Sava\c{s} [Generalized asymptotically $I$-lacunary equivalent of order $\alpha$ for sequences of sets, Filomat 31 (2017), 1507--1514] studied generalized asymptotically $I$-lacunary equivalent of order $\alpha$ for a sequence of sets. This article is completely based on a double sequence of sets by way of $n$-normed spaces. We firstly contrived an Orlicz extension of asymptotically Wijsman equivalence and asymptotically Wijsman lacunary equivalence. By using the hitherto defined concept, we further elongate these notions to asymptotically Orlicz-Wijsman statistical as well as lacunary statistical equivalence. Finally, we explain the concept of ideal extension of order $\alpha$ and present some inclusion relations.