Backfitting versus Profiling in General Criterion Functions

We study the backfitting and profile methods for general criterion functions that depend on a parameter of interest β and a nuisance function θ. We show that when different amounts of smoothing are employed for each method to estimate the function θ, the two estimation procedures produce asymptotically the same estimator of β, even when the criterion functions are non-smooth in β and/or θ. The results are applied to a partial linear median regression model and a change point model, both examples of non-smooth criterion functions.