An Improved Transformation-Based Kernel Estimator of Densities on the Unit Interval

Abstract

The kernel density estimator (KDE) suffers boundary biases when applied to densities on bounded supports, which are assumed to be the unit interval. Transformations mapping the unit interval to the real line can be used to remove boundary biases. However, this approach may induce erratic tail behaviors when the estimated density of transformed data is transformed back to its original scale. We propose a modified, transformation-based KDE that employs a tapered and tilted back-transformation. We derive the theoretical properties of the new estimator and show that it asymptotically dominates the naive transformation based estimator while maintains its simplicity. We then propose three automatic methods of smoothing parameter selection. Our Monte Carlo simulations demonstrate the good finite sample performance of the proposed estimator, especially for densities with poles near the boundaries. An example with real data is provided.

• Bandwidth selection
• Boundary bias
• Bounded support
• Kernel density estimation
• Transformation

Additional information

Notes on contributors

Kuangyu Wen

Kuangyu Wen, International School of Economics and Management, Capital University of Economics and Business, Beijing, 100070, PR China (E-mail: [email protected] ).

Ximing Wu

Ximing Wu, Department of Agricultural Economics, Texas A&M University, College Station, TX, 77843 (E-mail: [email protected] ). Ximing Wu is also affiliated to the School of Economics, Xiamen University, China.