Lattice polynomials of random variables

In many statistics and reliability theory models the object of interest is a random variable obtained from others by minimum and maximum operations. As a generalization, a random variable Y defined as a lattice polynomial of random arguments was introduced in Marichal [2006. Cumulative distribution function and moments of lattice polynomials. Statist. Probab. Lett. 76(12), 1273-1279] and studied in case of independent identically distributed arguments. Here, the cumulative distribution function of Y (in particular, order statistic) is studied for generally dependent arguments and special cases. A relation (presented in [Marichal, 2006. Cumulative distribution function and moments of lattice polynomials. Statist. Probab. Lett. 76(12), 1273-1279]) between Y and order statistics is proved to hold if and only if the arguments possess “cardinality symmetry”.