Assessing bandwidth selectors with visual error criteria
The kernel density estimator \(\widehat f_{h}(x)\) is defined by \(\widehat f_{h}(x)= n^{-1}\sum_{i=1}^{n}K_{h}(x-x_{i})\), where \(x_{1}, \ldots,x_{n}\) is a random sample from the probability density \(f(x)\); \(K_{h}(\cdot)=K({\cdot/h})\), \(K(x)\) is a Gaussian probability density, and \(h\) is a bandwidth. In this paper the bandwidth selection problem is studied with attention paid to the issue of how to assess the performance of various bandwidths. The visual error criterion \(VE_{2}(\widehat f_{h}\to f)\) is defined as follows: \[ VE_{2}(\widehat f_{h}\to f)=[\int_{a}^{b} d((x,\widehat f_{h}(x)),G_{f})^{2}dx]^{1/2}, \] \[ \text{where}\quad G_{f}=\{(x,y):x\in [a,b],\;y=f(x)\},\quad d((x,y),G)=\inf_{(x'y')\in G}\|(x,y)-(x',y')\|_{2}, \] where \(\|\cdot\|_{2}\) denotes the usual Euclidean distance. The author studies new properties of the visual error criterion. Different types of comparisons are made between the criteria \[ MISE(h)=E[\int_{-\infty}^{+\infty}(\widehat f_{h}(x)- f(x))^{2}dx]\quad \text{and}\quad EVE_{2}(\widehat f_{h}\to f). \] One of them is a study of their expected values as functions of \(n\) and \(h\).
scientific article; zbMATH DE number 1336642