CAS MAS-I Glossary: 28 Key Terms, Free (2026)

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Akaike Information Criterion (AIC): AIC compares non-nested statistical models by trading off goodness of fit against model complexity, penalizing additional parameters. The model with the lowest AIC is preferred among candidates fit to the same dataset. $\text{AIC} = 2k - 2 \ln(L)$

Bayesian Estimation: Bayesian estimation combines a prior distribution over the parameter with the likelihood of observed data via Bayes' theorem to produce a posterior distribution. Point estimates such as the posterior mean or mode are then derived from the posterior. $\pi(\theta | x) \propto f(x | \theta) \cdot \pi(\theta)$
Binomial Distribution: The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials with constant success probability. It is the canonical frequency distribution for count data with a known cap. $P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$
Bootstrap Method: Bootstrap resamples the observed data with replacement to generate the empirical sampling distribution of a statistic. It is used to construct confidence intervals and standard errors when analytical methods are intractable or assumptions are doubtful.

Compound Poisson Model: A compound Poisson model represents aggregate losses as the sum of N i.i.d. severities, where N follows a Poisson distribution. Aggregate moments combine the moments of N and the severity distribution via formulas such as the law of total expectation.
Confidence Interval: A confidence interval is a range of plausible values for a parameter, constructed so that the procedure produces an interval covering the true parameter with a specified frequency (often 95%) under repeated sampling.
Conjugate Prior: A conjugate prior is a prior distribution that, when combined with a specific likelihood, yields a posterior in the same parametric family. Beta-binomial and gamma-Poisson are canonical conjugate pairs heavily used on MAS-I.
Cumulative Distribution Function (CDF): The CDF of a random variable X is F(x) = P(X ≤ x), the probability that X takes a value at or below x. CDFs are non-decreasing, right-continuous, approach 0 as x → -∞ and 1 as x → ∞. $F(x) = P(X \le x)$

Deviance: Deviance is twice the difference between the log-likelihood of the saturated model and the log-likelihood of the fitted model. It is the GLM analogue of residual sum of squares and is used in goodness-of-fit testing.

Exponential Family: An exponential-family distribution can be written in the canonical form involving a natural parameter, sufficient statistic, and dispersion parameter. GLMs rely on this structure to unify estimation across normal, Poisson, binomial, gamma, and inverse Gaussian responses.

Generalized Linear Model (GLM): A GLM extends linear regression by allowing the response variable to follow any exponential-family distribution and the mean to be related to a linear predictor through a link function. GLMs are the workhorse model class for P&C insurance ratemaking and classification.

Hypothesis Test: A hypothesis test compares a null hypothesis (typically a parameter equality or constraint) against an alternative using a test statistic with a known sampling distribution. The null is rejected if the test statistic falls in the critical region or equivalently if the p-value is below the chosen significance level.

Identity Link: The identity link function specifies that the mean of the response equals the linear predictor, μ = Xβ. It is the link function in ordinary linear regression and is appropriate when the response is approximately normal.
Inverse Link: The inverse link function specifies that the mean of the response equals one over the linear predictor, μ = 1/(Xβ). It pairs naturally with the gamma distribution and is used when expected loss is inversely related to predictors.

Likelihood Ratio Test: The likelihood ratio test compares two nested models via twice the difference in log-likelihoods, which follows a chi-squared distribution under the null hypothesis. It tests whether the more complex model provides a statistically significant improvement. $\Lambda = 2 \cdot (\ln L_1 - \ln L_0) \sim \chi^2_k$
Link Function: A link function g connects the mean of the response variable to the linear predictor in a GLM via g(μ) = Xβ. Canonical links pair with each exponential-family distribution to simplify estimation.
Log Link: The log link specifies that log(μ) = Xβ, equivalently μ = exp(Xβ). It is the canonical link for the Poisson and gamma distributions and is the natural choice when the response is strictly positive.
Lognormal Distribution: A lognormal random variable X has the property that ln(X) follows a normal distribution. Lognormal is widely used in P&C insurance to model right-skewed loss severities with all-positive support.

Maximum Likelihood Estimation (MLE): MLE selects parameter values that maximize the likelihood of the observed data under the assumed model. Under regularity conditions, MLEs are asymptotically unbiased, consistent, and normally distributed with covariance equal to the inverse Fisher information.
Method of Moments: Method of moments estimates parameters by equating sample moments (mean, variance, etc.) to their theoretical counterparts under the assumed distribution. It is simpler than MLE but generally less efficient.
Monte Carlo Simulation: Monte Carlo simulation generates many random realizations from a probability model to approximate quantities such as expected values, percentiles, or tail probabilities. It is essential when closed-form solutions are unavailable or assumptions are too complex for analytical methods.

p-value: The p-value is the probability, under the null hypothesis, of observing a test statistic at least as extreme as the one realized in the sample. A p-value below the significance level (commonly 0.05) leads to rejecting the null hypothesis.
Pareto Distribution: The Pareto distribution is a heavy-tailed continuous distribution used to model right-skewed quantities such as large insurance losses. Its tail decays as a power law rather than exponentially, making it suitable for modeling extreme events.
Pearson Residual: Pearson residual is the difference between observed and fitted value, divided by the square root of the variance function. It standardizes residuals in GLMs so they can be compared across observations and used in diagnostic plots.
Posterior Distribution: The posterior distribution is the conditional distribution of the parameter given the observed data, obtained from Bayes' theorem by combining the prior with the likelihood. All Bayesian inference flows from the posterior.
Probability Density Function (PDF): The PDF of a continuous random variable specifies the density at each point such that the integral over any interval equals the probability the variable falls in that interval. The PDF is non-negative and integrates to one over the support.

Severity Distribution: Severity distribution models the size of an individual insurance loss given that a loss occurs. Common severity distributions on MAS-I include exponential, Pareto, lognormal, gamma, and Weibull.

Weibull Distribution: The Weibull distribution is a flexible continuous distribution with shape and scale parameters, used to model time-to-event and insurance severities. Depending on the shape parameter, hazard rate can be decreasing, constant, or increasing in time.