Operational Risk toward Basel III: Best Practices and Issues in Modeling, Management, and Regulation

CHAPTER 9

One-Sided Cross-Validation for Density Estimation with an Application to Operational Risk

Mar’a Dolores Mart’nez Miranda, Jens Perch Nielsen, and Stefan A. Sperlich

 

 

ABSTRACT

We introduce one-sided cross-validation to nonparametric kernel density estimation. The method is more stable than classical cross-validation, and it has a better overall performance compared to what we see in plug-in methods. One-sided cross-validation is a more direct data-driven method than plug-in methods with weaker assumptions of smoothness since it does not require a smooth pilot with consistent second derivatives. Our conclusions for one-sided kernel density cross-validation are similar to the conclusions obtained by Hart and Yi (1998) when they introduced one-sided cross-validation in the regression context, except that in our context of density estimation, the superiority of this new method is even much stronger. An extensive simulation study confirms that our one-sided cross-validation clearly outperforms the simple cross-validation. We conclude with real data applications.

This research was financially supported by knowledge company Festina Lente and the Direcci’n General de Investigaci’n del Ministerio de Ciencia y Tecnolog’a under research grant SEJ2004-04583/ECON.

9.1 INTRODUCTION

Suppose we have observed data X₁, X₂, . . . , Xn that are assumed to be independent and identically distributed with common density function, f (·). We want to estimate this common density nonparametrically ...