纵向数据下半参数回归模型的统计分析

对于纵向数据下半参数回归模型,基于广义估计方程和一般权函数方法构造了模型中参数分量和非参数分量的估计.在适当的条件下证明了参数估计量具有渐近正态性,并得到了非参数回归函数估计量的最优收敛速度.通过模拟研究说明了所提出的估计量在有限样本下的精确性.     

Bayesian Nonparametric Inference for a Multivariate Copula Function

The paper presents a general Bayesian nonparametric approach for estimating a high dimensional copula. We first introduce the skew–normal copula, which we then extend to an infinite mixture model. The skew–normal copula fixes some limitations in the Gaussian copula. An MCMC algorithm is developed to draw samples from the correct posterior distribution and the model is investigated using both simulated and real applications.     

Semiparametric estimation of average treatment effect through a random coefficient dummy endogenous variable model

This paper provides an estimation procedure for average treatment effect through a random coefficient dummy endogenous variable model. A leading example of the model is estimating the effect of a training program on earnings. The model is composed of two equations: an outcome equation and a decision equation. Given the linear restriction in outcome and decision equations, Chen (1999) provided a distribution-free estimation procedure under conditional symmetric error distributions. In this paper we extend Chen’s estimator by relaxing the linear index into a nonparametric function, which greatly reduces the risk of model misspecification. A two-step approach is proposed: the first step uses a nonparametric regression estimator for the decision variable, and the second step uses an instrumental variables approach to estimate average treatment effect in the outcome equation. The proposed estimator is shown to be consistent and asymptotically normally distributed. Furthermore, we investigate the finite performance of our estimator by a Monte Carlo study and also use our estimator to study the return of college education in different periods of China. The estimates seem more reasonable than those of other commonly used estimators.     

Estimation for a marginal generalized single-index longitudinal model

In this paper, we suggest an estimating equations based approach to study a general single-index model with a given out-layer link for longitudinal data and treat the classical one as its special case. Within a wide range of bandwidths which is for estimating the inner-layer nonparametric link, the root-n consistency of the estimator of the index can be proved. The estimation efficiency can be achieved even when there is an infinite-dimensional nuisance parameter to be estimated. The performance of the new method is assessed through the comparison with other existing methods and through an application to an epileptic seizure study.     

Smoothing combined generalized estimating equations in quantile partially linear additive models with longitudinal data

This paper develops a robust and efficient estimation procedure for quantile partially linear additive models with longitudinal data, where the nonparametric components are approximated by B spline basis functions. The proposed approach can incorporate the correlation structure between repeated measures to improve estimation efficiency. Moreover, the new method is empirically shown to be much more efficient and robust than the popular generalized estimating equations method for non-normal correlated random errors. However, the proposed estimating functions are non-smooth and non-convex. In order to reduce computational burdens, we apply the induced smoothing method for fast and accurate computation of the parameter estimates and its asymptotic covariance. Under some regularity conditions, we establish the asymptotically normal distribution of the estimators for the parametric components and the convergence rate of the estimators for the nonparametric functions. Furthermore, a variable selection procedure based on smooth-threshold estimating equations is developed to simultaneously identify non-zero parametric and nonparametric components. Finally, simulation studies have been conducted to evaluate the finite sample performance of the proposed method, and a real data example is analyzed to illustrate the application of the proposed method.     

Generalized partially linear models with missing covariates

In this article we study a semiparametric generalized partially linear model when the covariates are missing at random. We propose combining local linear regression with the local quasilikelihood technique and weighted estimating equation to estimate the parameters and nonparameters when the missing probability is known or unknown. We establish normality of the estimators of the parameter and asymptotic expansion for the estimators of the nonparametric part. We apply the proposed models and methods to a study of the relation between virologic and immunologic responses in AIDS clinical trials, in which virologic response is classified into binary variables. We also give simulation results to illustrate our approach.     

Empirical likelihood for balanced ranked-set sampled data

Ranked-set sampling (RSS) often provides more efficient inference than simple random sampling (SRS). In this article, we propose a systematic nonparametric technique, RSS-EL, for hypothesis testing and interval estimation with balanced RSS data using empirical likelihood (EL). We detail the approach for interval estimation and hypothesis testing in one-sample and two-sample problems and general estimating equations. In all three cases, RSS is shown to provide more efficient inference than SRS of the same size. Moreover, the RSS-EL method does not require any easily violated assumptions needed by existing rank-based nonparametric methods for RSS data, such as perfect ranking, identical ranking scheme in two groups, and location shift between two population distributions. The merit of the RSS-EL method is also demonstrated through simulation studies. This work was supported by National Natural Science Foundation of China (Grant No. 10871037)     

Joint semiparametric mean-covariance model in longitudinal study

Semiparametric regression models and estimating covariance functions are very useful for longitudinal study. To heed the positive-definiteness constraint, we adopt the modified Cholesky decomposition approach to decompose the covariance structure. Then the covariance structure is fitted by a semiparametric model by imposing parametric within-subject correlation while allowing the nonparametric variation function. We estimate regression functions by using the local linear technique and propose generalized es...     

Parametric distance functions vs. nonparametric neural networks for estimating road travel distances

Measuring and storing actual road travel distances between the points of a region is often not feasible and it is a common practice to estimate them. The usual approach is to use distance estimators which are parameterized functions of the coordinates of the points. We propose to use nonparametric approaches using neural networks for estimating actual distances. We consider multi-layer perceptrons trained with the back-propagation rule and regression neural networks implementing nonparametric regression using Gaussian kernels. We also consider training multiple estimators and combining them using voting and stacking. On a real-world study using cities drawn from Turkey, we found out that these nonparametric approaches are more accurate than the parametric distance functions. Estimating actual distances has many applications in location and distribution theory.     

A Generalized Estimating Equation Approach to Multivariate Adaptive Regression Splines

Multivariate adaptive regression splines (MARS) is a popular nonparametric regression tool often used for prediction and for uncovering important data patterns between the response and predictor variables. The standard MARS algorithm assumes responses are normally distributed and independent, but in this article we relax both of these assumptions by extending MARS to generalized estimating equations. We refer to this MARS-for-GEEs algorithm as “MARGE.” Our algorithm makes use of fast forward selection techniques, such that in the univariate case, MARGE has similar computation speed to a standard MARS implementation. Through simulation we show that the proposed algorithm has improved predictive performance than the original MARS algorithm when using correlated and/or nonnormal response data. MARGE is also competitive with alternatives in the literature, especially for problems with multiple interacting predictors. We apply MARGE to various ecological examples with different data types. Supplementary material for this article is available online.     

Statistical inference in partially-varying-coefficient single-index model

Consider a varying-coefficient single-index model which consists of two parts: the linear part with varying coefficients and the nonlinear part with a single-index structure, and are hence termed as varying-coefficient single-index models. This model includes many important regression models such as single-index models, partially linear single-index models, varying-coefficient model and varying-coefficient partially linear models as special examples. In this paper, we mainly study estimating problems of the varying-coefficient vector, the nonparametric link function and the unknown parametric vector describing the single-index in the model. A stepwise approach is developed to obtain asymptotic normality estimators of the varying-coefficient vector and the parametric vector, and estimators of the nonparametric link function with a convergence rate. The consistent estimator of the structural error variance is also obtained. In addition, asymptotic pointwise confidence intervals and confidence regions are constructed for the varying coefficients and the parametric vector. The bandwidth selection problem is also considered. A simulation study is conducted to evaluate the proposed methods, and real data analysis is also used to illustrate our methods.     

纵向数据广义部分线性模型的惩罚GMM估计

对纵向数据的部分线性模型,通常的做法是用样条方法或者核方法逼近非参数部分,然后再用广义估计方程的估计方法去估计参数部分.本文使用P-样条拟合非参数函数,对不同的矩条件用不同的广义矩方法对模型的参数和非参数进行估计,并且给出了估计量的大样本性质;并用计算机模拟和实例证明了当模型中存在不同的矩条件时,采用不同的惩罚广义矩方法可以显著地提高估计精度.     

Maximum likelihood estimation in the proportional hazards cure model

The proportional hazards cure model generalizes Cox’s proportional hazards model which allows that a proportion of study subjects may never experience the event of interest. Here nonparametric maximum likelihood approach is proposed to estimating the cumulative hazard and the regression parameters. The asymptotic properties of the resulting estimators are established using the modern empirical process theory. And the estimators for the regression parameters are shown to be semiparametric efficient.     

Using the bootstrap to estimate mean squared error and select smoothing parameter in nonparametric problems

We describe a bootstrap method for estimating mean squared error and smoothing parameter in nonparametric problems. The method involves using a resample of smaller size than the original sample. There are many applications, which are illustrated using the special cases of nonparametric density estimation, nonparametric regression, and tail parameter estimation.     

A machine learning approach for individual claims reserving in insurance

Accurate loss reserves are an important item in the financial statement of an insurance company and are mostly evaluated by macrolevel models with aggregate data in run‐off triangles. In recent years, a new set of literature has considered individual claims data and proposed parametric reserving models based on claim history profiles. In this paper, we present a nonparametric and flexible approach for estimating outstanding liabilities using all the covariates associated to the policy, its policyholder, and all the information received by the insurance company on the individual claims since its reporting date. We develop a machine learning–based method and explain how to build specific subsets of data for the machine learning algorithms to be trained and assessed on. The choice for a nonparametric model leads to new issues since the target variables (claim occurrence and claim severity) are right‐censored most of the time. The performance of our approach is evaluated by comparing the predictive values of the reserve estimates with their true values on simulated data. We compare our individual approach with the most used aggregate data method, namely, chain ladder, with respect to the bias and the variance of the estimates. We also provide a short real case study based on a Dutch loan insurance portfolio.     

Pyramid Quantile Regression

We describe a Bayesian model for simultaneous linear quantile regression at several specified quantile levels. More specifically, we propose to model the conditional distributions by using random probability measures, known as quantile pyramids, introduced by Hjort and Walker. Unlike many existing approaches, this framework allows us to specify meaningful priors on the conditional distributions, while retaining the flexibility afforded by the nonparametric error distribution formulation. Simulation studies demonstrate the flexibility of the proposed approach in estimating diverse scenarios, generally outperforming other competitive methods. We also provide conditions for posterior consistency. The method is particularly promising for modeling the extremal quantiles. Applications to extreme value analysis and in higher dimensions are also explored through data examples. Supplemental material for this article is available online.     

Estimation of risk-neutral density surfaces

Option price data is often used to infer risk-neutral densities for future prices of an underlying asset. Given the prices of a set of options on the same underlying asset with different strikes and maturities, we propose a nonparametric approach for estimating risk-neutral densities associated with several maturities. Our method uses bicubic splines in order to achieve the desired smoothness for the estimation and an optimization model to choose the spline functions that best fit the price data. Semidefinite programming is employed to guarantee the nonnegativity of the densities. We illustrate the process using synthetic option price data generated using log-normal and absolute diffusion processes as well as actual price data for options on the S&P 500 index. We also used the risk-neutral densities that we computed to price exotic options and observed that this approach generates prices that closely approximate the market prices of these options.     

Estimation in semiparametric models with missing data

This paper considers the problem of parameter estimation in a general class of semiparametric models when observations are subject to missingness at random. The semiparametric models allow for estimating functions that are non-smooth with respect to the parameter. We propose a nonparametric imputation method for the missing values, which then leads to imputed estimating equations for the finite dimensional parameter of interest. The asymptotic normality of the parameter estimator is proved in a general setting, and is investigated in detail for a number of specific semiparametric models. Finally, we study the small sample performance of the proposed estimator via simulations.     

Posterior contraction rates for the Bayesian approach to linear ill-posed inverse problems

We consider a Bayesian nonparametric approach to a family of linear inverse problems in a separable Hilbert space setting with Gaussian noise. We assume Gaussian priors, which are conjugate to the model, and present a method of identifying the posterior using its precision operator. Working with the unbounded precision operator enables us to use partial differential equations (PDE) methodology to obtain rates of contraction of the posterior distribution to a Dirac measure centered on the true solution. Our methods assume a relatively weak relation between the prior covariance, noise covariance and forward operator, allowing for a wide range of applications.     

Nonparametric conditional autoregressive expectile model via neural network with applications to estimating financial risk

The parametric conditional autoregressive expectiles (CARE) models have been developed to estimate expectiles, which can be used to assess value at risk and expected shortfall. The challenge lies in parametric CARE modeling is the specification of a parametric form. To avoid any model misspecification, we propose a nonparametric CARE model via neural network. The nonparametric CARE model can be estimated by a classical gradient based nonlinear optimization algorithm, and the consistency of nonparametric conditional expectile estimators is established. We then apply the nonparametric CARE model to estimating value at risk and expected shortfall of six stock indices. Empirical results for the new model is competitive with those classical models and parametric CARE models. Copyright © 2016 John Wiley & Sons, Ltd.