Search

A kernel density estimator for data on the polysphere $\mathbb{S}^{d_1}\times\cdots\times\mathbb{S}^{d_r}$, with $r,d_1,\ldots,d_r\geq 1$, is presented in this paper. We derive the main asymptotic properties of the estimator, including mean square error, normality, and optimal bandwidths. We address the kernel theory of the estimator beyond the von Mises-Fisher kernel, introducing new kernels that are more efficient and investigating normalizing constants, moments, and sampling methods thereof. Plug-in and cross-validated bandwidth selectors are also obtained. As a spin-off of the kernel density estimator, we propose a nonparametric $k$-sample test based on the Jensen-Shannon divergence. Numerical experiments illuminate the asymptotic theory of the kernel density estimator and demonstrate the superior performance of the $k$-sample test with respect to parametric alternatives in certain scenarios. Our smoothing methodology is applied to the analysis of the morphology of a sample of hippocampi of infants embedded on the high-dimensional polysphere $(\mathbb{S}^2)^{168}$ via skeletal representations ($s$-reps)., Comment: 20 pages, 4 figures. Supplementary material: 23 pages, 8 figures, 1 table

The problem of assessing a parametric regression model in the presence of spatial correlation is addressed in this work. For that purpose, a goodness-of-fit test based on a $L_2$-distance comparing a parametric and a nonparametric regression estimators is proposed. Asymptotic properties of the test statistic, both under the null hypothesis and under local alternatives, are derived. Additionally, a bootstrap procedure is designed to calibrate the test in practice. Finite sample performance of the test is analyzed through a simulation study, and its applicability is illustrated using a real data example., Comment: 49 pages, 7 figures

In oceanography, modeling wave fields requires the use of statistical tools capable of handling the circular nature of the {data measurements}. An important issue in ocean wave analysis is the study of height and direction waves, being direction values recorded as angles or, equivalently, as points on a unit circle. Hence, reconstruction of a wave direction field on the sea surface can be approached by the use of a linear-circular regression model, viewing wave directions as a realization of a circular spatial process whose trend should be estimated. In this paper, we consider a spatial regression model with a circular response and several real-valued predictors. Nonparametric estimators of the circular trend surface are proposed, accounting for the (unknown) spatial correlation. Some asymptotic results about these estimators as well as some guidelines for their practical implementation are also given. The performance of the proposed estimators is investigated in a simulation study. An application to wave directions in the Adriatic Sea is provided for illustration., Comment: 34 pages, 8 figures

Changes on temperature patterns, on a local scale, are perceived by individuals as the most direct indicators of global warming and climate change. As a specific example, for an Atlantic climate location, spring and fall seasons should present a mild transition between winter and summer, and summer and winter, respectively. By observing daily temperature curves along time, being each curve attached to a certain calendar day, a regression model for these variables (temperature curve as covariate and calendar day as response) would be useful for modeling their relation for a certain period. In addition, temperature changes could be assessed by prediction and observation comparisons in the long run. Such a model is presented and studied in this work, considering a nonparametric Nadaraya-Watson-type estimator for functional covariate and circular response. The asymptotic bias and variance of this estimator, as well as its asymptotic distribution are derived. Its finite sample performance is evaluated in a simulation study and the proposal is applied to investigate a real-data set concerning temperature curves., Comment: 28 pages, 11 figures

Testing procedures for assessing a parametric regression model with circular response and $\mathbb{R}^d$-valued covariate are proposed and analyzed in this work both for independent and for spatially correlated data. The test statistics are based on a circular distance comparing a (non-smoothed or smoothed) parametric circular estimator and a nonparametric one. Properly designed bootstrap procedures for calibrating the tests in practice are also presented. Finite sample performance of the tests in different scenarios with independent and spatially correlated samples, is analyzed by simulations., Comment: 38 pages, 2 figures

The analysis of continuously spatially varying processes usually considers two sources of variation, namely, the large-scale variation collected by the trend of the process, and the small-scale variation. Parametric trend models on latitude and longitude are easy to fit and to interpret. However, the use of simple parametric models for characterizing spatially varying processes may lead to misspecification problems if the model is not appropriate. Recently, Meil\'an-Vila et al. (2019) proposed a goodness-of-fit test based on an L2-distance for assessing a parametric trend model with correlated errors, under random design, comparing a parametric and a nonparametric trend estimators. The present work aims to provide a detailed computational analysis of the behavior of this approach using different bootstrap algorithms for calibration, under a fixed-design geostatistical framework. Asymptotic results for the test are provided and an extensive simulation study, considering complexities that usually arise in geostatistics, is carried out to illustrate the performance of the proposal., Comment: 29 pages, 7 figures

Nonparametric estimators of a regression function with circular response and Rd-valued predictor are considered in this work. Local polynomial type estimators are proposed and studied. Expressions for their asymptotic biases and variances are derived, and some guidelines to select asymptotically local optimal bandwidth matrices are also given. The finite sample behavior of the proposed estimators is assessed through simulations and their performance is also illustrated with a real data set., Comment: 30 pages, 16 figures