Remember the Curse of Dimensionality: The Case of Goodness-of-Fit Testing in Arbitrary Dimension

Despite a substantial literature on nonparametric two-sample goodness-of-fit testing in arbitrary dimensions spanning decades, there is no mention there of any curse of dimensionality. Only more recently Ramdas et al. (2015) have discussed this issue in the context of kernel methods by showing that their performance degrades with the dimension even when the underlying distributions are isotropic Gaussians. We take a minimax perspective and follow in the footsteps of Ingster (1987) to derive the minimax rate in arbitrary dimension when the discrepancy is measured in the L2 metric. That rate is revealed to be nonparametric and exhibit a prototypical curse of dimensionality. We further extend Ingster's work to show that the chi-squared test achieves the minimax rate. Moreover, we show that the test can be made to work when the distributions have support of low intrinsic dimension. Finally, inspired by Ingster (2000), we consider a multiscale version of the chi-square test which can adapt to unknown smoothness and/or unknown intrinsic dimensionality without much loss in power.Comment: This version comes after the publication of the paper in the Journal of Nonparametric Statistics. The main change is to cite the work of Ramdas et al. Some very minor typos were also correcte

A General Method for Calibrating Stochastic Radio Channel Models with Kernels

Publisher Copyright: CCBYCalibrating stochastic radio channel models to new measurement data is challenging when the likelihood function is intractable. The standard approach to this problem involves sophisticated algorithms for extraction and clustering of multipath components, following which, point estimates of the model parameters can be obtained using specialized estimators. We propose a likelihood-free calibration method using approximate Bayesian computation. The method is based on the maximum mean discrepancy, which is a notion of distance between probability distributions. Our method not only by-passes the need to implement any high-resolution or clustering algorithm, but is also automatic in that it does not require any additional input or manual pre-processing from the user. It also has the advantage of returning an entire posterior distribution on the value of the parameters, rather than a simple point estimate. We evaluate the performance of the proposed method by fitting two different stochastic channel models, namely the Saleh-Valenzuela model and the propagation graph model, to both simulated and measured data. The proposed method is able to estimate the parameters of both the models accurately in simulations, as well as when applied to 60 GHz indoor measurement data.Peer reviewe