On estimating the score function and the choice of the smoothing parameter with applications to adaptive estimations

Estimation of the derivative of the log density, or score, function is central to much of recent work on adaptive estimation of econometric models. Most existing score function estimation methods approach the problem by differentiating the logarithm of an estimated density function, such as the kernel estimate. Cox (1985) proposed a direct method of estimating score functions using smoothing spline techniques. Under mild regularity conditions, Cox's estimate is consistent and achieves the optimal rate of convergence. This approach is appealing not only because it is based directly on penalized likelihood methods for the score function rather than some other related quantity.