Improved local polynomial estimation in time series regression

We propose a modification of local polynomial estimation which improves the efficiency of the conventional method when the observation errors are correlated. The procedure is based on a pre-transformation of the data as a generalization of the pre-whitening procedure introduced by Xiao et al. [(2003), ‘More Efficient Local Polynomial Estimation in Nonparametric Regression with Autocorrelated Errors’, Journal of the American Statistical Association, 98, 980–992]. While these authors assumed a linear process representation for the error process, we avoid any structural assumption. We further allow the regressors and the errors to be dependent. More importantly, we show that the inclusion of both leading and lagged variables in the approximation of the error terms outperforms the best approximation based on lagged variables only. Establishing its asymptotic distribution, we show that the proposed estimator is more efficient than the standard local polynomial estimator. As a by-product we prove a suitable version of a central limit theorem which allows us to improve the asymptotic normality result for local polynomial estimators by Masry and Fan [(1997), ‘Local Polynomial Estimation of Regression Functions for Mixing Processes’, Scandinavian Journal of Statistics, 24, 165–179]. A simulation study confirms the efficiency of our estimator on finite samples. An application to climate data also shows that our new method leads to an estimator with decreased variability.     

Adaptive nonparametric estimation in the presence of dependence

We consider nonparametric estimation problems in the presence of dependent data, notably nonparametric regression with random design and nonparametric density estimation. The proposed estimation procedure is based on a dimension reduction. The minimax optimal rate of convergence of the estimator is derived assuming a sufficiently weak dependence characterised by fast decreasing mixing coefficients. We illustrate these results by considering classical smoothness assumptions. However, the proposed estimator requires an optimal choice of a dimension parameter depending on certain characteristics of the function of interest, which are not known in practice. The main issue addressed in our work is an adaptive choice of this dimension parameter combining model selection and Lepski's method. It is inspired by the recent work of Goldenshluger and Lepski [(2011), ‘Bandwidth Selection in Kernel Density Estimation: Oracle Inequalities and Adaptive Minimax Optimality’, The Annals of Statistics, 39, 1608–1632]. We show that this data-driven estimator can attain the lower risk bound up to a constant provided a fast decay of the mixing coefficients.     

On improving convergence rate of Bernstein polynomial density estimator

This paper is concerned with the Bernstein estimator [Vitale, R.A. (1975), ‘A Bernstein Polynomial Approach to Density Function Estimation’, in Statistical Inference and Related Topics, ed. M.L. Puri, 2, New York: Academic Press, pp. 87–99] to estimate a density with support [0, 1]. One of the major contributions of this paper is an application of a multiplicative bias correction [Terrell, G.R., and Scott, D.W. (1980), ‘On Improving Convergence Rates for Nonnegative Kernel Density Estimators’, The Annals of Statistics, 8, 1160–1163], which was originally developed for the standard kernel estimator. Moreover, the renormalised multiplicative bias corrected Bernstein estimator is studied rigorously. The mean squared error (MSE) in the interior and mean integrated squared error of the resulting bias corrected Bernstein estimators as well as the additive bias corrected Bernstein estimator [Leblanc, A. (2010), ‘A Bias-reduced Approach to Density Estimation Using Bernstein Polynomials’, Journal of Nonparametric Statistics, 22, 459–475] are shown to be O(n^?8/9) when the underlying density has a fourth-order derivative, where n is the sample size. The condition under which the MSE near the boundary is O(n^?8/9) is also discussed. Finally, numerical studies based on both simulated and real data sets are presented.     

Complete convergence for weighted sums of END random variables and its application to nonparametric regression models

In this article, the complete convergence for weighted sums of extended negatively dependent (END, for short) random variables is investigated. Some sufficient conditions for the complete convergence are provided. In addition, the Marcinkiewicz–Zygmund type strong law of large numbers for weighted sums of END random variables is obtained. The results obtained in the article generalise and improve the corresponding one of Wang et al. [(2014b), ‘On Complete Convergence for an Extended Negatively Dependent Sequence’, Communications in Statistics-Theory and Methods, 43, 2923–2937]. As an application, the complete consistency for the estimator of nonparametric regression model is established.     

Testing exchangeability of copulas in arbitrary dimension

A test for exchangeability of copulas for arbitrary dimensions is proposed, generalising and extending a result by Genest et al. [(2012), ‘Tests of Symmetry for Bivariate Copulas’, Annals of the Institute of Statistical Mathematics, 64, 811–834]. Three test statistics together with some modifications are presented and their asymptotical behaviour is analysed. Empirical p-values are computed by using a bootstrap-procedure proposed by Rémillard and Scaillet [(2009), ‘Testing for Equality between Two Copulas’, Journal of Multivariate Analysis, 100, 377–386] and suggested by Bücher and Dette [(2010), ‘A Note on Bootstrap Approximations for the Empirical Copula Process’, Statistics & Probability Letters, 80, 1925–1932], based on a multiplier central limit theorem by van der Vaart and Wellner [(1996), Weak Convergence and Empirical Processes, Springer Series in Statistics, New York: Springer]. Finally a simulation study compares various versions of the proposed tests.     

Asymptotic normality of locally modelled regression estimator for functional data

We focus on the nonparametric regression of a scalar response on a functional explanatory variable. As an alternative to the well-known Nadaraya-Watson estimator for regression function in this framework, the locally modelled regression estimator performs very well [cf. [Barrientos-Marin, J., Ferraty, F., and Vieu, P. (2010), ‘Locally Modelled Regression and Functional Data’, Journal of Nonparametric Statistics, 22, 617–632]. In this paper, the asymptotic properties of locally modelled regression estimator for functional data are considered. The mean-squared convergence as well as asymptotic normality for the estimator are established. We also adapt the empirical likelihood method to construct the point-wise confidence intervals for the regression function and derive the Wilk's phenomenon for the empirical likelihood inference. Furthermore, a simulation study is presented to illustrate our theoretical results.     

On shrinking minimax convergence in nonparametric statistics

‘?…?if we are prepared to assume that the unknown density has k derivatives, then?…?the optimal mean integrated squared error is of order n^{?2 k/(2 k+1)}?…?’ The citation is from Silverman [(1986), Density Estimation for Statistics and Data Analysis, London: Chapman &; Hall] and its assertion is based on a classical minimax lower bound which is the pillar of the modern nonparametric statistics. This paper proposes a new minimax methodology that implies a faster decreasing minimax lower bound that is attainable by a data-driven estimator, and the same estimator is also minimax under the classical approach. The recommendation is to test performance of estimators via the new and classical minimax approaches.     

Mann–Whitney test with empirical likelihood methods for pretest–posttest studies

Pretest–posttest studies are an important and popular method for assessing the effectiveness of a treatment or an intervention in many scientific fields. While the treatment effect, measured as the difference between the two mean responses, is of primary interest, testing the difference of the two distribution functions for the treatment and the control groups is also an important problem. The Mann–Whitney test has been a standard tool for testing the difference of distribution functions with two independent samples. We develop empirical likelihood-based (EL) methods for the Mann–Whitney test to incorporate the two unique features of pretest–posttest studies: (i) the availability of baseline information for both groups; and (ii) the structure of the data with missing by design. Our proposed methods combine the standard Mann–Whitney test with the EL method of Huang, Qin and Follmann [(2008), ‘Empirical Likelihood-Based Estimation of the Treatment Effect in a Pretest–Posttest Study’, Journal of the American Statistical Association, 103(483), 1270–1280], the imputation-based empirical likelihood method of Chen, Wu and Thompson [(2015), ‘An Imputation-Based Empirical Likelihood Approach to Pretest–Posttest Studies’, The Canadian Journal of Statistics accepted for publication], and the jackknife empirical likelihood method of Jing, Yuan and Zhou [(2009), ‘Jackknife Empirical Likelihood’, Journal of the American Statistical Association, 104, 1224–1232]. Theoretical results are presented and finite sample performances of proposed methods are evaluated through simulation studies.     

On the estimation of extropy

Recently, Lad, Sanfilippo, and Agro [(2015), ‘Extropy: Complementary Dual of Entropy’, Statistical Science, 30, 40–58.] showed the measure of entropy has a complementary dual, which is termed extropy. The present article introduces some estimators of the extropy of a continuous random variable. Properties of the proposed estimators are stated, and comparisons are made with Qiu and Jia’s estimators [(2018a), ‘Extropy Estimators with Applications in Testing uniformity’, Journal of Nonparametric Statistics, 30, 182–196]. The results indicate that the proposed estimators have a smaller mean squared error than competing estimators. A real example is presented and analysed.     

An alternative local polynomial estimator for the error-in-variables problem

We consider the problem of estimating a regression function when a covariate is measured with error. Using the local polynomial estimator of Delaigle et al. [(2009), ‘A Design-adaptive Local Polynomial Estimator for the Errors-in-variables Problem’, Journal of the American Statistical Association, 104, 348–359] as a benchmark, we propose an alternative way of solving the problem without transforming the kernel function. The asymptotic properties of the alternative estimator are rigorously studied. A detailed implementing algorithm and a computationally efficient bandwidth selection procedure are also provided. The proposed estimator is compared with the existing local polynomial estimator via extensive simulations and an application to the motorcycle crash data. The results show that the new estimator can be less biased than the existing estimator and is numerically more stable.     

Some maximum-indifference estimators for the slope of a univariate linear model

As known, the least-squares estimator of the slope of a univariate linear model sets to zero the covariance between the regression residuals and the values of the explanatory variable. To prevent the estimation process from being influenced by outliers, which can be theoretically modelled by a heavy-tailed distribution for the error term, one can substitute covariance with some robust measures of association, for example Kendall's tau in the popular Theil–Sen estimator. In a scarcely known Italian paper, Cifarelli [(1978), ‘La Stima del Coefficiente di Regressione Mediante l'Indice di Cograduazione di Gini’, Rivista di matematica per le scienze economiche e sociali, 1, 7–38. A translation into English is available at http://arxiv.org/abs/1411.4809 and will appear in Decisions in Economics and Finance] shows that a gain of efficiency can be obtained by using Gini's cograduation index instead of Kendall's tau. This paper introduces a new estimator, derived from another association measure recently proposed. Such a measure is strongly related to Gini's cograduation index, as they are both built to vanish in the general framework of indifference. The newly proposed estimator is shown to be unbiased and asymptotically normally distributed. Moreover, all considered estimators are compared via their asymptotic relative efficiency and a small simulation study. Finally, some indications about the performance of the considered estimators in the presence of contaminated normal data are provided.     

Cytometry inference through adaptive atomic deconvolution

In this paper, we consider a statistical estimation problem known as atomic deconvolution. Introduced in reliability, this model has a direct application when considering biological data produced by flow cytometers. From a statistical point of view, we aim at inferring the percentage of cells expressing the selected molecule and the probability distribution function associated with its fluorescence emission. We propose here an adaptive estimation procedure based on a previous deconvolution procedure introduced by Es, Gugushvili, and Spreij [(2008), ‘Deconvolution for an atomic distribution’, Electronic Journal of Statistics, 2, 265–297] and Gugushvili, Es, and Spreij [(2011), ‘Deconvolution for an atomic distribution: rates of convergence’, Journal of Nonparametric Statistics, 23, 1003–1029]. For both estimating the mixing parameter and the mixing density automatically, we use the Lepskii method based on the optimal choice of a bandwidth using a bias-variance decomposition. We then derive some convergence rates that are shown to be minimax optimal (up to some log terms) in Sobolev classes. Finally, we apply our algorithm on the simulated and real biological data.     

Nonparametric regression method for broad sense agreement

Characterising the correspondence between an ordinal measurement and a continuous measurement is often of interest in mental health studies. To this end Peng et al. [(2011), ‘A Framework for Assessing Broad Sense Agreement Between Ordinal and Continuous Measurements’, Journal of the American Statistical Association, 106, 1592–1601] introduced the concept of broad sense agreement (BSA) and developed nonparametric estimation and inference for a BSA measure. In this work, we propose a nonparametric regression framework for BSA, which provides a robust tool to further investigate population heterogeneity in BSA. We develop inferential procedures including regression function estimation and hypothesis testing. Extensive simulation studies demonstrate satisfactory performance of the proposed method. We also apply the new method to a recent Grady Trauma Study and reveal an interesting impact of depression severity on the alignment between a self-reported symptom instrument and clinician diagnosis in posttraumatic stress disorder patients.     

Adaptive confidence bands in the nonparametric fixed design regression model

In this note, we consider the problem of the existence of adaptive confidence bands in the fixed design regression model, adapting ideas in Hoffmann and Nickl [(2011), ‘On Adaptive Inference and Confidence Bands’, Annals of Statistics, 39, 2383–2409] to the present case. In the course of the proof, we show that sup-norm adaptive estimators exist as well in the regression setting.     

Improving estimation efficiency for semi-competing risks data with partially observed terminal event

Semi-competing risks data arise when two types of events, non-terminal and terminal, may be observed. When the terminal event occurs first, it censors the non-terminal event. Otherwise the terminal event is observable after the occurrence of the non-terminal event. In practice, it can be hard to ascertain all terminal event information after the non-terminal event. Yu and Yiannoutsos [(2015), ‘Marginal and Conditional Distribution Estimation from Double-Sampled Semi-Competing Risks Data’, Scandinavian Journal of Statistics, 42, 87–103] considered a setting when the terminal event is ascertained via double sampling from only a subset of patients who experienced the non-terminal event. They discussed estimation for marginal and conditional distributions under this double sampled semi-competing risk data framework. We propose a more efficient estimation method in the same setting by fully utilising the non-terminal event information. The efficiency gain can be substantial as observed in our simulation study.     

A note on the monotonicity of the ES algorithm

In the study of the robust nonparametric regression problem, Oh et al. [The role of pseudo data for robust smoothing with application to wavelet regression, Biometrika 94 (2007), pp. 893–904] developed and named the ES algorithm. In the event that the ES algorithm converges, the robust estimator can be obtained through a sequence of conventional penalized least-squares estimates, the computation of which is fast and straightforward. However, the convergence of the ES algorithm was not established theoretically in Oh et al. In this note, we show that under a certain simple condition, the ES algorithm is monotonic. In particular, the ES algorithm does converge globally in the setting of Oh et al.     

Correcting moments for goodness of fit tests based on two entropy estimates

The sample entropy (Vasicek, 1976) has been most widely used as a nonparametric entropy estimator due to its simplicity, but its underlying distribution function has not been known yet though its moments are required in establishing the entropy-based goodness of test statistic (Soofi et al., 1995). In this paper we derive the nonparametric distribution function of the sample entropy as a piece-wise uniform distribution in the lights of Theil (1980) and Dudwicz and van der Meulen (1987). Then we establish the entropy-based goodness of fit test statistics based on the nonparametric distribution functions of the sample entropy and modified sample entropy (Ebrahimi et al., 1994), and compare their performances for the exponential and normal distributions.     

Nonparametric beta kernel estimator for long and short memory time series

In this article we introduce a nonparametric estimator of the spectral density by smoothing the periodogram using beta kernel density. The estimator is proved to be bounded for short memory data and diverges at the origin for long memory data. The convergence in probability of the relative error and Monte Carlo simulations show that the proposed estimator automatically adapts to the long- and the short-range dependency of the process. A cross-validation procedure is studied in order to select the nuisance parameter of the estimator. Illustrations on historical as well as most recent returns and absolute returns of the S&P500 index show the performance of the beta kernel estimator. The Canadian Journal of Statistics 48: 582–595; 2020 © 2020 Statistical Society of Canada     

Spatial Mallows model averaging for geostatistical models

Important progress has been made with model averaging methods over the past decades. For spatial data, however, the idea of model averaging has not been applied well. This article studies model averaging methods for the spatial geostatistical linear model. A spatial Mallows criterion is developed to choose weights for the model averaging estimator. The resulting estimator can achieve asymptotic optimality in terms of L₂ loss. Simulation experiments reveal that our proposed estimator is superior to the model averaging estimator by the Mallows criterion developed for ordinary linear models [Hansen, 2007] and the model selection estimator using the corrected Akaike's information criterion, developed for geostatistical linear models [Hoeting et al., 2006]. The Canadian Journal of Statistics 47: 336–351; 2019 © 2019 Statistical Society of Canada     

Consistency of The MMGLE under the piecewise proportional hazards models with interval-censored data

Wong et al. [(2018), ‘Piece-wise Proportional Hazards Models with Interval-censored Data’, Journal of Statistical Computation and Simulation, 88, 140–155] studied the piecewise proportional hazards (PWPH) model with interval-censored (IC) data under the distribution-free set-up. It is well known that the partial likelihood approach is not applicable for IC data, and Wong et al. (2018) showed that the standard generalised likelihood approach does not work either. They proposed the maximum modified generalised likelihood estimator (MMGLE) and the simulation results suggest that the MMGLE is consistent. We establish the consistency and asymptotically normality of the MMGLE.