Adaptively Denoising Discrete Two-way Layouts

The two-way layout with ordinal or nominal factors is a fundamental data-type that is widespread in the sciences, engineering, and informatics. The unrestricted least squares estimator for the means of a two-way layout is usually inadmissible under quadratic loss and the model of homoscedastic independent Gaussian errors. In statistical practice, this least squares estimator may be modified by fitting hierarchical submodels and, for ordinal factors, by fitting polynomial submodels. ASP, an acronym for Adapative Shrinkage on Penalty bases, is an estimation (or denoising) strategy that chooses among submodel fits and more general shrinkage or smoothing fits to two-way layouts without assuming that any submodel is true. ASP fits distinguish between ordinal and nominal factors; respect the hierarchical decomposition of means into overall mean, main effects, and interaction terms; and are designed to reduce risk substantially over the unrestricted least squares estimator. For the balanced complete two-way layout, these points are developed through multiparametric asymptotics, in which the number of factor-level pairs tends to infinity, and through numerical case studies.