Conformal prediction for multivariate responses with Euclidean likelihood

Multivariate response analysis offers a more comprehensive understanding of the phenomena being studied than univariate response analysis. Despite the widespread popularity of conformal inference for a univariate response, relatively little research has been conducted on its application to multivariate responses. In this paper, we propose a novel conformal prediction method for multivariate response by taking the Euclidean likelihood ratio test statistic for a multivariate mean as a non-conformity score. To make full use of data information, we propose to calibrate the non-conformity score using the Jackknife method or a re-sampling technique in the absence and presence of covariate shift. Our approach can flexibly integrate pre-trained statistical or machine learning models and auxiliary information defined through estimating equations. Asymptotic coverage guarantees are established for the proposed conformal prediction regions. Our simulation and real analysis indicate that compared with the existing competitors, the proposed conformal prediction regions usually have desirable coverage probabilities with smaller volumes.