Bootstrap confidence intervals for conditional density function in Markov processes

Abstract

Bootstrap confidence intervals for the conditional density function in Markov processes are established. For this purpose, recent bootstrap algorithms constructed for prediction intervals are reviewed and extended to this context. These methods are: model-free bootstrap, predictive model-free, limit model-free, bootstrap for nonparametric autoregressive models with predictive and fitted residuals, local bootstrap and bootstrap based on estimates of the transition density. However, in order to achieve a good coverage probability, the choice of an appropriate smoothing parameter for conditional density estimation turns out to be of utmost importance. In this sense, cross validation and plug-in bandwidth parameter selectors are considered, as well as a deterministic one. An extensive simulation study is carried out to show the empirical behavior of these methods and to compare them. Finally, the methods are illustrated by applying them to a real data set.

Keywords:

• Model-free bootstrap
• kernel method
• local bootstrap
• bootstrap for autoregressive models

Acknowledgments

The first two authors acknowledge partial support by MINECO Grants MTM2014-52876-R and MTM2017-82724-R (EU ERDF support included). Additionally, financial support from the Xunta de Galicia (Centro Singular de Investigación de Galicia accreditation ED431G/01 2016-2019 and Grupos de Referencia Competitiva ED431C2016-015) and the European Union (European Regional Development Fund - ERDF), is gratefully acknowledged. The first author aknowledges financial support from the Xunta de Galicia and the European Union (European Social Fund - ESF), the reference of which is ED481A-2017/215. Additionally, the work of the first author has been carried out during a visit at the University of California, San Diego, financed by INDITEX, with reference INDITEX-UDC 2017.