Nonparametric Estimation of Trend in Directional Data
Consider measured positions of the paleomagnetic north pole over time. Each measured position may be viewed as a direction, expressed as a unit vector in three dimensions and incorporating some error. In this sequence, the true directions are expected to be close to one another at nearby times. A si...
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ftdatacite:10.48550/arxiv.1412.2315 2023-05-15T17:39:51+02:00 Nonparametric Estimation of Trend in Directional Data Beran, Rudolf 2014 https://dx.doi.org/10.48550/arxiv.1412.2315 https://arxiv.org/abs/1412.2315 unknown arXiv arXiv.org perpetual, non-exclusive license http://arxiv.org/licenses/nonexclusive-distrib/1.0/ Methodology stat.ME FOS Computer and information sciences 62H11 Preprint Article article CreativeWork 2014 ftdatacite https://doi.org/10.48550/arxiv.1412.2315 2022-04-01T12:27:31Z Consider measured positions of the paleomagnetic north pole over time. Each measured position may be viewed as a direction, expressed as a unit vector in three dimensions and incorporating some error. In this sequence, the true directions are expected to be close to one another at nearby times. A simple trend estimator that respects the geometry of the sphere is to compute a running average over the time-ordered observed direction vectors, then normalize these average vectors to unit length. This paper treats a considerably richer class of competing directional trend estimators that respect spherical geometry. The analysis relies on a nonparametric error model for directional data in $R^q$ that imposes no symmetry or other shape restrictions on the error distributions. Good trend estimators are selected by comparing estimated risks of competing estimators under the error model. Uniform laws of large numbers, from empirical process theory, establish when these estimated risks are trustworthy surrogates for the corresponding unknown risks. Report North Pole DataCite Metadata Store (German National Library of Science and Technology) North Pole |
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DataCite Metadata Store (German National Library of Science and Technology) |
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unknown |
topic |
Methodology stat.ME FOS Computer and information sciences 62H11 |
spellingShingle |
Methodology stat.ME FOS Computer and information sciences 62H11 Beran, Rudolf Nonparametric Estimation of Trend in Directional Data |
topic_facet |
Methodology stat.ME FOS Computer and information sciences 62H11 |
description |
Consider measured positions of the paleomagnetic north pole over time. Each measured position may be viewed as a direction, expressed as a unit vector in three dimensions and incorporating some error. In this sequence, the true directions are expected to be close to one another at nearby times. A simple trend estimator that respects the geometry of the sphere is to compute a running average over the time-ordered observed direction vectors, then normalize these average vectors to unit length. This paper treats a considerably richer class of competing directional trend estimators that respect spherical geometry. The analysis relies on a nonparametric error model for directional data in $R^q$ that imposes no symmetry or other shape restrictions on the error distributions. Good trend estimators are selected by comparing estimated risks of competing estimators under the error model. Uniform laws of large numbers, from empirical process theory, establish when these estimated risks are trustworthy surrogates for the corresponding unknown risks. |
format |
Report |
author |
Beran, Rudolf |
author_facet |
Beran, Rudolf |
author_sort |
Beran, Rudolf |
title |
Nonparametric Estimation of Trend in Directional Data |
title_short |
Nonparametric Estimation of Trend in Directional Data |
title_full |
Nonparametric Estimation of Trend in Directional Data |
title_fullStr |
Nonparametric Estimation of Trend in Directional Data |
title_full_unstemmed |
Nonparametric Estimation of Trend in Directional Data |
title_sort |
nonparametric estimation of trend in directional data |
publisher |
arXiv |
publishDate |
2014 |
url |
https://dx.doi.org/10.48550/arxiv.1412.2315 https://arxiv.org/abs/1412.2315 |
geographic |
North Pole |
geographic_facet |
North Pole |
genre |
North Pole |
genre_facet |
North Pole |
op_rights |
arXiv.org perpetual, non-exclusive license http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
op_doi |
https://doi.org/10.48550/arxiv.1412.2315 |
_version_ |
1766140628180140032 |