CovhatXY

CovhatXY is a statistical estimator for the covariance between two random variables X and Y. It is designed to be robust to outliers and to perform well under distributions that depart from normality, as well as in small samples.

Definition and method: Unlike the classical sample covariance, CovhatXY combines robust estimates of central tendency and dispersion for each variable with a robust cross-term. In practice, it uses robust location measures (such as medians) and robust scale measures (such as MAD), followed by a cross-term computed in a way that reduces influence from extreme observations. The resulting estimate aims to minimize sensitivity to outliers while preserving meaningful scale and sign.

Variants and computation: Several variants exist, differing in the choice of robust preliminaries and weighting scheme. Algorithms are typically designed to be computationally efficient and can be implemented in standard statistical software with options for tuning to data quality and sample size.

Properties and interpretation: CovhatXY is translation- and scale-equivariant and can be consistent under certain regularity conditions. It provides a measure of linear association that is less sensitive to outliers than the classical covariance but does not imply causal relationships. As with other robust estimators, its efficiency relative to the classical case depends on the underlying data distribution.

Applications: In finance, CovhatXY is used to estimate co-movement between assets when returns exhibit heavy tails or jumps. In genomics, it assists in detecting co-expression under noisy measurements. In engineering and signal processing, it supports cross-correlation analysis in sensor data with outliers or artifacts.

Limitations and reception: While robust in many settings, CovhatXY can incur efficiency loss under ideal Gaussian data and may require tuning of parameters for optimal performance. It is typically presented as a supplement to, rather than a replacement for, traditional covariance in statistical workflows.