Nonparametric estimation of a trend based upon sampled continuous processes

Degras, David

doi:10.1016/j.crma.2008.12.016

Cited by 5 publications

(9 citation statements)

References 15 publications

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“…In the same paper SCI are derived and compared to Bonferroni-and Scheffé-type intervals. Degras (2009) builds SCB for the regression function by coupling a functional central limit theorem [CLT] with a limit result on the supremum of a Gaussian process.…”

Section: Introductionmentioning

confidence: 99%

Simultaneous confidence bands for nonparametric regression with functional data

Degras¹

2011

STAT SINICA

104

119

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We consider nonparametric regression in the context of functional data, that is, when a random sample of functions is observed on a fine grid. We obtain a functional asymptotic normality result allowing to build simultaneous confidence bands (SCB) for various estimation and inference tasks. Two applications to a SCB procedure for the regression function and to a goodness-of-fit test for curvilinear regression models are proposed. The first one has improved accuracy upon the other available methods while the second can detect local departures from a parametric shape, as opposed to the usual goodness-of-fit tests which only track global departures. A numerical study of the SCB procedures and an illustration with a speech data set are provided.

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Section: Introductionmentioning

confidence: 99%

Simultaneous confidence bands for nonparametric regression with functional data

Degras¹

2011

STAT SINICA

104

119

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“…Using heuristic arguments similar to those of Degras (2009), we can also build asymptotic confidence bands in order to evaluate the global accuracy of our estimator. To do so, we make use of an asymptotic result from Landau & Shepp (1970), which states that the supremum of a centred Gaussian random function Z taking values in C[0, T ], with covariance function ρ(s, t) satisfies…”

Section: Asymptotic Normality and Confidence Bandsmentioning

confidence: 99%

Horvitz-Thompson estimators for functional data: asymptotic confidence bands and optimal allocation for stratified sampling

Cardot¹,

Josserand²

2011

Biometrika

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When dealing with very large datasets of functional data, survey sampling approaches are useful in order to obtain estimators of simple functional quantities, without being obliged to store all the data. We propose here a Horvitz-Thompson estimator of the mean trajectory. In the context of a superpopulation framework, we prove under mild regularity conditions that we obtain uniformly consistent estimators of the mean function and of its variance function. With additional assumptions on the sampling design we state a functional Central Limit Theorem and deduce asymptotic confidence bands. Stratified sampling is studied in detail, and we also obtain a functional version of the usual optimal allocation rule considering a mean variance criterion. These techniques are illustrated by means of a test population of N = 18902 electricity meters for which we have individual electricity consumption measures every 30 minutes over one week. We show that stratification can substantially improve both the accuracy of the estimators and reduce the width of the global confidence bands compared to simple random sampling without replacement.

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“…A large sample approximation to the distribution of the (scaled, centered) estimator

\sqrt{n} ((), \overset{μ}{true} - μ)

is then used to build the SCB, where n is the sample size. This approximation can be obtained numerically with resampling techniques such as parametric or nonparametric bootstrap or analytically using e.g., Gaussian random field theory or Karhunen–Loève expansions . Alongside dense functional data, SCB have been studied in the context of sparse functional data and survey sampling …”

Section: Introductionmentioning

confidence: 97%

Simultaneous confidence bands for the mean of functional data

Degras

2017

WIREs Computational Stats

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The mean function is a central object of inquiry in the analysis of functional data. Typical questions related to the mean function include quantifying estimation uncertainty, testing parametric models, and making comparisons between populations. To make probabilistic statements about the mean function over its entire domain, rather than at a single location, it is necessary to infer all of its values simultaneously. Pointwise inference is not appropriate for this task and indeed produces anticonservative results, i.e., the coverage level of confidence regions is too low and the significance level of hypothesis tests too high. In contrast, simultaneous confidence bands (SCB) provide a flexible framework for conducting simultaneous inference on the mean function and other functional parameters. They also offer powerful visualization tools for communicating analytic results to interdisciplinary audiences. The construction of SCB in the context of functional data requires specific theory and methods. In particular, it is not addressed by the nonparametric regression literature. Although software is available to perform individual steps of an SCB procedure, resources that provide end-to-end computations are scarce. Applications of SCB to one-and two-sample inferences are illustrated here with the R package SCBmeanfd.

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Nonparametric estimation of a trend based upon sampled continuous processes

Cited by 5 publications

References 15 publications

Simultaneous confidence bands for nonparametric regression with functional data

Simultaneous confidence bands for nonparametric regression with functional data

Horvitz-Thompson estimators for functional data: asymptotic confidence bands and optimal allocation for stratified sampling

Simultaneous confidence bands for the mean of functional data

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