The Properties of mode prediction using mean root error for regularization

While it is popular, estimating empirical distribution from observed data using MSE (Mean Squared Error) is often inefficient because it focuses on expectation. To address this problem, here we invest a new type of error term, named MRE (Mean Root Error). Different from MSE, MRE can predict the local mode point rather than the expectation. From numerical studies, we show that MRE models shows more robust and accurate prediction performance, which will be useful for complicated data such as finance data.