Tabular Survival Models – Estimates from complete data samples, Estimates from incomplete data samples (sample design, moments procedures, maximum likelihood procedures). Parametric Survival Models.
Net Premiums – Whole Life and Term Insurance, Endowments, Deferred Life Annuities, Premiums Paid m Times a Year, Policies with Premium Refund.
Net Premium Reserves – The Survival Risk, Net Reserve Premium for Whole Life, Net Premiums Reserve at Fractional Duration, Allocation of the Overall Loss to Policy Years,
Technical Gain, Procedure for Endowment and The Continuous Model. Multiple Decrements – An Introduction. Multiple Life Insurance.
Basic Mathematics of Life Contingencies, Effective and Nominal interest rates. The Future Lifetime of a Life Aged x (T) – Model, Force of Mortality, Analytic Distribution of T,
Curtate Future Lifetime of x, Life Tables, Probabilities of Death for Fractions of a Year.
Life Insurance – Whole Life and Term Insurances, Endowment, Insurance Payable at the Moment of Death, General Type of Life Insurance and Variable Life Insurance.
Life Annuities – Elementary, Payment made more Frequently than Once a Year, Variable Life Annuities, Standard Type of Life Annuities, Payment Starting at Non-integral Ages
Yield curves, spot rates, forward rates, duration, convexity, and immunization. Derivatives, forwards, futures, short and long positions, cal and put options, spread, collars, hedging, arbitrage, and swaps.
Inflation; rates of interest [simple, compound (interest and discount), real, nominal, effective, dollar-weighted, time-weighted, spot, forward], term structure of interest rates; force of interest (constant and varying); equivalent measures of interest; yield rate; principal; equation of value; present value; future value; current value; net present value; accumulation function; discount function; annuity certain (immediate and due); perpetuity (immediate and due); stocks ( common and preferred); bonds (including zero-coupon bonds); other financial instruments such as mutual funds, and guaranteed investment contracts. Determining equivalent measures of interest; discounting; accumulating; determining yield rates; estimation the rate of return on a fund; and amortization.
Survival function, hazard functions, cumulative hazard function, censoring. Kaplan-Meier survival curve, parametric models. Comparison of two groups – log-ranked test.
Inclusion of covariates – Cox P.H. model, application of model checking. Competing risks – extensions of Cox’s model.
Organisation and Planning: Protocol, patient selection, response Justification of method for randomisation: Uncontrolled trials, blind trials, Placebo’s, ethical issues. The size of a clinical trial: Maintaining trials progress: Forms and data management, protocol deviations. Methods of data analysis: Binary responses, cross-over trials, survival data prognostic factors. Testing Hypothesis, Statistical Models: Inferential statistics-creating statistical hypothesis, the Z-test; designing a single variable experiment; errors in statistical decision making. Power of test used in clinical trials/maximizing tests power. Significance testing: t-test. Conducting two-way experiments and trials. Interpreting overall results of Clinical Trials. Nonparametric Procedures/Tests & Ranked Data: , Mann-Whitney U test, Kruskal-Willis, Friedman.
Stationary and non-stationary series: removal of trend and seasonality by differencing. Moments and auto-correlation. Models: simple AR and MA models (mainly AR(1), MA(1)): moments and auto-correlations; the conditions of stationarity: invertibility. Mixed (ARMA) models, and the AR representation of MA and ARMA models. Yule-Walker equations and partial auto-correlations (showing forms for simple AR, MA models). Examples showing simulated series from such processes, and sample auto-correlations and partial auto-correlations. (Other models, e.g., trend and seasonal). Model identification: Elementary ideas of identification of models based on simple acf and pacf showing difficulties with real series. Estimation of parameter: initial estimate based on sample acf and pacf only (least squares estimates by iterative method). Result for standard error of sample acf, pacf and estimators. Forecasting: use of the AR representation for forecasting. Minimum mean square error forecasts. Updating.
Formulating Linear Programming Models: Goal programming, Transportation problem, Case study. Mathematical Programming: Project planning and control, Dynamic programming,
Integer programming. Probabilistic Models: Application of queuing theory, Forecasting and simulation, Decision analysis (making hard decisions), Multi-criteria decision making.
Multivariate data summary and graphical displays. Multivariate normal distributions: Estimation of mean and covariance, one- and two-sample problems, analysis of variance.
Reduction of dimensionality: principal components and factor analyses. Discrimination and classification. Correlation; partial, multiple and canonical. Non-metric problems: clustering and scaling.
