Skyline query has been a research issue attracting much attention in recent years. However, the need of dealing with attributes of unquantifiable values in such a query has not been noticed so far. These attributes of unquantifiable values (or unquantifiable attribute in short) usually contain important information that is unignorable in query processing. In this paper, we propose the notion of depth-k skyline query to address this issue. We specifically study this issue in a distributed system environment as it is the most common environment we are facing today. We propose two sifters to accelerate the query processing. The neural network technology is employed in the sifter, which significantly reduces the cost of the query processing. Extensive simulations demonstrate both the effectiveness and the efficiency of the proposed technique.