Maximum Likelihood Estimation¶

Definition¶

Given a specified family of observation laws indexed by parameter $\theta\in\Theta$ $Θ$ , a maximum likelihood estimate maximizes the likelihood of the observed data. For independent exact observations $x_i$ , $L(\theta)=\prod_{i=1}^nf(x_i;\theta)$ $\prod_{i = 1}^{n} f (x_{i}; θ)$ , where $f$ is the applicable density or mass function.

Interpretation¶

The parameter varies while the observations remain fixed. Insurance data may report intervals or censoring rather than exact losses, so the likelihood must describe what was actually observed.

Concave exponential log-likelihood 3 log λ − 150 λ for the losses 20, 30 and 100, with its maximum at λ = 0.02 marked by a green point and a horizontal tangent.

As an illustration, take the three exact losses 20, 30 and 100, so $n=3$ and $\displaystyle \sum_ix_i=150$ $150$ . The data stay fixed and the exponential log-likelihood $\ell(\lambda)=3\log\lambda-150\lambda$ is drawn as a function of the parameter. It is concave, its derivative $3/\lambda-150$ vanishes at $\hat\lambda=3/150=0.02$ $0.02$ , and that zero is the maximum because $\ell''(\lambda)=-3/\lambda^2<0$ $0$ , as in the general example below.

The independent random variables assumption makes the joint observation density or mass function a product. Taking its logarithm then gives the sum of individual log contributions used for estimation.

Key Results¶

Maximizing $L$ is equivalent to maximizing $\ell=\log L$ where $L>0$ $0$ .
For independent observations, $\displaystyle \ell(\theta)=\sum_i\log f(x_i;\theta)$ $i \sum lo g f (x_{i}; θ)$ .
A zero derivative identifies a candidate, not automatically a maximum. Check the parameter domain, boundaries and curvature or numerical objective.
Grouping, censoring and truncation require different observation contributions.

Examples¶

For positive i.i.d. exponential losses with rate $\lambda$ , $\displaystyle \ell=n\log\lambda-\lambda\sum_i x_i$ $λ i \sum x_{i}$ , so $\displaystyle \hat\lambda=n/\sum_i x_i$ $n / i \sum x_{i}$ . The second derivative is $-n/\lambda^2<0$ $0$ .

Prerequisites¶

Sources¶

Risk Insurance · Lecture 4

Aliases: MLE, log-likelihood maximisation, maximum likelihood, maximum likelihood estimator, maximum likelihood estimate, ML estimation, method of maximum likelihood Keywords: likelihood function, log-likelihood, zero derivative of the log-likelihood, exponential rate n / Σx_i, second-order condition Symbols: element of, lambda, product, sum, theta

Last updated: 17 October 2026 21:53