Nonparametric Curve Estimation Research Articles

Statistical inference for wavelet curve estimators of symmetric positive definite matrices

In this paper we treat statistical inference for a wavelet estimator of curves of symmetric positive definite (SPD) using the log-Euclidean distance. This estimator preserves positive-definiteness and enjoys permutation-equivariance, which is particularly relevant for covariance matrices. Our second-generation wavelet estimator is based on average-interpolation (AI) and allows the same powerful properties, including fast algorithms, known from nonparametric curve estimation with wavelets in standard Euclidean set-ups. The core of our work is the proposition of confidence sets for our AI wavelet estimator in a non-Euclidean geometry. We derive asymptotic normality of this estimator, including explicit expressions of its asymptotic variance. This opens the door for constructing asymptotic confidence regions which we compare with our proposed bootstrap scheme for inference. Detailed numerical simulations confirm the appropriateness of our suggested inference schemes.

Nonlinear semiparametric autoregressive models based on a class of two-piece distributions

In this study, we examined the well-known nonlinear autoregressive time series model in which innovations follow the flexible class of two-piece distributions based on the scale mixtures of normal (TP-SMN) family. The mentioned class of distributions is a rich class of distributions family that covers the robust symmetric/asymmetric light/heavy-tailed distributions. The nonlinear part of the autoregressive time series model is estimated via the semiparametric and nonparametric curve estimation based on the conditional least square method and nonparametric kernel approach. The maximum likelihood (ML) estimates of the model parameters, using a suitable hierarchical representation of the TP-SMN family on the model are obtained via an expectation-maximization type algorithm. Performances and usability of the proposed model and estimates are shown through simulation studies and a real dataset.

A robust class of nonlinear autoregressive models with regression function and dependent innovations using semiparametric kernel estimation

In this study, researchers examine a nonlinear autoregressive (NLAR) time-series model with regression function and dependent innovations in which the errors of the model follow the two-piece scale mixtures of normal (TP-SMN) distributions. Robustness and atypical forms of the proposed class of two-piece distributions along with the flexibility of the nonlinear autoregressive model develop desirable properties which can be applied to several types of datasets. The nonlinear regression function part of the autoregressive time-series model is estimated via the semiparametric and nonparametric curve estimation based on the conditional least square method and nonparametric kernel approach. The maximum likelihood (ML) estimates of the model parameters, using a suitable hierarchical representation of the TP-SMN family on the model are obtained via an expectation maximization (EM)-type algorithm. Performances and usefulness of the proposed model and estimates are shown via simulation studies and a real dataset.

On bandwidth selection problems in nonparametric trend estimation under martingale difference errors

In this paper, we are interested in the problem of smoothing parameter selection in nonparametric curve estimation under dependent errors. We focus on kernel estimation and the case when the errors form a general stationary sequence of martingale difference random variables where neither linearity assumption nor “all moments are finite” are required. We compare the behaviors of the smoothing bandwidths obtained by minimizing either the unknown average squared error, the theoretical mean average squared error, a Mallows-type criterion adapted to the dependent case and the family of criteria known as generalized cross validation (GCV) extensions of the Mallows’ criterion. We prove that these three minimizers and those based on the GCV family are first-order equivalent in probability. We give also a normal asymptotic behavior of the gap between the minimizer of the average squared error and that of the Mallows-type criterion. This is extended to the GCV family. Finally, we apply our theoretical results to a specific case of martingale difference sequence, namely the Auto-Regressive Conditional Heteroscedastic (ARCH(1)) process. A Monte-Carlo simulation study, for this regression model with ARCH(1) process, is conducted.

Recent advances in directional statistics

Mainstream statistical methodology is generally applicable to data observed in Euclidean space. There are, however, numerous contexts of considerable scientific interest in which the natural supports for the data under consideration are Riemannian manifolds like the unit circle, torus, sphere, and their extensions. Typically, such data can be represented using one or more directions, and directional statistics is the branch of statistics that deals with their analysis. In this paper, we provide a review of the many recent developments in the field since the publication of Mardia and Jupp (Wiley 1999), still the most comprehensive text on directional statistics. Many of those developments have been stimulated by interesting applications in fields as diverse as astronomy, medicine, genetics, neurology, space situational awareness, acoustics, image analysis, text mining, environmetrics, and machine learning. We begin by considering developments for the exploratory analysis of directional data before progressing to distributional models, general approaches to inference, hypothesis testing, regression, nonparametric curve estimation, methods for dimension reduction, classification and clustering, and the modelling of time series, spatial and spatio-temporal data. An overview of currently available software for analysing directional data is also provided, and potential future developments are discussed.

Semiparametric Estimation of the Distribution of Episodically Consumed Foods Measured With Error

Dietary data collected from 24-hour dietary recalls are observed with significant measurement errors. In the nonparametric curve estimation literature, much of the effort has been devoted to designing methods that are consistent under contamination by noise, and which have been traditionally applied for analyzing those data. However, some foods such as alcohol or fruits are consumed only episodically, and may not be consumed during the day when the 24-hour recall is administered. These so-called excess zeros make existing nonparametric estimators break down, and new techniques need to be developed for such data. We develop two new consistent semiparametric estimators of the distribution of such episodically consumed food data, making parametric assumptions only on some less important parts of the model. We establish its theoretical properties and illustrate the good performance of our fully data-driven method in simulated and real data. Supplementary materials for this article are available online.

Bayesian wavelet shrinkage with logistic prior

Consider the nonparametric curve estimation problem. Wavelet shrinkage methods are applied to the data in the wavelet domain for noise reduction. After denoising, the function can be estimated by wavelet basis expansion. The present paper proposes a Bayesian approach for wavelet shrinkage with the use of a mixture of a point mass function at zero and the logistic distribution symmetric around zero as the prior distribution to the wavelet coefficients of an unknown function in models with additive Gaussian errors. Statistical properties such as bias, classical and Bayesian risks of the rules are analyzed and performances of the proposed rules are obtained in simulations studies involving the Donoho-Johnstone test functions. Application in a mass spectrometry real data set is done.

Nonparametric curve estimation and bootstrap bandwidth selection

AbstractOver the last four decades, the bootstrap method has been considered so as to define data‐driven bandwidth selectors for nonparametric curve estimation. An extensive and updated review of bootstrap methods used to select the smoothing parameter for the nonparametric estimation of several curves has been carried out. Different data generating processes have been profoundly reviewed, such as the classical independent and identically distributed setup as well as dependent, censored, length‐biased, grouped, missing or directional data, among others. Several curves have also been considered, such as the density, regression, hazard rate, intensity, latency, incidence, or distribution functions, among many others. It is worth mentioning that there exist situations where Monte Carlo methods are not needed when using the bootstrap for bandwidth selections. This idea has also been exploited to find closed expressions for the bootstrap version of criterion functions for some proxy of the real curve estimator.This article is categorized under: Statistical and Graphical Methods of Data Analysis > Bootstrap and Resampling Statistical and Graphical Methods of Data Analysis > Nonparametric Methods Statistical and Graphical Methods of Data Analysis > Density Estimation

Investigation of uncertainty quantification methods for constitutive models and the application to LOFT LBLOCA

Best estimate (BE) codes are developed to carry out realistic safety analysis of the nuclear reactor, and generally hundreds of constitutive models are comprised in a BE code. Nevertheless, uncertainties of these constitutive models are often not properly handled in the best estimate plus uncertainty (BEPU) analysis. It is not sufficient to fully evaluate the uncertainties of different sorts of models with only one method. Thereby a structural method for uncertainty quantification (UQ) of constitutive models is proposed and relevant description is presented. Based on the method, constitutive models will be classified into two categories according to the characteristics, namely the independent model and the dependent model, and different methods will be adopted for different sorts of model. Several statistical methods for UQ of constitutive models such as the non-parametric curve estimation method, the Bayesian calibration method and the coverage calibration method are evaluated and the characteristics of these methods are discussed. In addition, several methods for the construction of surrogate model are utilized to reduce the computational cost, and a model selection technique is adopted to opt the optimal model among all alternative models. The large break loss of coolant accident (LBLOCA) experiment LOFT LP-02-6 is utilized to verify the proposed structural method, and the BEPU analysis of the LP-02-6 experiment is carried out. The results show that uncertainty intervals of the identified models obtained through the structural method are reasonable, uncertainties of the peak cladding temperature (PCT) as well as the accumulator injection time (AIT) are quantified, and the sensitivity analysis is carried out to evaluate the influence of different input parameters on the 1st PCT, 2nd PCT and the AIT.

Nonlinear semiparametric autoregressive model with finite mixtures of scale mixtures of skew normal innovations

ABSTRACTWe propose data generating structures which can be represented as the nonlinear autoregressive models with single and finite mixtures of scale mixtures of skew normal innovations. This class of models covers symmetric/asymmetric and light/heavy-tailed distributions, so provide a useful generalization of the symmetrical nonlinear autoregressive models. As semiparametric and nonparametric curve estimation are the approaches for exploring the structure of a nonlinear time series data set, in this article the semiparametric estimator for estimating the nonlinear function of the model is investigated based on the conditional least square method and nonparametric kernel approach. Also, an Expectation–Maximization-type algorithm to perform the maximum likelihood (ML) inference of unknown parameters of the model is proposed. Furthermore, some strong and weak consistency of the semiparametric estimator in this class of models are presented. Finally, to illustrate the usefulness of the proposed model, some simulation studies and an application to real data set are considered.

Nonlinear wavelet shrinkage estimator of nonparametric regularity regression function via cross-validation with simulation study

Nonparametric regression techniques provide a very effective and simple way of finding structure in data sets without the imposition of a parametric regression model. Wavelet theory has the potential to provide statisticians with powerful new techniques for nonparametric inference. In this paper, we consider the wavelet shrinkage kernel estimator of regression function with a common one-dimensional probability density function. We investigate a new nonparametric curve estimator and convergence ratio of given estimator by using cross-validation method to choice of wavelet threshold when the observations are taken on the regular grid. At the end we used simulation study to examine our proposed estimator. We survey the theoretical outcomes with numerical computation by using [Formula: see text] software to compare purpose estimator with another estimators.

Missing, Modified, and Large-p-small-nData in Nonparametric Curve Estimation

Nonparametric curve estimation, which makes no assumptions about shape of estimated functions, is one of the main pillars of the modern statistical science. It is used when no adequate parametric or semi-parametric model is available. Asymptotic results on adaptive estimation of nonparametric curves, under both traditional and shrinking minimaxes, are presented. The latter approach allows us to explain the phenomenon of superefficiency when a function can be estimated with a rate faster than the minimax one. Resent results on sequential nonparametric estimation, which yields an assigned risk with minimal average stopping time, are also presented. Then it is explained how the theory can be used in practically important cases of missing data, modified data in survival analysis, and Big Data. Wavelet applications in the analysis of microarrays and fMRI illustrate feasibility of the proposed nonparametric estimation.

Wavelet Estimation: Minimax Theory and Application

For now wavelet estimation is an important part of the nonparametric curve estimation. The minimax theory is well developed and there are many interesting practical applications where wavelets perfectly fit purposes of the data analysis. This paper presents a new theoretical result and a new application. It is well known in the statistical wavelet theory that for a Besov space of order the minimax rate of the mean integrated squared error (MISE) convergence is . The developed in the paper local minimax approach shows that this rate is too slow and that any Besov’s function of order can be estimated with a faster rate. Furthermore, an adaptive estimator is proposed that is minimax under the classical and new minimax approaches. Then an interesting and challenging application of the adaptive wavelet estimator in the fMRI study of neuroplasticity is discussed. DOI: http://dx.doi.org/10.4038/sljastats.v5i4.7782

Statistical inference in the partial linear models with the double smoothing local linear regression method

Statistical inference about a partial linear model includes two parts, one linear and the other nonparametric. The double smoothing method proposed by He and Huang (2009) is a progressive local smoothing method for nonparametric curve estimation. In this paper, we discuss its extension to partial linear models, accompanied with difference-based estimation method for the linear part. Asymptotic theory of the proposed method is developed. The results of simulation studies and real data examples demonstrate that our approach is effective even for data with moderate sample sizes.

Nonparametric estimation of the ROC curve based on smoothed empirical distribution functions

The receiver operating characteristic (ROC) curve is a graphical representation of the relationship between false positive and true positive rates. It is a widely used statistical tool for describing the accuracy of a diagnostic test. In this paper we propose a new nonparametric ROC curve estimator based on the smoothed empirical distribution functions. We prove its strong consistency and perform a simulation study to compare it with some other popular nonparametric estimators of the ROC curve. We also apply the proposed method to a real data set.

Nonparametric analysis of high wind speed data

In this paper, nonparametric curve estimation methods are applied to analyze time series of wind speeds, focusing on the extreme events exceeding a chosen threshold. Classical parametric statistical approaches in this context consist in fitting a generalized Pareto distribution (GPD) to the tail of the empirical cumulative distribution, using maximum likelihood or the method of the moments to estimate the parameters of this distribution. Additionally, confidence intervals are usually computed to assess the uncertainty of the estimates. Nonparametric methods to estimate directly some quantities of interest, such as the probability of exceedance, the quantiles or return levels, or the return periods, are proposed. Moreover, bootstrap techniques are used to develop pointwise and simultaneous confidence intervals for these functions. The proposed models are applied to wind speed data in the Gulf Coast of US, comparing the results with those using the GPD approach, by means of a split-sample test. Results show that nonparametric methods are competitive with respect to the standard GPD approximations. The study is completed generating synthetic data sets and comparing the behavior of the parametric and the nonparametric estimates in this framework.

The foxconn suicides and their media prominence: is the werther effect applicable in china?

BackgroundMedia reporting of suicide and its relationship with actual suicide has rarely been investigated in Mainland China. The "Foxconn suicides" is a description referring to a string of suicides/attempts during 2010, all of which were related to a giant electrical manufacturing company, Foxconn. This study aimed to examine the clustering and copycat effects of the Foxconn suicides, and to investigate temporal patterns in how they were reported by the media in Mainland China, Hong Kong (HK), and Taiwan (TW).MethodsRelevant articles were collected from representative newspapers published in three big cities in Mainland China (Beijing (BJ), Shenzhen (SZ), and Guangzhou (GZ)), HK, and TW, together with searching intensity data on the topic conducted using the Baidu search engine in Mainland China. The temporal clustering effects of the Foxconn suicides and their media prominence were assessed using the Kolmogorov-Smirnov test. The media reports of the Foxconn suicides' temporal patterns were explored using a nonparametric curve estimation method (that is, the local linear method). The potential mutual interactions between the Foxconn suicides and their media prominence were also examined, using logistic and Poisson regression methods.ResultsThe results support a temporal clustering effect for the Foxconn suicides. The BJ-based newspapers' reporting and the occurrence of a Foxconn suicide/attempt are each found to be associated with an elevated chance of a further Foxconn suicide 3 days later. The occurrence of a Foxconn suicide also immediately influenced the intensity of both Baidu searching and newspaper reporting. Regional diversity in suicide reporting tempo-patterns within Mainland China, and similarities between HK and TW, are also demonstrated.ConclusionsThe Foxconn suicides were temporally clustered. Their occurrences were influenced by the reporting of BJ-based newspapers, and contagion within the company itself. Further suicide research and prevention work in China should consider its special media environment.

Dose-response curve estimation: a semiparametric mixture approach.

In the estimation of a dose-response curve, parametric models are straightforward and efficient but subject to model misspecifications; nonparametric methods are robust but less efficient. As a compromise, we propose a semiparametric approach that combines the advantages of parametric and nonparametric curve estimates. In a mixture form, our estimator takes a weighted average of the parametric and nonparametric curve estimates, in which a higher weight is assigned to the estimate with a better model fit. When the parametric model assumption holds, the semiparametric curve estimate converges to the parametric estimate and thus achieves high efficiency; when the parametric model is misspecified, the semiparametric estimate converges to the nonparametric estimate and remains consistent. We also consider an adaptive weighting scheme to allow the weight to vary according to the local fit of the models. We conduct extensive simulation studies to investigate the performance of the proposed methods and illustrate them with two real examples.

A Wavelet Estimator of the Intensity Function with Censored Data

In reliability life-testing experiments and in medical studies, the interest often focuses on estimating the intensity function of the time to occurrence of some event of interest (such as engine failure, heart attack). Numerous methods have been developed for that purpose. Wavelets have recently become a widely used tool for nonparametric curve estimation. Their use for estimating intensity functions in the industrial and medical settings is however still limited. In this paper, we attempt to open the wavelet techniques to a broader audience, by providing a short introduction to the basic foundations of wavelets, and by describing a simple wavelet estimator for the intensity function of a random failure time. We consider the case where the random time is subject to right-censoring, as is usual in industrial life-testing and medical follow up. Asymptotic properties of this estimator are discussed using new arguments.

Nonparametric functional data estimation applied to ozone data: Prediction and extreme value analysis

The study of extreme values and prediction of ozone data is an important topic of research when dealing with environmental problems. Classical extreme value theory is usually used in air-pollution studies. It consists in fitting a parametric generalised extreme value (GEV) distribution to a data set of extreme values, and using the estimated distribution to compute return levels and other quantities of interest. Here, we propose to estimate these values using nonparametric functional data methods. Functional data analysis is a relatively new statistical methodology that generally deals with data consisting of curves or multi-dimensional variables. In this paper, we use this technique, jointly with nonparametric curve estimation, to provide alternatives to the usual parametric statistical tools. The nonparametric estimators are applied to real samples of maximum ozone values obtained from several monitoring stations belonging to the Automatic Urban and Rural Network (AURN) in the UK. The results show that nonparametric estimators work satisfactorily, outperforming the behaviour of classical parametric estimators. Functional data analysis is also used to predict stratospheric ozone concentrations. We show an application, using the data set of mean monthly ozone concentrations in Arosa, Switzerland, and the results are compared with those obtained by classical time series ( ARIMA) analysis.