Accurate approximation to the extreme order statistics of Gaussian samples

Evaluation of the integral properties of Gaussian Statistics is problematic because the Gaussian function is not analytically integrable. We show that the expected value of the greatest order statistics in Gaussian samples (the max distribution) can be accurately approximated by the expression Phi(-1)(0.5264(1/n)), where n is the sample size and Phi(-1) is the inverse of the Gaussian cumulative distribution function. The expected value of the least order statistics in Gaussian samples (the min distribution) is correspondingly approximated by -Phi(-1)(0.5264(1/n)). The standard deviation of both extreme order distributions can be approximated by the expression 0.5[Phi(-1)(0.8832(1/n)) - Phi(-1)(0.2142(1/n))]. We also show that the probability density function of the extreme order distribution can be well approximated by gamma distributions with appropriate parameters. These approximations are accurate, computationally efficient, and readily implemented by build-in functions in many commercial mathematical software packages such as MATLAB, Mathematica, and Excel.