Maximum Likelihood Estimation

An insurer has 200 loss observations: 40 exact amounts, 80 grouped, 50 right-censored at a policy limit, and 30 left-truncated by a deductible. Each data type demands a different likelihood contribution. Build it wrong for even one observation, and your parameter estimates are biased. The FAM exam tests whether you can construct the correct likelihood for every data scenario.

Given independent observations from a distribution with parameters $\boldsymbol{\theta}$:

$$L(\boldsymbol{\theta}) = \prod_{i=1}^n f(x_i; \boldsymbol{\theta}), \quad \ell(\boldsymbol{\theta}) = \sum_{i=1}^n \ln f(x_i; \boldsymbol{\theta})$$

Maximize by solving $\partial \ell / \partial \theta_j = 0$ for each parameter.

KEY: Constructing the likelihood for different data types is the core testable skill, every observation type has a distinct contribution.

HIGH-FREQUENCY: Constructing the likelihood for different data types is the core testable skill.

Each observation contributes $f(x_i; \theta)$.

Exponential (mean $\theta$): $\hat{\theta} = \bar{x}$. Normal: $\hat{\mu} = \bar{x}$, $\hat{\sigma}^2 = \frac{1}{n}\sum(x_i - \bar{x})^2$, divides by $n$, not $n-1$.

Common mistakes

• Dividing by $n$ instead of $r$ for censored exponential data. With 14 uncensored and 6 censored, the denominator is 14. Using 20 gives 241.0 instead of 344.3. Trap: 241.0 on any problem with censoring.
• Forgetting to condition on truncation. Without $S(d)$ in the denominator, the MLE ignores the deductible's selection effect. For the exponential, this means using $\bar{x}$ instead of $\bar{x} - d$. Trap: the unconditional mean when the problem specifies a deductible.
• Using $n-1$ in the normal variance MLE. MLE divides by $n$, not $n-1$. With $n = 20$ and $\sum(x_i - \bar{x})^2 = 1{,}900$: MLE gives 95.0, unbiased gives 100.0. Trap: 100.0.

Bottom line

• MLE maximizes the log-likelihood over the parameter domain. Compare stationary points and boundaries; negative second derivative is sufficient for a strict local maximum, not necessary or proof of a global maximum.
• Complete data: each observation contributes $f(x_i; \theta)$
• Grouped in $(c_{j-1}, c_j]$: contributes $F(c_j) - F(c_{j-1})$, never a midpoint density
• Right-censored at $c$: contributes $S(c) = 1 - F(c)$

Exam shortcut

When an MLE problem mixes data types, label each observation's contribution before doing algebra: "density for uncensored, survival for censored, CDF difference for grouped, divided by survival for truncated." For the exponential, look for the shortcut: total exposure over uncensored count. Remember: "Ceiling vs. Floor". censoring puts a ceiling (multiply by $S(c)$), truncation puts a floor (divide by $S(d)$). The operations are opposites.

The full lesson (about 2,563 words, 17 min read) adds 6 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 3a

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