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Pattern-based block motion estimation (PBME) is one of the most widely adopted compression tools in the contemporary video coding systems. However, despite that many researches have studied PBME, few have yet attempted to construct an analytical model that can explain the underneath principle and mechanism of various PBME algorithms. In this paper, we propose a statistical PBME model that consists of two components: 1) a statistical probability distribution for motion vectors and 2) the minimal number of search points (so-called weighting function) achieved by a search algorithm. We first verify the accuracy of the proposed model by checking the experimental data. Then, an application example using this model is shown. Starting from an ideal weighting function, we devise a novel genetic rhombus pattern search (GRPS) to match the design target. Simulations show that, comparing to the other popular search algorithms, GRPS reduces the average search points for more than 20% and, in the meanwhile, it maintains a similar level of coded image quality.