Detecting influential observations in single-index models with metric-valued response objects

Regression with random data objects is becoming increasingly common in modern data analysis. Unfortunately, like the traditional regression setting with Euclidean data, random response regression is not immune to the trouble caused by unusual observations. A metric Cook's distance extending the classical Cook's distances of Cook (1977) to general metric-valued response objects is proposed. The performance of the metric Cook's distance in both Euclidean and non-Euclidean response regression with Euclidean predictors is demonstrated in an extensive experimental study. A real data analysis of county-level COVID-19 transmission in the United States also illustrates the usefulness of this method in practice.