Functional inference in semiparametric models using the piggyback bootstrap

This paper introduces the “piggyback bootstrap.” Like the weighted bootstrap, this bootstrap procedure can be used to generate random draws that approximate the joint sampling distribution of the parametric and nonparametric maximum likelihood estimators in various semiparametric models, but the dimension of the maximization problem for each bootstrapped likelihood is smaller. This reduction results in significant computational savings in comparison to the weighted bootstrap. The procedure can be stated quite simply. First obtain a valid random draw for the parametric component of the model. Then take the draw for the nonparametric component to be the maximizer of the weighted bootstrap likelihood with the parametric component fixed at the parametric draw. We prove the procedure is valid for a class of semiparametric models that includes frailty regression models airsing in survival analysis and biased sampling models that have application to vaccine efficacy trials. Bootstrap confidence sets from the piggyback, and weighted bootstraps are compared for biased sampling data from simulated vaccine efficacy trials.     

Robustifying Generalized Linear Mixed Models Using a New Class of Mixtures of Multivariate Polya Trees

In applied sciences, generalized linear mixed models have become one of the preferred tools to analyze a variety of longitudinal and clustered data. Due to software limitations, the analyses are often restricted to the setting in which the random effects terms follow a multivariate normal distribution. However, this assumption may be unrealistic, obscuring important features of among-unit variation. This work describes a widely applicable semiparametric Bayesian approach that relaxes the normality assumption by using a novel mixture of multivariate Polya trees prior to define a flexible nonparametric model for the random effects distribution. The nonparametric prior is centered on the commonly used parametric normal family. We allow this parametric family to hold only approximately, thereby providing a robust alternative for modeling. We discuss and implement practical procedures for addressing the computational challenges that arise under this approach. We illustrate the methodology by applying it to real-life examples.

Supplemental materials for this paper are available online.     

Bootstrapping Pseudolikelihood Models for Clustered Binary Data

Asymptotic properties of the parametric bootstrap procedure for maximum pseudolikelihood estimators and hypothesis tests are studied in the general framework of associated populations. The technique is applied to the analysis of toxicological experiments which, based on pseudolikelihood inference for clustered binary data, fits into this framework. It is shown that the bootstrap approximation can be used as an interesting alternative to the classical asymptotic distribution of estimators and test statistics. Finite sample simulations for clustered binary data models confirm the asymptotic theory and indicate some substantial improvements.     

Restricted Likelihood Ratio Testing for Zero Variance Components in Linear Mixed Models

The goal of our article is to provide a transparent, robust, and computationally feasible statistical platform for restricted likelihood ratio testing (RLRT) for zero variance components in linear mixed models. This problem is nonstandard because under the null hypothesis the parameter is on the boundary of the parameter space. Our proposed approach is different from the asymptotic results of Stram and Lee who assumed that the outcome vector can be partitioned into many independent subvectors. Thus, our methodology applies to a wider class of mixed models, which includes models with a moderate number of clusters or nonparametric smoothing components. We propose two approximations to the finite sample null distribution of the RLRT statistic. Both approximations converge weakly to the asymptotic distribution obtained by Stram and Lee when their assumptions hold. When their assumptions do not hold, we show in extensive simulation studies that both approximations outperform the Stram and Lee approximation and the parametric bootstrap. We also identify and address numerical problems associated with standard mixed model software. Our methods are motivated by and applied to a large longitudinal study on air pollution health effects in a highly susceptible cohort. Relevant software is posted as an online supplement.     

Use of the Probability Integral Transformation to Fit Nonlinear Mixed-Effects Models With Nonnormal Random Effects

This article describes a simple computational method for obtaining the maximum likelihood estimates (MLE) in nonlinear mixed-effects models when the random effects are assumed to have a nonnormal distribution. Many computer programs for fitting nonlinear mixed-effects models, such as PROC NLMIXED in SAS, require that the random effects have a normal distribution. However, there is often interest in either fitting models with nonnormal random effects or assessing the sensitivity of inferences to departures from the normality assumption for the random effects. When the random effects are assumed to have a nonnormal distribution, we show how the probability integral transform can be used, in conjunction with standard statistical software for fitting nonlinear mixed-effects models (e.g., PROC NLMIXED in SAS), to obtain the MLEs. Specifically, the probability integral transform is used to transform a normal random effect to a nonnormal random effect. The method is illustrated using a gamma frailty model for clustered survival data and a beta-binomial model for clustered binary data. Finally, the results of a simulation study, examining the impact of misspecification of the distribution of the random effects, are presented.     

On testing equality of pairwise rank correlations in a multivariate random vector

Spearman’s rank-correlation coefficient (also called Spearman’s rho) represents one of the best-known measures to quantify the degree of dependence between two random variables. As a copula-based dependence measure, it is invariant with respect to the distribution’s univariate marginal distribution functions. In this paper, we consider statistical tests for the hypothesis that all pairwise Spearman’s rank correlation coefficients in a multivariate random vector are equal. The tests are nonparametric and their asymptotic distributions are derived based on the asymptotic behavior of the empirical copula process. Only weak assumptions on the distribution function, such as continuity of the marginal distributions and continuous partial differentiability of the copula, are required for obtaining the results. A nonparametric bootstrap method is suggested for either estimating unknown parameters of the test statistics or for determining the associated critical values. We present a simulation study in order to investigate the power of the proposed tests. The results are compared to a classical parametric test for equal pairwise Pearson’s correlation coefficients in a multivariate random vector. The general setting also allows the derivation of a test for stochastic independence based on Spearman’s rho.     

Scenarios for Multistage Stochastic Programs

A major issue in any application of multistage stochastic programming is the representation of the underlying random data process. We discuss the case when enough data paths can be generated according to an accepted parametric or nonparametric stochastic model. No assumptions on convexity with respect to the random parameters are required. We emphasize the notion of representative scenarios (or a representative scenario tree) relative to the problem being modeled.     

基于随机加权的空间聚集性检验及其在基础教育师资均衡性评价中的应用

使用空间统计检验方法研究北京基础教育资源分配的均衡性问题.对于空间分布均匀性的检验,常用的统计量是Moran's I统计量.但基于Moran's I统计量做推断的时候,人们往往用渐进正态分布或者用Bootstrap反复抽样得到经验分布来进行.提出使用随机加权法进行统计量的经验检验.Jin和Lee(2014)文中得出基于Bootstrap的Moran's I统计量满足一致逼近和渐进正态等性质.采用类似的统计工具证明了基于随机加权得到的统计量的渐进分布也满足这些良好性质.填补了用随机加权法在空间统计量的推断中理论保证的空白.通过模拟研究,证明了所提算法的有效性.方法应用于北京基础教育的师资-适龄儿童数比例,师资-在校生数比例的空间聚集性检验中得到了良好的应用,并与其它检验方法所得结论进行比较.结论显示在不同相邻概念(地理相邻、政策空间相邻)下,方法得到的结论符合常理.     

变系数混合模型的光滑样条推断

为了拟合纵向数据和其他相关数据,本文提出了变系数混合效应模型(VCMM).该模型运用变系数线性部分来表示协变量对响应变量的影响,而用随机效应来描述纵向数据组内的相关性, 因此,该模型允许协变量和响应变量之间存在十分灵活的泛函关系.文中运用光滑样条来估计均值部分的系数函数,而用限制最大似然的方法同时估计出光滑参数和方差成分,我们还得到了所提估计的计算方法.大量的模拟研究表明对于具有各种协方差结构的变系数混合效应模型,运用本文所提出的方法都能够十分有效地估计出模型中的系数函数和方差成分.     

随机缺失数据下样本分位数估计

分位数的估计在生物医学、社会经济调查等领域有着广泛的应用,然而在实际问题的研究中,往往由于各种人为或不可控因素造成数据收集不完全.本文在随机缺失(MAR)假设条件下,利用非参数核补法和局部多重插补法给出了响应变量缺失时样本分位数的估计,并利用经验过程等理论证明了由这两种方法得到的分位数估计的大样本性质,同时,使用重抽样方法给出了估计的渐近方差的估计,模拟结果验证了这两种方法的有效性.文章所提两种方法的优点在于:首先,所提出的缺失修正方法不需要对缺失概率的模型做任何假设;其次,方法亦适用于其他有关参数不可微的估计目标函数;最后,方法很容易地推广到一般M估计的情况,并可以对多个分位数同时进行估计.     

Time series of functional data with application to yield curves

We develop time series analysis of functional data observed discretely, treating the whole curve as a random realization from a distribution on functions that evolve over time. The method consists of principal components analysis of functional data and subsequently modeling the principal component scores as vector autoregressive moving averag (VARMA) process. We justify the method by showing that an underlying ARMAH structure of the curves leads to a VARMA structure on the principal component scores. We derive asymptotic properties of the estimators, fits, and forecast. For term structures of interest rates, these provide a unified framework for studying the time and maturity components of interest rates under one setup with few parametric assumptions. We apply the method to the yield curves of USA and India. We compare our forecasts to the parametric model that is based on Nelson‐Siegel curves. In another application, we study the dependence of long term interest rate on the short term interest rate using functional regression.     

The pairwise beta distribution: A flexible parametric multivariate model for extremes

We present a new parametric model for the angular measure of a multivariate extreme value distribution. Unlike many parametric models that are limited to the bivariate case, the flexible model can describe the extremes of random vectors of dimension greater than two. The novel construction method relies on a geometric interpretation of the requirements of a valid angular measure. An advantage of this model is that its parameters directly affect the level of dependence between each pair of components of the random vector, and as such the parameters of the model are more interpretable than those of earlier parametric models for multivariate extremes. The model is applied to air quality data and simulated spatial data.     

Bootstrap for estimating the MSE of the Spatial EBLUP

This work assumes that the small area quantities of interest follow a Fay–Herriot model with spatially correlated random area effects. Under this model, parametric and nonparametric bootstrap procedures are proposed for estimating the mean squared error of the empirical best linear unbiased predictor (EBLUP). A simulation study based on the Italian Agriculture Census 2000 compares bootstrap and analytical estimates of the MSE and studies their robustness to non-normality. Results indicate lower bias for the non-parametric bootstrap under specific departures from normality.      

Some new constructions of bivariate Weibull models

In this article, several approaches are advanced towards the construction of bivariate Weibull models from the consideration of failure behaviors of the components of a two-component system. First, a general method of construction of bivariate life models is developed in the setting of random environmental effects. Some new bivariate Weibull models are derived as special cases and added insights are provided for some of the existing ones. In the course of model formulation in terms of the dependence structure, a new bivariate family of life distributions is constructed so as to incorporate both positive and negative quadrant dependence in the same parametric setting, and a bivariate Weibull model is obtained as a special case. Finally, some distributional properties are presented for a bivariate Weibull model derived from the consideration of random hazards.     

基于B样条和聚类分析的非参数轮廓监控方法研究

在轮廓监控中,产品或过程的质量特征可以由一种特定的函数关系表示。如果轮廓的函数形式是已知的,则可以使用参数化方法来监控轮廓。然而,当轮廓形态复杂时,继续使用参数方法则可能导致由于模型设定不准确而无法正确识别异常轮廓的问题。因此本文提出了一种基于非参数回归的新方法以解决制造过程中常见的复杂轮廓监控问题。所提方法将基于非参数回归的B样条与迭代的聚类分析过程相结合,在应用过程中不需要对轮廓的形式进行限制性假设。仿真研究评估了该监控方法在不同变异情况下的性能,并且通过与现有方法的比较分析,验证了该方法的有效性和优越性。最后通过轮廓监控领域的一个经典案例说明了新方法的实际应用效果。     

Pricing longevity risk with the parametric bootstrap: A maximum entropy approach

In recent years, there has been significant development in the securitization of longevity risk. Various methods for pricing longevity risk have been proposed. In this paper we present an alternative pricing method, which is based on the maximization of the Shannon entropy in physics. Specifically, we propose implementing this pricing method with the parametric bootstrap (Brouhns et al., 2005), which is highly flexible and can be performed under different model assumptions. Through this pricing method we also quantify the impact of cohort effects and parameter uncertainty on prices of mortality-linked securities. Numerical illustrations based on longevity bonds with different maturities are provided.     

Mixed effects regression trees for clustered data

This paper presents an extension of the standard regression tree method to clustered data. Previous works extending tree methods to accommodate correlated data are mainly based on the multivariate repeated-measures approach. We propose a “mixed effects regression tree” method where the correlated observations are viewed as nested within clusters rather than as vectors of multivariate repeated responses. The proposed method can handle unbalanced clusters, allows observations within clusters to be split, and can incorporate random effects and observation-level covariates. We implemented the proposed method using a standard tree algorithm within the framework of the expectation-maximization (EM) algorithm. The simulation results show that the proposed regression tree method provides substantial improvements over standard trees when the random effects are non negligible. A real data example is used to illustrate the method.     

Modeling clustered binary data with nonparametric unobserved heterogeneity: An application to stock crash analysis

Various random effects models have been developed for clustered binary data; however, traditional approaches to these models generally rely heavily on the specification of a continuous random effect distribution such as Gaussian or beta distribution. In this article, we introduce a new model that incorporates nonparametric unobserved random effects on unit interval (0,1) into logistic regression multiplicatively with fixed effects. This new multiplicative model setup facilitates prediction of our nonparametric random effects and corresponding model interpretations. A distinctive feature of our approach is that a closed-form expression has been derived for the predictor of nonparametric random effects on unit interval (0,1) in terms of known covariates and responses. A quasi-likelihood approach has been developed in the estimation of our model. Our results are robust against random effects distributions from very discrete binary to continuous beta distributions. We illustrate our method by analyzing recent large stock crash data in China. The performance of our method is also evaluated through simulation studies.     

Bootstrapping for Penalized Spline Regression

We describe and contrast several different bootstrap procedures for penalized spline smoothers. The bootstrap methods considered are variations on existing methods, developed under two different probabilistic frameworks. Under the first framework, penalized spline regression is considered as an estimation technique to find an unknown smooth function. The smooth function is represented in a high-dimensional spline basis, with spline coefficients estimated in a penalized form. Under the second framework, the unknown function is treated as a realization of a set of random spline coefficients, which are then predicted in a linear mixed model. We describe how bootstrap methods can be implemented under both frameworks, and we show theoretically and through simulations and examples that bootstrapping provides valid inference in both cases. We compare the inference obtained under both frameworks, and conclude that the latter generally produces better results than the former. The bootstrap ideas are extended to hypothesis testing, where parametric components in a model are tested against nonparametric alternatives.

Datasets and computer code are available in the online supplements.     

The bootstrap and Bayesian bootstrap method in assessing bioequivalence

Parametric method for assessing individual bioequivalence (IBE) may concentrate on the hypothesis that the PK responses are normal. Nonparametric method for evaluating IBE would be bootstrap method. In 2001, the United States Food and Drug Administration (FDA) proposed a draft guidance. The purpose of this article is to evaluate the IBE between test drug and reference drug by bootstrap and Bayesian bootstrap method. We study the power of bootstrap test procedures and the parametric test procedures in FDA (2001). We find that the Bayesian bootstrap method is the most excellent.