Description

In this course, we will study three important asymptotic expansions that improve upon the normal distribution approximation provided by the central limit theorem - the Bahadur, Edgeworth, and Cornish-Fisher expansions. A key focus will be using these expansions for estimating quantiles, which are widely used on risk measures.

The central limit theorem shows the distribution of sample statistics converges to normal, but the approximation can be poor for small samples or non-normal distributions. The expansions we study address this by incorporating higher order terms and cumulants.

The Bahadur expansion provides an approximation for sample quantiles using an asymptotic expansion incorporating higher order terms related to sample size. The Edgeworth expansion expands the cumulative distribution function itself using cumulants. The Cornish-Fisher expansion directly approximates the quantile function by adjusting the normal quantiles based on skewness and kurtosis.

In this course, we will derivate, compare, and apply these expansions for better understanding the quantile estimation. Students will have strong conceptual and practical skills for using asymptotic theory. The techniques will be broadly applicable across statistics and data science.