2017/02/02 by Anil Aswani, Aswani, Anil · 1 citation
Decision Sciences · Mathematics · #Advanced Statistical Methods and Models #FOS: Mathematics #Fuzzy Systems and Optimization #Optimization and Control (math.OC) #Probabilistic and Robust Engineering Design #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1702.00708
openalex publication_date 2017/02/02 · openalex created_date 2022/09/30 · openalex updated_date 2026/07/28
Much of statistics relies upon four key elements: a law of large numbers, a\ncalculus to operationalize stochastic convergence, a central limit theorem, and\na framework for constructing local approximations. These elements are\nwell-understood for objects in a vector space (e.g., points or functions);\nhowever, much statistical theory does not directly translate to sets because\nthey do not form a vector space. Building on probability theory for random\nsets, this paper uses variational analysis to develop operational tools for\nstatistics with set-valued functions. These tools are first applied to\nnonparametric estimation (kernel regression of set-valued functions). The\nsecond application is to the problem of inverse approximate optimization, in\nwhich approximate solutions (corrupted by noise) to an optimization problem are\nobserved and then used to estimate the amount of suboptimality of the solutions\nand the parameters of the optimization problem that generated the solutions. We\nshow that previous approaches to this problem are statistically inconsistent\nwhen the data is corrupted by noise, whereas our approach is consistent under\nmild conditions.\n