Simple General Approximations for a Random Variable and its Inverse Distribution Function Based on Linear Transformations of a Nonskewed Variate

Abstract

Linear transformations of a nonskewed random variable are employed to derive simple general approximations for a random variable having known cumulants. Introducing the unit normal variate, these become linear normal approximations. Some nonskewed variates with explicit inverse cumulative density function are then used to derive general approximations for the inverse DF of the approximated variable. The approximations are applied to the binomial, Poisson, Fisher’s z and F, gamma (chi-square in particular) and the t distributions and their accuracy examined. Simple general approximations for the loss function of a random variable either continuous or discrete are developed. A simple approximation for the loss function of the Poisson distribution is then derived and demonstrated by an example from inventory analysis. Two further examples from interval estimation and from hypothesis testing highlight the usefulness of the new approximations.