Grunnleggende konsepter
A scalable and provably valid test for determining whether a multivariate distribution is log-concave, using a combination of universal inference and random projections.
Sammendrag
The key insights and highlights of the content are:
Log-concavity is an important shape constraint in density estimation, with applications across economics, survival modeling, and reliability theory. However, there has not been a valid test for log-concavity in finite samples and arbitrary dimensions.
The authors develop a universal likelihood ratio test (LRT) approach that provides a valid test for log-concavity, controlling the type I error rate in finite samples. This universal LRT approach can be applied to any class of models, including the class of log-concave densities.
To address the curse of dimensionality in computing the high-dimensional log-concave maximum likelihood estimate (MLE), the authors propose a random projections approach. This converts the d-dimensional testing problem into many one-dimensional problems, where the log-concave MLE is computationally efficient to compute.
Simulations show that the random projections approach combined with the universal LRT outperforms existing methods, maintaining high power even as the dimension increases, while provably controlling the type I error rate.
The authors also provide theoretical results on the power of the universal LRT for testing log-concavity, showing that the power only exhibits a moderate curse of dimensionality, in contrast to the full oracle approach.
Overall, the paper develops a scalable and statistically efficient method for testing log-concavity in high dimensions, with strong theoretical guarantees.