統計推論生產技術之凸性性質

Abstract

生產技術（production technology）用來描述利用不同投入資源生產得到相異產出的生產過程，而生產技術的前緣（frontier）表示生產過程的最理想狀態。一般來說，生產技術通常滿足幾項基本性質，其中一項為凸性（convexity）性質，此性質因此常被用來作為使用觀測資料推估生產技術的基本假設。例如，Charnes等人在1978年提出的資料包絡分析法（data envelopment analysis, DEA）是藉由觀測收集的資料來推估生產技術，並利用此推估的生產技術前緣來衡量效率值，而其推估過程中一重要假設為資料符合凸性假設。然而，在實際應用上，受測資料卻可能因為規模經濟（economies of scale）或規模報酬遞增（increasing returns to scale）等情況而違反凸性假設，而使推估結果錯誤，導致錯誤的結論。 本研究主要探討生產技術前緣的型態特別是在規模（scale）上是否滿足凸性假設，提出一統計檢定方法利用樣本受測資料來做推論；同時，本研究將假設檢定所得的結果加以視覺化（visualize），使得由受測資料所構成的前緣可利用簡單圖形表達，提供決策者了解邊際規模變化的資訊，以協助決策之訂定。所得的結果除了做為DEA研究的事前檢驗外，更可依不同組織目的而有不同的應用，特別是在產能規劃（capacity planning）上，本研究所得的視覺化的前緣型態可做為決定產能時的重要參考。

Production technology describes a process transforming various resources to different outputs. Production frontier represents the ideal situation of the transformation process. Convexity is one of the properties that a production technology generally satisfies, and it is used as the basic assumptions in many methods estimating the underlying but unknown production technology. For example, data envelopment analysis (DEA), first introduced by Charnes et al. in 1978, computes the efficiency of a record by estimating the production technology according to a set of records. Convexity is one of the important assumptions in most well known DEA models. In reality, however, the underlying technology may not follow convexity due to economics of scale and/or increasing returns to scale. Adopting the convexity assumption in these cases may leads to biased estimations and incorrect conclusions. This study investigates the shape of a production technology, particularly in scale. A statistical procedure is proposed to estimate and inference the underlying but unknown technology by a sample data set generated from itself. In addition, the inference results are visualized so that the shape of ideal technology frontier can be easily accessed for a decision makers. This work not only provides a way to examine the convexity of data for DEA studies. Understanding and visualizing frontier shape have broad applications such as decision aids for capacity planning.