Statistical Inference and Regression
Domain 2: Statistical Inference and Regression
CB2 tests fitting probability models to data, estimating parameters, testing hypotheses and using regression or time-series models in an actuarial context. You must interpret output, not merely compute it.
Estimation
Maximum likelihood estimation chooses parameters that make the observed data most likely. Method of moments equates sample moments to theoretical moments. Report estimates with standard errors and confidence intervals; an estimate without uncertainty is incomplete.
Hypothesis testing
Set a null and alternative hypothesis before seeing the data. The p-value is the probability, assuming the null is true, of a result at least as extreme as observed; it is not the probability that the null is true. A significance level controls Type I error (rejecting a true null); power is the chance of rejecting a false null. Type II error is failing to reject a false null.
Regression and time series
Linear regression assumes a linear mean, independent errors, constant variance and no severe collinearity. A GLM extends this with a link function: Poisson for counts, logistic for binary outcomes. In time series, separate trend, seasonality and irregular components, and check autocorrelation because it invalidates independence assumptions.
Worked example
Claim sizes are modelled as exponential with mean θ. A random sample of 100 claims has mean £200. The exponential MLE is the sample mean, so θ̂ = 200. The variance of an exponential is θ², so the standard error of θ̂ is θ/√n = 200/10 = 20. An approximate 95% confidence interval is 200 ± 1.96 × 20, giving £160.80 to £239.20. The wide interval reflects the heavy tail and the modest sample size; a larger sample would narrow it.
Traps
- Saying a p-value is the probability that the null hypothesis is true.
- Ignoring overdispersion when fitting a Poisson GLM to claim counts.
- Extrapolating a regression far beyond the observed data range.
- Fitting a time-series model without checking residual autocorrelation.
- Reporting a point estimate without a standard error or interval.