Abstract
We estimate the density and its derivatives using a local polynomial approximation to the logarithm of an unknown density function f. The estimator is guaranteed to be non-negative and achieves the same optimal rate of convergence in the interior as on the boundary of the support of f. The estimator is therefore well-suited to applications in which non-negative density estimates are required, such as in semiparametric maximum likelihood estimation. In addition, we show that our estimator compares favorably with other kernel-based methods, both in terms of asymptotic performance and computational ease. Simulation results confirm that our method can perform similarly or better in finite samples compared to these alternative methods when they are used with optimal inputs, that is, an Epanechnikov kernel and optimally chosen bandwidth sequence. We provide code in several languages.
Highlights
We propose a new nonparametric estimator for of a density function and its derivatives that attain the optimal rate of convergence both in the interior and at the boundary of the support
We show that our method with the cubic choice g(t) = (t + z)(1 − t)2 attains the same asymptotic mean squared error (AMSE) in the interior and is more efficient at the boundary than the estimator in Loader (1996) with an Epanechnikov kernel
When the bandwidth sequence is optimally scaled by a multiplicative factor that depends on the roughness and second moment of the kernel, one can show that the Epanechnikov kernel minimizes the AMSE of the estimator for the density among all non-negative second-order kernels
Summary
Licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited. An advantage of using a polynomial approximation to the log-density instead of the density is that the estimated density can be guaranteed to be positive, which is not true for alternative boundary correction methods that use boundary kernels or a local polynomial approximation of f (Cheng, Fan, and Marron, 1997; Zhang and Karunamuni, 1998) or F (Lejeune and Sarda, 1992; Cattaneo, Jansson, and Ma, 2020). . Our local polynomial approximation to L can be expected to outperform alternative estimators for f in finite samples when our bias is smaller, for example, when the log-density is polynomial. Our estimator realizes this improved performance without sacrificing non-negativity of the estimated density, as would be required to achieve the same asymptotic distribution using alternative methods.. We provide code in several languages at https://github.com/kschurter/logdensity
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