On semiparametric pivotal bayesian inference for quantiles

In semiparametric inference we distinguish between the parameter of interest which may be a location parameter, and a nuisance parameter that determines the remaining shape of the sampling distribution. As was pointed out by Diaconis and Freedman the main problem in semiparametric Bayesian inference is to obtain a consistent posterior distribution for the parameter of interest. The present paper considers a semiparametric Bayesian method based on a pivotal likelihood function. It is shown that when the parameter of interest is the median, this method produces a consistent posterior distribution and is easily implemented, Numerical comparisons with classical methods and with Bayesian methods based on a Dirichlet prior are provided. It is also shown that in the case of symmetric intervals, the classical confidence coefficients have a Bayesian interpretation as the limiting posterior probability of the interval based on the Dirichlet prior with a parameter that converges to zero.     

Semiparametric regression of clustered current status data

This paper discusses regression analysis of clustered current status data under semiparametric additive hazards models. In particular, we consider the situation when cluster sizes can be informative about correlated failure times from the same cluster. To address the problem, we present estimating equation-based estimation procedures and establish asymptotic properties of the resulting estimates. Finite sample performance of the proposed method is assessed through an extensive simulation study, which indicates the procedure works well. The method is applied to a motivating data set from a lung tumorigenicity study.     

Total least squares solution for compositional data using linear models

The restrictive properties of compositional data, that is multivariate data with positive parts that carry only relative information in their components, call for special care to be taken while performing standard statistical methods, for example, regression analysis. Among the special methods suitable for handling this problem is the total least squares procedure (TLS, orthogonal regression, regression with errors in variables, calibration problem), performed after an appropriate log-ratio transformation. The difficulty or even impossibility of deeper statistical analysis (confidence regions, hypotheses testing) using the standard TLS techniques can be overcome by calibration solution based on linear regression. This approach can be combined with standard statistical inference, for example, confidence and prediction regions and bounds, hypotheses testing, etc., suitable for interpretation of results. Here, we deal with the simplest TLS problem where we assume a linear relationship between two errorless measurements of the same object (substance, quantity). We propose an iterative algorithm for estimating the calibration line and also give confidence ellipses for the location of unknown errorless results of measurement. Moreover, illustrative examples from the fields of geology, geochemistry and medicine are included. It is shown that the iterative algorithm converges to the same values as those obtained using the standard TLS techniques. Fitted lines and confidence regions are presented for both original and transformed compositional data. The paper contains basic principles of linear models and addresses many related problems.     

Semiparametric estimation of the cumulative incidence function under competing risks and left-truncated sampling

In this article, we propose semiparametric methods to estimate the cumulative incidence function of two dependent competing risks for left-truncated and right-censored data. The proposed method is based on work by Huang and Wang (1995). We extend previous model by allowing for a general parametric truncation distribution and a third competing risk before recruitment. Based on work by Vardi (1989), several iterative algorithms are proposed to obtain the semiparametric estimates of cumulative incidence functions. The asymptotic properties of the semiparametric estimators are derived. Simulation results show that a semiparametric approach assuming the parametric truncation distribution is correctly specified produces estimates with smaller mean squared error than those obtained in a fully nonparametric model.     

Semiparametric model average prediction in panel data analysis

Forecasting in economic data analysis is dominated by linear prediction methods where the predicted values are calculated from a fitted linear regression model. With multiple predictor variables, multivariate nonparametric models were proposed in the literature. However, empirical studies indicate the prediction performance of multi-dimensional nonparametric models may be unsatisfactory. We propose a new semiparametric model average prediction (SMAP) approach to analyse panel data and investigate its prediction performance with numerical examples. Estimation of individual covariate effect only requires univariate smoothing and thus may be more stable than previous multivariate smoothing approaches. The estimation of optimal weight parameters incorporates the longitudinal correlation and the asymptotic properties of the estimated results are carefully studied in this paper.     

NEW EFFICIENT ESTIMATION AND VARIABLE SELECTION METHODS FOR SEMIPARAMETRIC VARYING-COEFFICIENT PARTIALLY LINEAR MODELS

The complexity of semiparametric models poses new challenges to statistical inference and model selection that frequently arise from real applications. In this work, we propose new estimation and variable selection procedures for the semiparametric varying-coefficient partially linear model. We first study quantile regression estimates for the nonparametric varying-coefficient functions and the parametric regression coefficients. To achieve nice efficiency properties, we further develop a semiparametric composite quantile regression procedure. We establish the asymptotic normality of proposed estimators for both the parametric and nonparametric parts and show that the estimators achieve the best convergence rate. Moreover, we show that the proposed method is much more efficient than the least-squares-based method for many non-normal errors and that it only loses a small amount of efficiency for normal errors. In addition, it is shown that the loss in efficiency is at most 11.1% for estimating varying coefficient functions and is no greater than 13.6% for estimating parametric components. To achieve sparsity with high-dimensional covariates, we propose adaptive penalization methods for variable selection in the semiparametric varying-coefficient partially linear model and prove that the methods possess the oracle property. Extensive Monte Carlo simulation studies are conducted to examine the finite-sample performance of the proposed procedures. Finally, we apply the new methods to analyze the plasma beta-carotene level data.     

SEMIPARAMETRIC ADDITIVE RISKS REGRESSION FOR TWO-STAGE DESIGN SURVIVAL STUDIES

In this article we study a semiparametric additive risks model (McKeague and Sasieni (1994)) for two-stage design survival data where accurate information is available only on second stage subjects, a subset of the first stage study. We derive two-stage estimators by combining data from both stages. Large sample inferences are developed. As a by-product, we also obtain asymptotic properties of the single stage estimators of McKeague and Sasieni (1994) when the semiparametric additive risks model is misspecified. The proposed two-stage estimators are shown to be asymptotically more efficient than the second stage estimators. They also demonstrate smaller bias and variance for finite samples. The developed methods are illustrated using small intestine cancer data from the SEER (Surveillance, Epidemiology, and End Results) Program.     

Testing in semiparametric models with interaction, with applications to gene–environment interactions

Summary.  Motivated from the problem of testing for genetic effects on complex traits in the presence of gene–environment interaction, we develop score tests in general semiparametric regression problems that involves Tukey style 1 degree-of-freedom form of interaction between parametrically and non-parametrically modelled covariates. We find that the score test in this type of model, as recently developed by Chatterjee and co-workers in the fully parametric setting, is biased and requires undersmoothing to be valid in the presence of non-parametric components. Moreover, in the presence of repeated outcomes, the asymptotic distribution of the score test depends on the estimation of functions which are defined as solutions of integral equations, making implementation difficult and computationally taxing. We develop profiled score statistics which are unbiased and asymptotically efficient and can be performed by using standard bandwidth selection methods. In addition, to overcome the difficulty of solving functional equations, we give easy interpretations of the target functions, which in turn allow us to develop estimation procedures that can be easily implemented by using standard computational methods. We present simulation studies to evaluate type I error and power of the method proposed compared with a naive test that does not consider interaction. Finally, we illustrate our methodology by analysing data from a case–control study of colorectal adenoma that was designed to investigate the association between colorectal adenoma and the candidate gene NAT2 in relation to smoking history.     

Analysing competing risks data with transformation models

We present a flexible class of marginal models for the cumulative incidence function. The semiparametric transformation model is utilized in a decomposition for the marginal failure probabilities which extends previous work on Farewell's cure model. Novel estimation, inference and prediction procedures are developed, with large sample properties derived from the theory of martingales and U-statistics. A small simulation study demonstrates that the methods are appropriate for practical use. The methods are illustrated with a thorough analysis of a prostate cancer clinical trial. Simple graphical displays are used to check for the goodness of fit.     

A gradient-based algorithm for semiparametric models with missing covariates

In the parametric regression model, the covariate missing problem under missing at random is considered. It is often desirable to use flexible parametric or semiparametric models for the covariate distribution, which can reduce a potential misspecification problem. Recently, a completely nonparametric approach was developed by [H.Y. Chen, Nonparametric and semiparametric models for missing covariates in parameter regression, J. Amer. Statist. Assoc. 99 (2004), pp. 1176–1189; Z. Zhang and H.E. Rockette, On maximum likelihood estimation in parametric regression with missing covariates, J. Statist. Plann. Inference 47 (2005), pp. 206–223]. Although it does not require a model for the covariate distribution or the missing data mechanism, the proposed method assumes that the covariate distribution is supported only by observed values. Consequently, their estimator is a restricted maximum likelihood estimator (MLE) rather than the global MLE. In this article, we show the restricted semiparametric MLE could be very misleading in some cases. We discuss why this problem occurs and suggest an algorithm to obtain the global MLE. Then, we assess the performance of the proposed method via some simulation experiments.     

Semiparametric method for detecting multiple change points model in financial time series

The problem of multiple change points has been discussed in these years on the background of financial shocks. In order to decrease the damage, it is worthy to find a more available model for the problem as precise as possible by the information from data set. This paper proposes the problem of detecting the change points by semiparametric test. The change points estimations are obtained by empirical likelihood method. Then some asymptotic results for multiple change points are obtained by loglikelihood ratio test and law of large numbers. Furthermore, the consistency of change points estimations is presented. Indeed, the method and steps to find the change points are derived. The simulation experiments prove that the semiparametric test is more efficient than nonparametric test. The diagnosis with simulation and the applications for multiple change points also illustrates the proposed model well.     

Article: 2

Summary. Searching for an effective dimension reduction space is an important problem in regression, especially for high dimensional data. We propose an adaptive approach based on semiparametric models, which we call the (conditional) minimum average variance estimation (MAVE) method, within quite a general setting. The MAVE method has the following advantages. Most existing methods must undersmooth the nonparametric link function estimator to achieve a faster rate of consistency for the estimator of the parameters (than for that of the nonparametric function). In contrast, a faster consistency rate can be achieved by the MAVE method even without undersmoothing the nonparametric link function estimator. The MAVE method is applicable to a wide range of models, with fewer restrictions on the distribution of the covariates, to the extent that even time series can be included. Because of the faster rate of consistency for the parameter estimators, it is possible for us to estimate the dimension of the space consistently. The relationship of the MAVE method with other methods is also investigated. In particular, a simple outer product gradient estimator is proposed as an initial estimator. In addition to theoretical results, we demonstrate the efficacy of the MAVE method for high dimensional data sets through simulation. Two real data sets are analysed by using the MAVE approach.     

Penalized models to estimate customer survival

In this paper we propose a novel procedure, for the estimation of semiparametric survival functions. The proposed technique adapts penalized likelihood survival models to the context of lifetime value modeling. The method extends classical Cox model by introducing a smoothing parameter that can be estimated by means of penalized maximum likelihood procedures. Markov Chain Monte Carlo methods are employed to effectively estimate such smoothing parameter, using an algorithm which combines Metropolis–Hastings and Gibbs sampling. Our proposal is contextualized and compared with conventional models, with reference to a marketing application that involves the prediction of customer’s lifetime value estimation.     

A SEMIPARAMETRIC MODEL FOR A FUNCTIONAL BEHAVIOURAL RESPONSE TO CAPTURE IN CAPTURE–RECAPTURE EXPERIMENTS

Capture–recapture experiments are commonly used to estimate the size of a closed population. However, the associated estimators of the population size are well known to be highly sensitive to misspecification of the capture probabilities. To address this, we present a general semiparametric framework for the analysis of capture–recapture experiments when the capture probability depends on individual characteristics, time effects and behavioural response. This generalizes well‐known general parametric capture–recapture models and extends previous semiparametric models in which there is no time dependence or behavioural response. The method is evaluated in simulations and applied to two real data sets.     

Semiparametric models: a generalized self-consistency approach

Summary. In semiparametric models, the dimension d of the maximum likelihood problem is potentially unlimited. Conventional estimation methods generally behave like O ( d ³). A new O ( d ) estimation procedure is proposed for a large class of semiparametric models. Potentially unlimited dimension is handled in a numerically efficient way through a Nelson–Aalen-like estimator. Discussion of the new method is put in the context of recently developed minorization–maximization algorithms based on surrogate objective functions. The procedure for semiparametric models is used to demonstrate three methods to construct a surrogate objective function: using the difference of two concave functions, the EM way and the new quasi-EM (QEM) approach. The QEM approach is based on a generalization of the EM-like construction of the surrogate objective function so it does not depend on the missing data representation of the model. Like the EM algorithm, the QEM method has a dual interpretation, a result of merging the idea of surrogate maximization with the idea of imputation and self-consistency. The new approach is compared with other possible approaches by using simulations and analysis of real data. The proportional odds model is used as an example throughout the paper.     

Recent Two-Stage Sample Selection Procedures With an Application to the Gender Wage Gap

Recently developed two-stage estimation methods of sample selection models are used, in the context of data from the 1989 Labor Market Activity Survey, to examine labor supply decisions and wage outcomes for employed men and women. Recent hypothesis test procedures are used to test for no sample selection and to test for a parametric against a semiparametric selection-correction procedure. We conclude that selection is indeed an issue for the sample at hand and that the semiparametric specification is appropriate. We also present the standard decomposition of the gender wage gap into its explained and unexplained portions.     

Semiparametric Analysis of Truncated Data

Randomly truncated data are frequently encountered in many studies where truncation arises as a result of the sampling design. In the literature, nonparametric and semiparametric methods have been proposed to estimate parameters in one-sample models. This paper considers a semiparametric model and develops an efficient method for the estimation of unknown parameters. The model assumes that K populations have a common probability distribution but the populations are observed subject to different truncation mechanisms. Semiparametric likelihood estimation is studied and the corresponding inferences are derived for both parametric and nonparametric components in the model. The method can also be applied to two-sample problems to test the difference of lifetime distributions. Simulation results and a real data analysis are presented to illustrate the methods.     

Estimation in the Cox cure model with covariates missing not at random,with application to disease screening/prediction

In an attempt to provide a statistical tool for disease screening and prediction, we propose a semiparametric approach to analysis of the Cox proportional hazards cure model in situations where the observations on the event time are subject to right censoring and some covariates are missing not at random. To facilitate the methodological development, we begin with semiparametric maximum likelihood estimation (SPMLE) assuming that the (conditional) distribution of the missing covariates is known. A variant of the EM algorithm is used to compute the estimator. We then adapt the SPMLE to a more practical situation where the distribution is unknown and there is a consistent estimator based on available information. We establish the consistency and weak convergence of the resulting pseudo-SPMLE, and identify a suitable variance estimator. The application of our inference procedure to disease screening and prediction is illustrated via empirical studies. The proposed approach is used to analyze the tuberculosis screening study data that motivated this research. Its finite-sample performance is examined by simulation.     

Semiparametric estimation of employment duration models

There are a variety of economic areas, such as studies of employment duration and of the durability of capital goods, in which data on important variables typically are censored. The standard techinques for estimating a model from censored data require the distributions of unobservable random components of the model to be specified a priori up to a finite set of parameters, and misspecification of these distributions usually leads to inconsistent parameter estimates. However, economic theory rarely gives guidance about distributions and the standard estimation techniques do not provide convenient methods for identifying distributions from censored data. Recently, several distribution-free or semiparametric methods for estimating censored regression models have been developed. This paper presents the results of using two such methods to estimate a model of employment duration. The paper reports the operating characteristics of the semiparametric estimators and compares the semiparametric estimates with those obtained from a standard parametric model.     

Some asymptotic results for semiparametric nonlinear mixed-effects models with incomplete data

In modeling complex longitudinal data, semiparametric nonlinear mixed-effects (SNLME) models are very flexible and useful. Covariates are often introduced in the models to partially explain the inter-individual variations. In practice, data are often incomplete in the sense that there are often measurement errors and missing data in longitudinal studies. The likelihood method is a standard approach for inference for these models but it can be computationally very challenging, so computationally efficient approximate methods are quite valuable. However, the performance of these approximate methods is often based on limited simulation studies, and theoretical results are unavailable for many approximate methods. In this article, we consider a computationally efficient approximate method for a class of SNLME models with incomplete data and investigate its theoretical properties. We show that the estimates based on the approximate method are consistent and asymptotically normally distributed.