Nonparametric estimation of multivariate extreme-value copulas

Extreme-value copulas arise in the asymptotic theory for componentwise maxima of independent random samples. An extreme-value copula is determined by its Pickands dependence function, which is a function on the unit simplex subject to certain shape constraints that arise from an integral transform of an underlying measure called spectral measure. Multivariate extensions are provided of certain rank-based nonparametric estimators of the Pickands dependence function. The shape constraint that the estimator should itself be a Pickands dependence function is enforced by replacing an initial estimator by its best least-squares approximation in the set of Pickands dependence functions having a discrete spectral measure supported on a sufficiently fine grid. Weak convergence of the standardized estimators is demonstrated and the finite-sample performance of the estimators is investigated by means of a simulation experiment.     

A comparison of dependence function estimators in multivariate extremes

Various nonparametric and parametric estimators of extremal dependence have been proposed in the literature. Nonparametric methods commonly suffer from the curse of dimensionality and have been mostly implemented in extreme-value studies up to three dimensions, whereas parametric models can tackle higher-dimensional settings. In this paper, we assess, through a vast and systematic simulation study, the performance of classical and recently proposed estimators in multivariate settings. In particular, we first investigate the performance of nonparametric methods and then compare them with classical parametric approaches under symmetric and asymmetric dependence structures within the commonly used logistic family. We also explore two different ways to make nonparametric estimators satisfy the necessary dependence function shape constraints, finding a general improvement in estimator performance either (i) by substituting the estimator with its greatest convex minorant, developing a computational tool to implement this method for dimensions \(D\ge 2\) or (ii) by projecting the estimator onto a subspace of dependence functions satisfying such constraints and taking advantage of Bernstein–Bézier polynomials. Implementing the convex minorant method leads to better estimator performance as the dimensionality increases.     

Testing and Estimating Shape-Constrained Nonparametric Density and Regression in the Presence of Measurement Error

In many applications we can expect that, or are interested to know if, a density function or a regression curve satisfies some specific shape constraints. For example, when the explanatory variable, X, represents the value taken by a treatment or dosage, the conditional mean of the response, Y , is often anticipated to be a monotone function of X. Indeed, if this regression mean is not monotone (in the appropriate direction) then the medical or commercial value of the treatment is likely to be significantly curtailed, at least for values of X that lie beyond the point at which monotonicity fails. In the case of a density, common shape constraints include log-concavity and unimodality. If we can correctly guess the shape of a curve, then nonparametric estimators can be improved by taking this information into account. Addressing such problems requires a method for testing the hypothesis that the curve of interest satisfies a shape constraint, and, if the conclusion of the test is positive, a technique for estimating the curve subject to the constraint. Nonparametric methodology for solving these problems already exists, but only in cases where the covariates are observed precisely. However in many problems, data can only be observed with measurement errors, and the methods employed in the error-free case typically do not carry over to this error context. In this paper we develop a novel approach to hypothesis testing and function estimation under shape constraints, which is valid in the context of measurement errors. Our method is based on tilting an estimator of the density or the regression mean until it satisfies the shape constraint, and we take as our test statistic the distance through which it is tilted. Bootstrap methods are used to calibrate the test. The constrained curve estimators that we develop are also based on tilting, and in that context our work has points of contact with methodology in the error-free case.     

Nonparametric maximum likelihood estimation for shifted curves

The analysis of a sample of curves can be done by self-modelling regression methods. Within this framework we follow the ideas of nonparametric maximum likelihood estimation known from event history analysis and the counting process set-up. We derive an infinite dimensional score equation and from there we suggest an algorithm to estimate the shape function for a simple shape invariant model. The nonparametric maximum likelihood estimator that we find turns out to be a Nadaraya–Watson-like estimator, but unlike in the usual kernel smoothing situation we do not need to select a bandwidth or even a kernel function, since the score equation automatically selects the shape and the smoothing parameter for the estimation. We apply the method to a sample of electrophoretic spectra to illustrate how it works.     

超越对数生产函数的半参数变系数估计模型

提出超越对数生产函数的半参数变系数模型,利用Profile方法给出产出弹性函数系数的局部加权最小二乘估计,并利用非参数条件自助法对有限样本的近似分布进行模拟,给出相对精确的广义似然比检验。规模报酬约束下中国1953--2008年的实证结果拒绝超越对数生产函数模型假设,产出弹性不可简单线性化而是对数劳均资本的非线性函数,时变资本弹性表现为倒U型变化趋势,时变劳动力弹性表现为U型变化趋势。     

Comparing Nonparametric Estimators for Length-Biased Data

Length-biased data appear when sampling lifetimes by cross-section. Right-censoring may affect the sampled information due to time limitation in following-up, lost to follow-up cases, etc. In this article, we compare by simulations two alternative nonparametric estimators of the lifetime distribution function when the data are length-biased and right-censored. These estimates, recently introduced in the literature, are based on nonparametric maximum-likelihood and moment-based principles. It is shown that the relative benefits associated to each estimator depend on several factors, such as the shape of the underlying distribution, sample size, or censoring level.     

Frontier estimation using kernel smoothing estimators with data transformation

In economics, a production frontier function is a graph that shows the maximum output of production units such as firms, industries, or economies, as a function of their inputs. Practically, estimating production frontiers often requires imposition of constraints such as monotonicity or monotone concavity. However, few constrained estimators of production frontier have been proposed in the literature. They are based on simple envelopment techniques which often suffer from lack of precision and smoothness. Motivated by this observation, we propose a smooth constrained nonparametric frontier estimator respecting constraints by considering kernel smoothing estimators from a transformed data. It is particularly appealing to practitioners who would like to use smooth estimates that, in addition, satisfy theoretical axioms of production. The utility of this method is illustrated through application to one real dataset and simulation evidences are also presented to show its superiority over the most known methods.     

Bayesian regression under combinations of constraints

A Bayesian method for regression under several types of constraints is proposed. The constraints can be range-restricted and include shape restrictions, constraints on the value of the regression function, smoothness conditions and combinations of these types of constraints. The support of the prior distribution is included in the set of piecewise linear functions. It is shown that the proposed prior can be arbitrarily close to the distribution induced by the addition of a polynomial plus an (m−1)-fold integrated Brownian motion. Hence, despite its piecewise linearity, the regression function behaves (approximately) like an m−1 times continuously differentiable random function. Furthermore, thanks to the piecewise linear property, many combinations of constraints can easily be considered. The regression function is estimated by the posterior mode computed by a simulated annealing algorithm. The constraints on the shape and the values of the regression function are taken into account thanks to the proposal distribution, while the smoothness condition is handled by the acceptation step. Simulations from the posterior distribution are obtained by a Gibbs sampling algorithm.     

Seemingly Unrelated Ridge Regression in Semiparametric Models

This article is concerned with the problem of multicollinearity in the linear part of a seemingly unrelated semiparametric (SUS) model. It is also suspected that some additional non stochastic linear constraints hold on the whole parameter space. In the sequel, we propose semiparametric ridge and non ridge type estimators combining the restricted least squares methods in the model under study. For practical aspects, it is assumed that the covariance matrix of error terms is unknown and thus feasible estimators are proposed and their asymptotic distributional properties are derived. Also, necessary and sufficient conditions for the superiority of the ridge-type estimator over the non ridge type estimator for selecting the ridge parameter K are derived. Lastly, a Monte Carlo simulation study is conducted to estimate the parametric and nonparametric parts. In this regard, kernel smoothing and cross validation methods for estimating the nonparametric function are used.     

Nonparametric likelihood and doubly robust estimating equations for marginal and nested structural models

This article considers Robins's marginal and nested structural models in the cross‐sectional setting and develops likelihood and regression estimators. First, a nonparametric likelihood method is proposed by retaining a finite subset of all inherent and modelling constraints on the joint distributions of potential outcomes and covariates under a correctly specified propensity score model. A profile likelihood is derived by maximizing the nonparametric likelihood over these joint distributions subject to the retained constraints. The maximum likelihood estimator is intrinsically efficient based on the retained constraints and weakly locally efficient. Second, two regression estimators, named hat and tilde, are derived as first‐order approximations to the likelihood estimator under the propensity score model. The tilde regression estimator is intrinsically and weakly locally efficient and doubly robust. The methods are illustrated by data analysis for an observational study on right heart catheterization. The Canadian Journal of Statistics 38: 609–632; 2010 © 2010 Statistical Society of Canada     

Nonparametric quasi-likelihood for right censored data

Quasi-likelihood was extended to right censored data to handle heteroscedasticity in the frame of the accelerated failure time (AFT) model. However, the assumption of known variance function in the quasi-likelihood for right censored data is usually unrealistic. In this paper, we propose a nonparametric quasi-likelihood by replacing the specified variance function with a nonparametric variance function estimator. This nonparametric variance function estimator is obtained by smoothing a function of squared residuals via local polynomial regression. The rate of convergence of the nonparametric variance function estimator and the asymptotic limiting distributions of the regression coefficient estimators are derived. It is demonstrated in simulations that for finite samples the proposed nonparametric quasi-likelihood method performs well. The new method is illustrated with one real dataset.     

Some results on random design regression with long memory errors and predictors

This paper studies nonparametric regression with long memory (LRD) errors and predictors. First, we formulate general conditions which guarantee the standard rate of convergence for a nonparametric kernel estimator. Second, we calculate the mean integrated squared error (MISE). In particular, we show that LRD of errors may influence MISE. On the other hand, an estimator for a shape function is typically not influenced by LRD in errors. Finally, we investigate properties of a data-driven bandwidth choice. We show that averaged squared error (ASE) is a good approximation of MISE; however, this is not the case for a cross-validation criterion.     

Fourth-order kernel method for simple linear degradation model

Degradation analysis is a useful technique when life tests result in few or even no failures. The degradation measurements are recorded over time and the estimation of time-to-failure distribution plays a vital role in degradation analysis. The parametric method to estimate the time-to-failure distribution assumed a specific parametric model with known shape for the random effects parameter. To avoid any assumption about the model shape, a nonparametric method can be used. In this paper, we suggest to use the nonparametric fourth-order kernel method to estimate the time-to-failure distribution and its percentiles for the simple linear degradation model. The performances of the proposed method are investigated and compared with the classical kernel; maximum likelihood and ordinary least squares methods via simulation technique. The numerical results show the good performance of the fourth-order kernel method and demonstrate its superiority over the parametric method when there is no information about the shape of the random effect parameter distribution.     

Quantile regression in functional linear semiparametric model

This paper proposes nonparametric estimation methods for functional linear semiparametric quantile regression, where the conditional quantile of the scalar responses is modelled by both scalar and functional covariates and an additional unknown nonparametric function term. The slope function is estimated using the functional principal component basis and the nonparametric function is approximated by a piecewise polynomial function. The asymptotic distribution of the estimators of slope parameters is derived and the global convergence rate of the quantile estimator of unknown slope function is established under suitable norm. The asymptotic distribution of the estimator of the unknown nonparametric function is also established. Simulation studies are conducted to investigate the finite-sample performance of the proposed estimators. The proposed methodology is demonstrated by analysing a real data from ADHD-200 sample.     

Mixed effects smoothing spline analysis of variance

We propose a general family of nonparametric mixed effects models. Smoothing splines are used to model the fixed effects and are estimated by maximizing the penalized likelihood function. The random effects are generic and are modelled parametrically by assuming that the covariance function depends on a parsimonious set of parameters. These parameters and the smoothing parameter are estimated simultaneously by the generalized maximum likelihood method. We derive a connection between a nonparametric mixed effects model and a linear mixed effects model. This connection suggests a way of fitting a nonparametric mixed effects model by using existing programs. The classical two-way mixed models and growth curve models are used as examples to demonstrate how to use smoothing spline analysis-of-variance decompositions to build nonparametric mixed effects models. Similarly to the classical analysis of variance, components of these nonparametric mixed effects models can be interpreted as main effects and interactions. The penalized likelihood estimates of the fixed effects in a two-way mixed model are extensions of James–Stein shrinkage estimates to correlated observations. In an example three nested nonparametric mixed effects models are fitted to a longitudinal data set.     

Bias-corrected estimations in varying-coefficient partially nonlinear models with measurement error in the nonparametric part

In this paper, we consider the statistical inference for the varying-coefficient partially nonlinear model with additive measurement errors in the nonparametric part. The local bias-corrected profile nonlinear least-squares estimation procedure for parameter in nonlinear function and nonparametric function is proposed. Then, the asymptotic normality properties of the resulting estimators are established. With the empirical likelihood method, a local bias-corrected empirical log-likelihood ratio statistic for the unknown parameter, and a corrected and residual adjusted empirical log-likelihood ratio for the nonparametric component are constructed. It is shown that the resulting statistics are asymptotically chi-square distribution under some suitable conditions. Some simulations are conducted to evaluate the performance of the proposed methods. The results indicate that the empirical likelihood method is superior to the profile nonlinear least-squares method in terms of the confidence regions of parameter and point-wise confidence intervals of nonparametric function.     

Detection of Change Point in Nonparametric Function with Unit-Root Noise by Wavelet

A wavelet method is proposed to detect jumps in a function which is observed with unit-root noise. We obtain critical values at any scale and prove the consistency of wavelet detection when the nonparametric function is smooth. It shows that the estimation of the number and locations of change points are consistent when there are change points in the nonparametric function. Simulation study supports our method.     

Comparison of renewal function estimators for censored data

The computation of the renewal function when the distribution function is completely known has received much attention in the literature. However, in many cases the form of the distribution function is unknown and has to be estimated nonparametrically. A nonparametric estimator for the renewal function for complete data was suggested by Frees (1986). In many cases, however, censoring of the lifetime might occur. We shall present parametric and nonparametric estimators of the renewal function based on censored data. In a simulation study we compare the nonparametric estimators with parametric estimators for the Weibull and lognormal distribution. The study suggests that the nonparametric estimator is a viable alternative to the parametric estimators when the lifetime distribution is unknown. Also, the nonparametric estimator is computationally simpler than the parametric estimator.     

Smoothing Splines and Shape Restrictions

Constrained smoothing splines are discussed under order restrictions on the shape of the function m . We consider shape constraints of the type m ^{( r )}≥ 0, i.e. positivity, monotonicity, convexity, .... (Here for an integer r ≥ 0, m ^{( r )} denotes the r th derivative of m .) The paper contains three results: (1) constrained smoothing splines achieve optimal rates in shape restricted Sobolev classes; (2) they are equivalent to two step procedures of the following type: (a) in a first step the unconstrained smoothing spline is calculated; (b) in a second step the unconstrained smoothing spline is "projected" onto the constrained set. The projection is calculated with respect to a Sobolev-type norm; this result can be used for two purposes, it may motivate new algorithmic approaches and it helps to understand the form of the estimator and its asymptotic properties; (3) the infinite number of constraints can be replaced by a finite number with only a small loss of accuracy, this is discussed for estimation of a convex function.     

Hazard-based nonparametric survivor function estimation

Summary.  A representation is developed that expresses the bivariate survivor function as a function of the hazard function for truncated failure time variables. This leads to a class of nonparametric survivor function estimators that avoid negative mass. The transformation from hazard function to survivor function is weakly continuous and compact differentiable, so that such properties as strong consistency, weak convergence to a Gaussian process and bootstrap applicability for a hazard function estimator are inherited by the corresponding survivor function estimator. The set of point mass assignments for a survivor function estimator is readily obtained by using a simple matrix calculation on the set of hazard rate estimators. Special cases arise from a simple empirical hazard rate estimator, and from an empirical hazard rate estimator following the redistribution of singly censored observations within strips. The latter is shown to equal van der Laan's repaired nonparametric maximum likelihood estimator, for which a Greenwood-like variance estimator is given. Simulation studies are presented to compare the moderate sample performance of various nonparametric survivor function estimators.