Convergence of Estimated Roughness by the Shift Residual Method

Abstract

The roughness of a function to the given design is introduced. The conditions to ensure the convergence of the roughness of the model functions associated with designs to the roughness of the limit model function are indicated. Since then, the paper confirms the almost sure convergence of the roughness estimated by the method of shift residual to the roughness of the theoretical model function. The conditions to ensure convergence is quite extensive and easy to meet in real data analysis. Simulation studies show the appropriateness of the theoretical conclusions. The quality of the estimate is satisfactory even when the number of observations is relatively small, the roughness of the original model function is not very large, but the variance of the noise needs to be small. As the number of observations increases, the roughness of the original model may decrease, and the variance of the noise may not need to be very small, but the obtained estimate can still be quite satisfactory