In this paper, the problem of determining the maximum of independent but non-identically distributed (i.n.i.d.) double α-μ random variables (RVs) is investigated. In particular, we present a novel analysis to calculate the moments of the maximum of i.n.i.d. double α-μ RVs. To this end, an exact analytical expression for the n-th moment is first derived, which is then used to study the performance of selection combining receivers in terms of channel quality estimation index, average bit error rate, and average (ergodic) channel capacity. The derived analytical results are compared with Monte-Carlo simulations, which validates the analysis and provides useful insights on the influence of fading and shadowing on the system performance.