Cornish–Fisher Expansions

Introduction

In statistical inference it is of fundamental importance to obtain the sampling distribution of statistics. However, we often encounter situations where the exact distribution cannot be obtained in closed form, or even if it is obtained, it might be of little use because of its complexity. One practical way of getting around the problem is to provide reasonable approximations of the distribution function and its quantiles, along with extra information on their possible errors. This can be accomplished with the help of Edgeworth and Cornish–Fisher expansions. Recently, interest in Cornish–Fisher expansions has increased because of intensive study of VaR (Value at Risk) models in financial mathematics and financial risk management (see Jaschke (2002)).

Expansion Formulas

Let X be a univariate random variable with a continuous distribution function F. For α : 0 < α < 1, there exists x such that F(x) = α, which is called the (lower) 100α%point of F. If Fis strictly...