Accurate approximation of the expected value, standard deviation, and probability density function of extreme order statistics from Gaussian samples

We show that the expected value of the largest order statistic in Gaussian samples can be accurately approximated as $\left(0.2069\hspace{0.17em}ln\hspace{0.17em}\left(ln\hspace{0.17em}\left(n\right)\right)+0.942\right)4,$ where $n\in [2,108]$ is the sample size, while the standard deviation of the largest order statistic can be approximated as $-0.4205\arctan \left(0.5556\left[ln\left(ln\hspace{0.17em}\left(n\right)\right)-0.9148\right]\right)+0.5675.$ We also provide an approximation of the probability density function of the largest order statistic which in turn can be used to approximate its higher order moments. The proposed approximations are computationally efficient, and improve previous approximations of the mean and standard deviation given by Chen and Tyler (1999).