IFoA CM2 key terms

The assumption that the future evolution of a process depends only on its present state and not on the path taken to reach that state. It makes the transition probabilities sufficient for prediction.

A property of a process whose statistical behaviour, such as mean, variance and autocovariance, does not change when the time origin is shifted. Weak stationarity restricts this to the first two moments.

A model in which the output is fully determined once the inputs and parameters are fixed, leaving no random variation. It contrasts with a stochastic model, where outcomes follow probability distributions.

Generating artificial random outcomes from a fitted model to study its behaviour, estimate quantities with no closed form, or assess uncertainty. It is widely used for ruin, option pricing and projection work.

The sampling variability in a simulation-based estimate that arises because only a finite number of random scenarios are generated. It falls roughly in proportion to the inverse square root of the number of simulations.

The distribution of the number of successes in a fixed number of independent trials, each with the same probability of success. Its mean is np and its variance np(1-p).

A distribution for counts of events occurring randomly in a fixed interval, where events are independent and occur at a constant average rate. Mean and variance are both equal to the rate parameter.

A counting process in which events occur singly, with independent increments and a constant rate, so that counts in any interval follow a Poisson distribution. It underlies models of claim arrivals and mortality.

For a non-negative integer-valued random variable, the function of a dummy variable equal to the expected value of that variable raised to the power of the random variable. It yields probabilities and moments by differentiation.

The expected value of the exponential of a dummy variable times a random variable. Its derivatives at zero give the moments, and it identifies the distribution when it exists in a neighbourhood of zero.

The result that the sum or average of many independent random variables, suitably standardised, tends to a standard normal distribution as the number of terms grows. It justifies normal approximations in estimation and testing.

A continuous symmetric bell-shaped distribution defined by a mean and variance, with about 95% of its mass within two standard deviations of the mean. Many estimators are approximately normal in large samples.

The distribution of a variable whose logarithm is normal. It is positively skewed and used for quantities that cannot be negative, such as claim sizes and future asset prices.

A symmetric heavy-tailed distribution used when a normal mean is estimated with an unknown variance replaced by its sample estimate. Its tails approach the normal as the degrees of freedom increase.

An estimator whose expected value equals the true parameter for every possible parameter value. Unbiasedness concerns the average over repeated samples, not the accuracy of any single estimate.

The expected squared difference between an estimator and the true parameter. It decomposes into the estimator's variance plus the square of its bias, allowing trade-offs between the two.

A property whereby an estimator converges in probability to the true parameter as the sample size grows without limit. Consistency says nothing about behaviour at any finite sample size.

The parameter value that maximises the likelihood, the joint probability or density of the observed data viewed as a function of the parameter. It is usually consistent and asymptotically efficient under regularity conditions.

A measure of how much a sample carries about an unknown parameter, equal to the negative expected second derivative of the log-likelihood. Its reciprocal bounds the variance of unbiased estimators.

The smallest variance achievable by any unbiased estimator of a parameter, equal to the reciprocal of the Fisher information. Estimators attaining it are called efficient.

An estimation approach that sets sample moments equal to their theoretical counterparts and solves for the parameters. It is simple and often consistent, though usually less efficient than maximum likelihood.

The default claim tested in a hypothesis test, typically that a parameter takes a specified value or that no effect exists. Evidence against it is measured by the p-value.

The probability, assuming the null hypothesis is true, of observing a test statistic at least as extreme as the one obtained. It is not the probability that the null hypothesis is true.

The pre-chosen probability of rejecting a true null hypothesis, often 5% or 1%. It fixes the rejection region before the data are examined and controls the type I error rate.

The probability of rejecting the null hypothesis when a particular alternative is true. Power increases with larger samples, larger effect sizes and lower variability.

Failing to reject the null hypothesis when it is in fact false. Its probability is denoted beta, and one minus beta is the power of the test.

A test comparing the maximised log-likelihood under a restricted null model with that under the full model. Twice the difference is approximately chi-squared with degrees of freedom equal to the number of restrictions.

A regression fitting method that chooses coefficients minimising the sum of squared residuals. Under standard assumptions it gives the best linear unbiased estimators of the regression parameters.

The difference between an observed response and its value predicted by the fitted model. Residual plots reveal non-constant variance, curvature and outliers that summary statistics can hide.

The proportion of total variation in the response explained by the fitted regression, ranging from zero to one. A high value does not by itself prove the model is appropriate.

A regression framework in which the response follows an exponential family distribution and its mean is linked to a linear combination of predictors by a link function.

The function connecting the mean of the response distribution to the linear predictor in a generalised linear model. Common choices include the logit for binary data and the log for counts.

Twice the difference between the maximised log-likelihood of a saturated model and that of the fitted model. It measures goodness of fit and is used to compare nested generalised linear models.

The covariance between values of a time series separated by a given lag. It describes the dependence structure and, together with the mean and variance, characterises weakly stationary series.

The correlation between values of a time series at different lags, obtained by scaling the autocovariance by the variance. Its plot helps identify appropriate orders in ARMA models.

The correlation between observations at a given lag after removing the linear effect of all intervening lags. It is used alongside the autocorrelation function to identify autoregressive order.

A time series model in which the current value is a linear function of previous values plus a random error. The order gives the number of past values included.

A time series model in which the current value depends on the current random error and a finite number of past errors. It is stationary for any finite order.

A stochastic process moving among a discrete set of states, where the probability of the next state depends only on the current state. Transition probabilities are collected in a transition matrix.

The instantaneous rate at which a continuous-time Markov process moves from one state to another. It defines the generator matrix from which transition probabilities over an interval are derived.

A system of differential equations describing how transition probabilities of a Markov jump process evolve over time. They are solved to obtain probabilities of being in each state.

A continuous-time process that stays in each state for an exponentially distributed random time before jumping to another state. It is specified by transition intensities between states.

The probability that a lifetime exceeds a given age, as a function of age. It is non-increasing, equals one at age zero and relates directly to the distribution of the future lifetime.

The instantaneous rate of dying at a given age conditional on survival to that age. It equals the negative derivative of the log survival function and is the hazard function in survival analysis.

A non-parametric estimate of the survival function that updates at each observed death time, allowing for individuals censored before death. It is also called the product-limit estimator.

A situation where the exact time of an event is not observed, only that it occurred after, or sometimes before, a known time. Standard survival methods adjust for censored observations.

The total time lived by a group of lives within a specified age interval and observation period. It is the denominator of the mortality rate estimated from that interval.

The process of smoothing crude mortality rates into a smooth curve or table, while retaining the underlying pattern. It reduces sampling noise and produces a usable set of rates.

A reference mortality table used as a baseline when comparing or graduating the experience of a particular population. Comparisons are often expressed as ratios or through adjusted ages.

The estimation of how mortality rates will change in future years, typically by extrapolating past improvements. Projections are needed for pricing and reserving long-term life and pension business.