Prediction in a trivariate normal distribution via a linear combination of order statistics

In this paper, by considering a trivariate normal distribution, we derive the exact joint distribution of one variable and a linear combination of order statistics from the other two variables. We show that this joint distribution is a mixture of unified bivariate skew-normal distributions. This mixture form enables us to predict the variable based on a linear combination of order statistics from the other two variables. We finally illustrate the usefulness of these results by using a real-life data.