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Many statistical methods require solutions to optimization problems. When theglobal solution is hard to attain, statisticians always use the better if thereare two solutions for chosen, where the word "better" is understood in thesense of optimization. This seems reasonable in that the better solution ismore likely to be the global solution, whose statistical properties of interestusually have been well established. From the statistical perspective, we usethe better solution because we intuitively believe the principle, called bettersolution principle (BSP) in this paper, that a better solution to a statisticaloptimization problem also has better statistical properties of interest. BSPdisplays some concordance between optimization and statistics, and is expectedto widely hold. Since theoretical study on BSP seems to be neglected bystatisticians, this paper aims to establish a framework for discussing BSP invarious statistical optimization problems. We demonstrate several simple buteffective comparison theorems as the key results of this paper, and apply themto verify BSP in commonly encountered statistical optimization problems,including maximum likelihood estimation, best subsample selection, and bestsubset regression. It can be seen that BSP for these problems holds underreasonable conditions, i.e., a better solution indeed has better statisticalproperties of interest. In addition, guided by the BSP theory, we develop a newbest subsample selection method that performs well when there are clusteredoutliers.

Better Solution Principle: A Facet of Concordance between Optimization and Statistics