Detecting influential observations in single-index Fréchet regression

Regression with random data objects is becoming increasingly common in modern data analysis. Unfortunately, this novel regression method is not immune to the trouble caused by unusual observations. A metric Cook's distance extending the original Cook's distances of Cook (1977) to regression between metric-valued response objects and Euclidean predictors is proposed. The performance of the metric Cook's distance is demonstrated in regression across four different response spaces in an extensive experimental study. Two real data applications involving the analyses of distributions of COVID-19 transmission in the State of Texas and the analyses of the structural brain connectivity networks are provided to illustrate the utility of the proposed method in practice.