Expected Values of Order Statistics in Finite Samples from Normal Mixture Distributions

In the real life, order statistics from mixture distributions arise naturally, for example, the extreme wind speed during changes of seasons, the highest water level at the confluence of two rivers, and SAT Math scores from a racially mixed population. However, their properties are still not well-studied in the literature due to the complexity of the statistics. In this paper, we consider order statistics in finite samples from the mixtures of two normal distributions. The exact expected values of order statistics in a random sample of size less than or equal to four are carried out and expressed in terms of the error function, Owen’s T function as well as Steck’s S function. For the random sample of size equal to five, closed forms of the expected values, however, are not available for the time being, because some of the related normal integrals still remain unsolved. On the other hand, for large sample size, the asymptotic result about the expected values of maximum order statistics from normal mixture distributions is discussed. High-order moments of the order statistics are briefly discussed.