Inference and Computation for Sparsely Sampled Random Surfaces

TL;DR

This paper introduces a new covariance estimator for sparsely observed 2D surfaces that improves inference accuracy and computational efficiency, especially when traditional separability assumptions are relaxed.

Contribution

It proposes a local linear smoother-based covariance estimator for sparse 2D functional data, maintaining optimal convergence rates and enabling effective regularization.

Findings

Estimator achieves minimax-optimal convergence rates.

Simulation shows favorable bias-variance trade-off and speed-up.

Application to implied volatility surfaces improves out-of-sample prediction.

Abstract

Non-parametric inference for functional data over two-dimensional domains entails additional computational and statistical challenges, compared to the one-dimensional case. Separability of the covariance is commonly assumed to address these issues in the densely observed regime. Instead, we consider the sparse regime, where the latent surfaces are observed only at few irregular locations with additive measurement error, and propose an estimator of covariance based on local linear smoothers. Consequently, the assumption of separability reduces the intrinsically four-dimensional smoothing problem into several two-dimensional smoothers and allows the proposed estimator to retain the classical minimax-optimal convergence rate for two-dimensional smoothers. Even when separability fails to hold, imposing it can be still advantageous as a form of regularization. A simulation study reveals a favorable bias-variance trade-off and massive speed-ups achieved by our approach. Finally, the proposed methodology is used for qualitative analysis of implied volatility surfaces corresponding to call options, and for prediction of the latent surfaces based on information from the entire data set, allowing for uncertainty quantification. Our cross-validated out-of-sample quantitative results show that the proposed methodology outperforms the common approach of pre-smoothing every implied volatility surface separately.