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 $X_1,\ldots,X_n$ 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.

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

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: $\hat\sigma^2_{MLE}=\tfrac{1}{n}\sum(x_i-\bar x)^2$ 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 2,274 words, 15 min read) adds 8 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• B8

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