years had a $10,000 deductible. MLE handles all of it inside one likelihood, and the exam expects you to know which criteria reward you for that choice.

Building the likelihood. Given iid sample $X_1,\dots ,X_n$ from density $f(x\mid \theta )$, the likelihood is $L(\theta )=\prod f(x_i\mid \theta )$ and the log-likelihood is $\ell (\theta )=\sum \log f(x_i\mid \theta )$. The MLE $\hat{\theta }$ maximizes $\ell$. For interior maxima of smooth models, set the score to zero and check the second derivative is negative.

$$\frac{\partial \ell (\theta )}{\partial \theta }=0,\frac{{\partial }^2\ell (\theta )}{\partial {\theta }^2}<0$$

Invariance. If $\hat{\theta }$ is the MLE of $\theta$, then $g(\hat{\theta })$ is the MLE of $g(\theta )$ for any function $g$. No re-optimization required.

Consistency. Under standard regularity, $\hat{\theta }{\to }_p\theta$ as $n\to \infty$. Bias and variance both shrink; the MLE concentrates on the truth.

Unbiasedness. $E[\hat{\theta }]=\theta$ exactly. MLEs are usually only asymptotically unbiased. The classic case: ${\hat{\sigma }}_{MLE}^2=\frac{1}{n}\sum (x_i-\bar{x}{)}^2$ has expectation $(n-1){\sigma }^2/n$, so it is biased low.

Common mistakes

• Treating censored observations as exact. Plugging the censoring limit into the density understates the tail and biases the MLE toward zero variance.
• Forgetting to divide by $S(d)$ under truncation. Using raw density for observations above a deductible ignores conditioning on exceedance, biasing mean severity low.
• Confusing CRLB applicability. The bound $1/[nI(\theta )]$ is for unbiased estimators. A biased MLE may legitimately have lower variance than CRLB; direct comparison is meaningless.

Bottom line

• MLE maximizes $\ell (\theta )=\sum \log f(x_i\mid \theta )$; solve $\partial \ell /\partial \theta =0$ and verify the second derivative is negative.
• Cramér-Rao lower bound is $1/[nI(\theta )]$ with $I(\theta )=-E[{\partial }^2\log f/\partial {\theta }^2]$; MLEs are asymptotically efficient.
• MLEs are consistent, asymptotically normal, asymptotically unbiased, and functions of any sufficient statistic, but often biased in finite samples.
• MSE $=$ variance $+$ bias${}^2$; a biased estimator can still beat an unbiased one on MSE.

Exam shortcut

For exponential severity with deductible $d$ and censoring at $u$, the MLE of the mean is total observed exposure above $d$, divided by the count of fully observed losses; memorize this and skip the score derivation. When asked for asymptotic variance, compute $I(\theta )$ for one observation, divide by $n$, and plug in $\hat{\theta }$ last.

The full lesson (about 1,505 words, 10 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• B8

Browse all free MAS-I lessons or jump into free MAS-I practice questions.