Efficient Density Estimation using Fejer-Type Kernel Functions

We consider a class of kernel-type density estimators with Fejér-type kernels and theoretical smoothing parameters. In theory, the estimator under consideration dominates in $\mathbb{L}_p$, $1\le p<\infty$, all other estimators from the literature, in the locally asymptotic minimax sense. We demonstrate via simulations that the kernel-type estimator is good by comparing its performance to other fixed kernel estimators. We also consider two empirical bandwidth selection methods, namely, the common cross-validation and the less-known method based on the Fourier analysis of kernel density estimators. The common $\mathbb{L}_2$-risk is used to assess the quality of estimation. The kernel-type estimator is then tried to real financial data for a risk measure that is widely used in many applications. The simulation results show that the theoretical estimator under study provides very good finite sample performance and the bandwidth obtained by using the Fourier analysis techniques performs better than the one from cross-validation in most settings.