"Local polynomial regression has received extensive attention for the non-parametric estimation of regression functions when both the response and the covariate are in Euclidean space. However, little has been done when the response is in a Riemannian manifold. We develop an intrinsic local polynomial regression estimate for the analysis of symmetric positive definite matrices as responses that lie in a Riemannian manifold with covariate in Euclidean space. The primary motivation and application of the methodology proposed is in computer vision and medical imaging. We examine two commonly used metrics, including the trace metric and the log-Euclidean metric on the space of symmetric positive definite matrices. For each metric, we develop a cross-validation bandwidth selection method, derive the asymptotic bias, variance and normality of the intrinsic local constant and local linear estimators, and compare their asymptotic mean-square errors. Simulation studies are further used to compare the estimators under the two metrics and to examine their finite sample performance. We use our method to detect diagnostic differences between diffusion tensors along fibre tracts in a study of human immunodeficiency virus."

"We discuss the use of multivariate kernel smoothing methods to date manuscripts dating from the 11th to the 15th centuries, in the English county of Essex. The dataset consists of some 3300 dated and 5000 undated manuscripts, and the former are used as a training sample for imputing dates for the latter. It is assumed that two manuscripts that are ``close'', in a sense that may be defined by a vector of measures of distance for documents, will have close dates. Using this approach, statistical ideas are used to assess ``similarity'', by smoothing among distance measures, and thus to estimate dates for the 5000 undated manuscripts by reference to the dated ones."

Can we get data?

"Conditional independence testing is an important problem, especially in Bayesian network learning and causal discovery. Due to the curse of dimensionality, testing for conditional independence of continuous variables is particularly challenging. We propose a Kernel-based Conditional Independence test (KCI-test), by constructing an appropriate test statistic and deriving its asymptotic distribution under the null hypothesis of conditional independence. The proposed method is computationally efficient and easy to implement. Experimental results show that it outperforms other methods, especially when the conditioning set is large or the sample size is not very large, in which case other methods encounter difficulties."

"Recent results about the robustness of kernel methods involve the analysis of influence functions. By definition the influence function is closely related to leave-one-out criteria. In statistical learning, the latter is often used to assess the generalization of a method. In statistics, the influence function is used in a similar way to analyze the statistical efficiency of a method. Links between both worlds are explored. The influence function is related to the first term of a Taylor expansion. Higher order influence functions are calculated. A recursive relation between these terms is found characterizing the full Taylor expansion. It is shown how to evaluate influence functions at a specific sample distribution to obtain an approximation of the leave-one-out error. A specific implementation is proposed using a L1 loss in the selection of the hyperparameters and a Huber loss in the estimation procedure. The parameter in the Huber loss controlling the degree of robustness is optimized as well. The resulting procedure gives good results, even when outliers are present in the data."

"We define a numerical method that provides a non-parametric estimation of the kernel shape in symmetric multivariate Hawkes processes. This method relies on second order statistical properties of Hawkes processes that relate the covariance matrix of the process to the kernel matrix. The square root of the correlation function is computed using a minimal phase recovering method. We illustrate our method on some examples and provide an empirical study of the estimation errors. Within this framework, we analyze high frequency financial price data modeled as 1D or 2D Hawkes processes. We find slowly decaying (power-law) kernel shapes suggesting a long memory nature of self-excitation phenomena at the microstructure level of price dynamics."

"We consider nonparametric functional regression when both predictors and responses are functions. More specifically, we let $(X_1,Y_1),...,(X_n,Y_n)$ be random elements in $mathcal{F}timesmathcal{H}$ where $mathcal{F}$ is a semi-metric space and $mathcal{H}$ is a separable Hilbert space. Based on a recently introduced notion of weak dependence for functional data, we showed the almost sure convergence rates of both the Nadaraya-Watson estimator and the nearest neighbor estimator, in a unified manner. Several factors, including functional nature of the responses, the assumptions on the functional variables using the Orlicz norm and the desired generality on weakly dependent data, make the theoretical investigations more challenging and interesting."

"In this paper we present a generalization of kernel density estimation called Convex Adaptive Kernel Density Estimation (CAKE) that replaces single bandwidth se- lection by a convex aggregation of kernels at all scales, where the convex aggregation is allowed to vary from one training point to another, treating the fundamental problem of heterogeneous smoothness in a novel way. Learning the CAKE estimator given a training set reduces to solving a single con- vex quadratic programming problem. We derive rates of convergence of CAKE like estimator to the true underlying density under smoothness assumptions on the class and show that given a sufficiently large sample the mean squared error of such estimators is optimal in a minimax sense. We also give a risk bound of the CAKE estimator in terms of its empirical risk. We empirically compare CAKE to other density estimators proposed in the statistics literature for handling heterogeneous smoothness on different synthetic and natural distributions. "

"We introduce an estimator for the population mean based on maximizing likelihoods formed by parameterizing a kernel density estimate. Due to these origins, we have dubbed the estimator the maximum kernel likelihood estimate (MKLE). A speedy computational method to compute the MKLE based on binning is implemented in a simulation study which shows that the MKLE at an optimal bandwidth is decidedly superior in terms of efficiency to the sample mean and other measures of location for heavy tailed symmetric distributions. An empirical rule and a computational method to estimate this optimal bandwidth are developed and used to construct bootstrap confidence intervals for the population mean. We show that the intervals have approximately nominal coverage and have significantly smaller average width than the standard t and z intervals. Finally, we develop some mathematical properties for a very close approximation to the MKLE called the kernel mean. In particular, we demonstrate that the kernel mean is indeed unbiased for the population mean for symmetric distributions."

The abstract is great: "Few would deny that the most powerful statistical tool is graph paper. When however there are many observations (and/or many variables) graphical procedures become tedious. It seems to the author that the most characteristic problem for statisticians at the moment is the development of methods for analyzing the data poured out by electronic observing systems. The present paper gives a simple computer method for obtaining a "graph" from a large number of observations."

Kernel smoothing when both inputs and outputs are functions.  Sounds cool.

"We present a new adaptive kernel density estimator based on linear diffusion processes. The proposed estimator builds on existing ideas for adaptive smoothing by incorporating information from a pilot density estimate. In addition, we propose a new plug-in bandwidth selection method that is free from the arbitrary normal reference rules used by existing methods. We present simulation examples in which the proposed approach outperforms existing methods in terms of accuracy and reliability."

" kernel estimation of marginal densities and regression functions of stationary processes. It is shown that for a wide class of time series, with proper centering and scaling, the maximum deviations of kernel density and regression estimates are asymptotically Gumbel. Our results substantially generalize earlier ones which were obtained under independence or beta mixing assumptions. The asymptotic results can be applied to assess patterns of marginal densities or regression functions via the construction of simultaneous confidence bands for which one can perform goodness-of-fit tests"

"We propose an adaptive varying-coefficient spatiotemporal model for data that are observed irregularly over space and regularly in time. The model is capable of catching possible non-linearity (both in space and in time) and non-stationarity (in space) by allowing the auto-regressive coefficients to vary with both spatial location and an unknown index variable. We suggest a two-step procedure to estimate both the coefficient functions and the index variable, which is readily implemented and can be computed even for large spatiotemporal data sets. Our theoretical results indicate that, in the presence of the so-called nugget effect, the errors in the estimation may be reduced via the spatial smoothing—the second step in the estimation procedure proposed. The simulation results reinforce this finding. As an illustration, we apply the methodology to a data set of sea level pressure in the North Sea."

"This paper presents a set of rate of uniform consistency results for kernel estimators of density functions and regressions functions. We generalize the existing literature by allowing for stationary strong mixing multivariate data with infinite support, and kernels with unbounded support, and general bandwidth sequences. These results are useful for semiparametric estimation based on a first stage nonparametric estimator."

A recursive/on-line kernel regression estimator, with proofs that it's about as efficient as the off-line/batch-mode Nadaraya-Watson estimator. Sounds cool...