Statistica Sinica 28 (2018), 2289-2307

WAVELET METHODS FOR ERRATIC REGRESSION

MEANS IN THE PRESENCE OF MEASUREMENT ERROR

Peter Hall ^{1}, Spiridon Penev ^{2} and Jason Tran ^{1}

Abstract: In nonparametric regression with errors in the explanatory variable, the regression function is typically assumed to be smooth, and in particular not to have a rapidly changing derivative. Not all data applications have this property. When the property fails, conventional techniques, usually based on kernel methods, have unsatisfactory performance. We suggest an adaptive, wavelet-based approach, founded on the concept of explained sum of squares, and using matrix regularisation to reduce noise. This non-standard technique is used because conventional wavelet methods fail to estimate wavelet coefficients consistently in the presence of measurement error. We assume that the measurement error distribution is known. Our approach enjoys very good performance, especially when the regression function is erratic. Pronounced maxima and minima are recovered more accurately than when using conventional methods that tend to flatten peaks and troughs. We also show that wavelet techniques have advantages when estimating conventional, smooth functions since they require less sophisticated smoothing parameter choice. That problem is particularly challenging in the setting of measurement error. A data example is discussed and a simulation study is presented.

Key words and phrases: Chirp, cross-validation, deconvolution, discontinuity, errors in variables, error sum of squares, explained sum of squares, kernel methods.