## Abstract

We study the problem of testing the null hypothesis that X and Y are conditionally independent given Z, where each of X, Y and Z may be functional random variables. This generalises testing the significance of X in a regression model of scalar response Y on functional regressors X and Z. We show, however, that even in the idealised setting where additionally (X, Y, Z) has a Gaussian distribution, the power of any test cannot exceed its size. Further modelling assumptions are needed and we argue that a convenient way of specifying these assumptions is based on choosing methods for regressing each of X and Y on Z. We propose a test statistic involving inner products of the resulting residuals that is simple to compute and calibrate: type I error is controlled uniformly when the in-sample prediction errors are sufficiently small. We show this requirement is met by ridge regression in functional linear model settings without requiring any eigen-spacing conditions or lower bounds on the eigenvalues of the covariance of the functional regressor. We apply our test in constructing confidence intervals for truncation points in truncated functional linear models and testing for edges in a functional graphical model for EEG data.

Originalsprog | Engelsk |
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Tidsskrift | Journal of the Royal Statistical Society. Series B: Statistical Methodology |

Vol/bind | 84 |

Udgave nummer | 5 |

Sider (fra-til) | 1821-1850 |

Antal sider | 30 |

ISSN | 1369-7412 |

DOI | |

Status | Udgivet - 2022 |

### Bibliografisk note

Funding Information:RDS was supported by an EPSRC Programme Grant EP/N031938/1 and an EPSRC First Grant EP/R013381/1. ‡

Funding Information:

Part of this work was done while ARL was at the University of Copenhagen. ARL was supported by the Cantab Capital Institute for the Mathematics of Information. †

Funding Information:

JP was supported by the Carlsberg Foundation and a research grant (18968) from the VILLUM Foundation.

Publisher Copyright:

© 2022 The Authors. Journal of the Royal Statistical Society: Series B (Statistical Methodology) published by John Wiley & Sons Ltd on behalf of Royal Statistical Society.