IAI CS1 flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Survival function S(x)

Probability that a newborn survives beyond age x; equals 1 minus the distribution function, is non-increasing and right-continuous, and satisfies S(0)=1 and S(infinity)=0.

Force of mortality μ(x)

Instantaneous hazard rate at age x, equal to f(x)/S(x) and to minus the derivative of log S(x); it fully determines the survival distribution.

Complete expectation of life

Expected future lifetime e-dot-x, computed as the integral from zero to infinity of t-year survival probabilities; it exceeds the curtate expectation by roughly half a year.

Curtate future lifetime K

Integer part of the future lifetime random variable; its probability mass at k equals k-year survival times the one-year death probability at age x plus k.

Uniform distribution of deaths

Assumption that deaths are spread evenly within each year of age; it implies t-year death probability equals t times the annual death probability for t between zero and one.

Constant force of mortality

Assumption that the hazard rate is a fixed value mu over an interval; survival over t years then equals the exponential of minus mu times t.

Whole life insurance Āx

Expected present value of a unit paid at the moment of death; under constant force mu and force of interest delta it simplifies to mu divided by mu plus delta.

Whole life annuity äx

Expected present value of a unit paid annually in advance while the life survives; it relates to the insurance value by the identity one equals d times annuity plus insurance.

Discount factor v

Present value of one unit payable one year hence, equal to the exponential of minus the force of interest; the discount rate d equals one minus v.

Net premium equivalence principle

Premium is set so expected present value of premiums equals expected present value of benefits; for whole life this gives premium equal to insurance value divided by annuity value.

Gross premium with expenses

Premium including expense loadings, found by equating expected present values of premiums with benefits plus initial, renewal and claim expenses, often expressed as percentages.

Prospective reserve

Expected present value of future benefits minus future net premiums at duration t, conditional on survival to that duration; it equals the accumulated past premiums minus past benefits.

Thiele's differential equation

Ordinary differential equation describing how the reserve changes continuously, involving the force of interest, the premium rate and the force of mortality times the sum at risk.

Select mortality

Mortality that depends on duration since underwriting as well as attained age; select rates are lower than ultimate rates, which apply after the select period ends.

Life table radix

Conventional number of survivors at age zero, typically one hundred thousand; all life table functions such as deaths and exposure are scaled relative to it.

Central death rate mx

Deaths in an age interval divided by average population exposed to risk; under uniform deaths it approximates the force of mortality at the middle of the interval.

Gompertz law

Model in which the force of mortality grows exponentially with age; it produces survival curves of a double exponential form and is used for old-age mortality graduation.

Makeham law

Extension of Gompertz adding a constant term to the force of mortality, capturing age-independent background mortality in addition to exponential ageing.

Kaplan-Meier estimator

Nonparametric estimator of the survival function for censored data; it multiplies one minus the death fraction across successive event times, treating censored observations as removed.

Cox proportional hazards model

Semi-parametric regression model where the hazard is a baseline hazard multiplied by an exponential linear predictor; covariate effects are constant hazard ratios over time.

Exponential family form

Density written as exponential of a term involving the natural parameter times a statistic, minus a cumulant function, plus a scale term; it underpins generalised linear models.

Link and variance functions

In a generalised linear model the link maps the mean to the linear predictor, while the variance function specifies how variance depends on the mean.

Maximum likelihood estimator

Parameter value maximising the likelihood; it solves the score equation, is asymptotically normal, and its variance follows from the inverse observed or expected information.

Cramer-Rao lower bound

Lower bound on the variance of any unbiased estimator, given by the reciprocal of the expected Fisher information for a single parameter.

Sufficient statistic

Statistic containing all information in the sample about the parameter; by the factorisation theorem the likelihood splits into a part depending on it and a part free of the parameter.

Method of moments

Estimation by equating sample moments to their theoretical counterparts and solving the resulting equations for the parameters; simple but often less efficient than maximum likelihood.

Delta method

Approximation of the variance of a smooth function of an estimator using its derivative squared times the estimator variance; used to build approximate confidence intervals.

Likelihood ratio test

Test comparing maximised log-likelihoods of nested models; twice the log-likelihood difference is approximately chi-squared with degrees of freedom equal to the parameter restriction count.

Wald and score tests

Wald test uses the standardised estimate; score test uses the derivative of the log-likelihood at the null. Both are asymptotically chi-squared and equivalent to the likelihood ratio test.

Poisson regression for mortality

Model with log link and variance proportional to the mean, applied to death counts with exposure offsets; coefficients give log mortality rate ratios.

Binomial model for mortality

Model treating deaths as binomial with the number exposed as trials and the death probability as parameter; the logit link is commonly used for mortality rates.

Graduation of mortality rates

Smoothing of crude rates to produce a smooth, plausible mortality curve; assessed by goodness of fit, smoothness and adherence to underlying mortality patterns.