A corrected normal approximation for the Student’s t distribution

Abstract

For the Student’s t cumulative distribution function F( x; n) with n≥3 degrees of freedom, a corrected normal approximation, Φ( λx), is proposed as an extension of the well-known ordinary normal approximation Φ( x), where Φ( x) is the standard normal cumulative distribution and λ= λ( x, n) is a shrinking factor (0< λ<1). This approximation has a theoretical error O(1/ n 2) uniformly in x. Numerical results show that it can give satisfactory accuracy for even very small n. Thus, it provides a competitive alternative because of its reasonable balance between accuracy and simplicity.