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Given noisy data observed at certain intervals, the aim is to approximate the data by a function without restricting ourselves to functions from a small family like linear or polynomial models. Smoothness or simplicity assumptions are made instead.
Many methods have been suggested and studied, the most popular ones are kernel estimators, spline smoothing, local polynomial regression and wavelet thresholding.
Local extreme values play an important role in many applications of nonparametric statistics because their positions have often meaningful interpretations.
So recent methods based on minimising total variation, like the taut string method, try to fit the data with a function that contains local extreme values only at positions where indicated by the data.
Practical applications of nonparametric regression include image decompression and signal cleaning, and general problems of dealing with missing data.