Proposed Estimation Procedure Research Articles

Model structure selection in single-index-coefficient regression models

Single-index-coefficient regression models (SICRM) have been proposed and used in the literature for avoiding the “curse of dimensionality”. However, there is no efficient model structure determination methodology for the SICRM. This may cause a tendency to use models that are much larger than required. In this paper, we propose a new procedure for model structure determination in the SICRM; that is, the penalized estimating equations (PEE) for variable selection that combines the “delete-one-component” method and the smoothly clipped absolute deviation penalty. The proposed PEE method can simultaneously identify significant variables of the index and estimate the nonzero coefficients of the index parameters. We also further study testing for nonparametric index-coefficient functions. Asymptotic properties for the proposed estimation procedure have been established. Under the appropriate conditions, we demonstrate that the proposed estimators have the oracle properties. Monte Carlo simulation studies are conducted to assess the finite sample performance of the proposed methods. A real example is analyzed as an illustration.

Kernel Density-Based Linear Regression Estimate

For linear regression models with non normally distributed errors, the least squares estimate (LSE) will lose some efficiency compared to the maximum likelihood estimate (MLE). In this article, we propose a kernel density-based regression estimate (KDRE) that is adaptive to the unknown error distribution. The key idea is to approximate the likelihood function by using a nonparametric kernel density estimate of the error density based on some initial parameter estimate. The proposed estimate is shown to be asymptotically as efficient as the oracle MLE which assumes the error density were known. In addition, we propose an EM type algorithm to maximize the estimated likelihood function and show that the KDRE can be considered as an iterated weighted least squares estimate, which provides us some insights on the adaptiveness of KDRE to the unknown error distribution. Our Monte Carlo simulation studies show that, while comparable to the traditional LSE for normal errors, the proposed estimation procedure can have substantial efficiency gain for non normal errors. Moreover, the efficiency gain can be achieved even for a small sample size.

Maximum-likelihood estimation for multi-aspect multi-baseline SAR interferometry of urban areas

The reconstruction of digital surface models (DSMs) of urban areas from interferometric synthetic aperture radar (SAR) data is a challenging task. In particular the SAR inherent layover and shadowing effects need to be coped with by sophisticated processing strategies. In this paper, a maximum-likelihood estimation procedure for the reconstruction of DSMs from multi-aspect multi-baseline InSAR imagery is proposed. In this framework, redundant as well as contradicting observations are exploited in a statistically optimal way. The presented method, which is especially suited for single-pass SAR interferometers, is examined using test data consisting of experimental airborne millimeterwave SAR imagery. The achievable accuracy is evaluated by comparison to LiDAR-derived reference data. It is shown that the proposed estimation procedure performs better than a comparable non-statistical reconstruction method.

Nonparametric spectral analysis of heart rate variability through penalized sum of squares

Researchers in a variety of biomedical fields have utilized frequency domain properties of heart rate variability (HRV), or the elapsed time between consecutive heart beats. HRV is measured from the electrocardiograph signal through the interbeat interval series. Popular approaches for estimating power spectra from these interval data apply common spectral analysis methods that are designed for the analysis of evenly sampled time series. The application of these methods to the interbeat interval series, which is indexed over an uneven time grid, requires a bias-inducing transformation. The goal of this article is to explore the use of penalized sum of squares for nonparametric estimation of the spectrum of HRV directly from the interbeat intervals. A novel cross-validation procedure is introduced for smoothing parameter selection. Empirical properties of the proposed estimation procedure are explored and compared with popular methods in a simulation study. The proposed method is used in an analysis of data from an insomnia study, which seeks to illuminate the association between the power spectrum of HRV during different periods of sleep with response to behavioral therapy.

Nonparametric Mixture of Regression Models

Motivated by an analysis of U.S. house price index (HPI) data, we propose nonparametric finite mixture of regression models. We study the identifiability issue of the proposed models, and develop an estimation procedure by employing kernel regression. We further systematically study the sampling properties of the proposed estimators, and establish their asymptotic normality. A modified EM algorithm is proposed to carry out the estimation procedure. We show that our algorithm preserves the ascent property of the EM algorithm in an asymptotic sense. Monte Carlo simulations are conducted to examine the finite sample performance of the proposed estimation procedure. An empirical analysis of the U.S. HPI data is illustrated for the proposed methodology.

Controlling the bias of robust small-area estimators

Sinha & Rao (2009) proposed estimation procedures designed for small-area means, based on robustified maximum likelihood estimators and robust empirical best linear unbiased predictors. Their methods are of the plug-in type and may be biased. Bias-corrected estimators have been proposed by Chambers et al. (2013). Here, we investigate two new approaches: one relying on the work of Chambers (1986), and the second using the concept of conditional bias to measure the influence of units in the population. These two classes of estimators also include correction terms for the bias but are both fully bias-corrected, in the sense that the corrections account for the potential impact of the other domains on the small area of interest. Monte Carlo simulations suggest that the Sinha--Rao method and the bias-adjusted estimator of Chambers et al. (2013) may exhibit a large bias, while the new procedures often offer lower bias and mean squared error. A parametric bootstrap procedure is considered for constructing confidence intervals. Copyright 2013, Oxford University Press.

On using angular cross-correlations to determine source redshift distributions

We investigate how well the redshift distribution of a population of extragalactic objects can be reconstructed using angular cross-correlations with a sample whose redshifts are known. We derive the minimum variance quadratic estimator, which has simple analytic representations in very applicable limits and is significantly more sensitive than earlier proposed estimation procedures. This estimator is straightforward to apply to observations, it robustly finds the likelihood maximum, and it conveniently selects angular scales at which fluctuations are well approximated as independent between redshift bins and at which linear theory applies. We find that the linear bias times number of objects in a redshift bin generally can be constrained with cross-correlations to fractional error (10^2 n/N)^1/2, where N is the total number of spectra per dz and n is the number of redshift bins spanned by the bulk of the unknown population. The error is often independent of the sky area and sampling fraction. Furthermore, we find that sub-percent measurements of the angular source density per unit redshift, dN/dz, are in principle possible, although cosmic magnification needs to be accounted for at fractional errors of <~ 10 per cent. We discuss how the sensitivity to dN/dz changes as a function of photometric and spectroscopic depth and how to optimize the survey strategy to constrain dN/dz. We also quantify how well cross-correlations of photometric redshift bins can be used to self-calibrate a photometric redshift sample. Simple formulae that can be quickly applied to gauge the utility of cross correlating different samples are given.

Robust estimation and variable selection for semiparametric partially linear varying coefficient model based on modal regression

The semiparametric partially linear varying coefficient models (SPLVCM) are frequently used in statistical modelling, but most existing methods were built on either the least-square or likelihood-based methods, which are very sensitive to the outliers and their efficiency may be significantly reduced for heavy tail error distribution. This paper proposes a new efficient and robust estimation procedure for the SPLVCM based on modal regression. 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 develop a variable selection procedure to select significant parametric components for the SPLVCM and prove the method possessing the oracle property. We also discuss the bandwidth selection and propose an expectation–maximisation-type algorithm for the proposed estimation procedure. Some simulation results and real data analysis confirm that the newly proposed method works very competitively compared to other existing methods.

Logistic‐AFT location‐scale mixture regression models with nonsusceptibility for left‐truncated and general interval‐censored data

In conventional survival analysis there is an underlying assumption that all study subjects are susceptible to the event. In general, this assumption does not adequately hold when investigating the time to an event other than death. Owing to genetic and/or environmental etiology, study subjects may not be susceptible to the disease. Analyzing nonsusceptibility has become an important topic in biomedical, epidemiological, and sociological research, with recent statistical studies proposing several mixture models for right-censored data in regression analysis. In longitudinal studies, we often encounter left, interval, and right-censored data because of incomplete observations of the time endpoint, as well as possibly left-truncated data arising from the dissimilar entry ages of recruited healthy subjects. To analyze these kinds of incomplete data while accounting for nonsusceptibility and possible crossing hazards in the framework of mixture regression models, we utilize a logistic regression model to specify the probability of susceptibility, and a generalized gamma distribution, or a log-logistic distribution, in the accelerated failure time location-scale regression model to formulate the time to the event. Relative times of the conditional event time distribution for susceptible subjects are extended in the accelerated failure time location-scale submodel. We also construct graphical goodness-of-fit procedures on the basis of the Turnbull-Frydman estimator and newly proposed residuals. Simulation studies were conducted to demonstrate the validity of the proposed estimation procedure. The mixture regression models are illustrated with alcohol abuse data from the Taiwan Aboriginal Study Project and hypertriglyceridemia data from the Cardiovascular Disease Risk Factor Two-township Study in Taiwan.

Partly Conditional Estimation of the Effect of a Time‐Dependent Factor in the Presence of Dependent Censoring

We propose semiparametric methods for estimating the effect of a time-dependent covariate on treatment-free survival. The data structure of interest consists of a longitudinal sequence of measurements and a potentially censored survival time. The factor of interest is time-dependent. Treatment-free survival is of interest and is dependently censored by the receipt of treatment. Patients may be removed from consideration for treatment, temporarily or permanently. The proposed methods combine landmark analysis and partly conditional hazard regression. A set of calendar time cross-sections is specified, and survival time (from cross-section date) is modeled through weighted Cox regression. The assumed model for death is marginal in the sense that time-varying covariates are taken as fixed at each landmark, with the mortality hazard function implicitly averaging across future covariate trajectories. Dependent censoring is overcome by a variant of inverse probability of censoring weighting (IPCW). The proposed estimators are shown to be consistent and asymptotically normal, with consistent covariance estimators provided. Simulation studies reveal that the proposed estimation procedures are appropriate for practical use. We apply the proposed methods to pre-transplant mortality among end-stage liver disease (ESLD) patients.

Estimated relative vocal tract lengths from vowel spectra based on fundamental frequency adaptive analyses and their relations to relevant physical data of speakers

A Japanese vowel database of males, females and children speakers (385 speakers in total) along with relevant physical data [Deguchi et al. (2011)] was analyzed using a set of F0 adaptive procedures, which were developed for a speech analysis, modification and synthesis framework TANDEM-STRAIGHT [Kawahara et al. (2008)] and its extensions. By restricting spectral region in estimating ratios for vocal tract length normalization (VTLN), cepstrally weighted distance measure yielded estimates with 2% standard error, by assuming relative vocal tract lengths estimated using regression analysis as the ground truth. A set of regression analyses of the estimated relative vocal tract lengths, average F0s, ages, gender, and physical data (speakers' height, weight) were conducted. The results suggest that the proposed analysis procedures applied to the Japanese five vowels may provide sufficient information for estimating speakers' physical data. Possible applications of the proposed estimation procedures will also be discussed.

On Bayesian estimation of regression models subject to uncertainty about functional constraints

In this paper, we provide a Bayesian estimation procedure for the regression models when the constraint of the regression function needs to be incorporated in modeling but such a restriction is uncertain. For this purpose, we consider a family of rectangle screened multivariate Gaussian prior distributions in order to reflect uncertainty about the functional constraint, and propose the Bayesian estimation procedure of the regression models based on two stages of a prior hierarchy of the functional constraint, referred to as hierarchical screened Gaussian regression models (HSGRM). Specifically, we explore theoretical properties of the proposed estimation procedure by deriving the posterior distribution and predictive distribution of the unknown parameters under HSGRM in analytic forms, and discuss specific applications to regression models with uncertain functional constraints that can be explained in the context of HSGRM.

Quantile Regression for Recurrent Gap Time Data

Evaluating covariate effects on gap times between successive recurrent events is of interest in many medical and public health studies. While most existing methods for recurrent gap time analysis focus on modeling the hazard function of gap times, a direct interpretation of the covariate effects on the gap times is not available through these methods. In this article, we consider quantile regression that can provide direct assessment of covariate effects on the quantiles of the gap time distribution. Following the spirit of the weighted risk-set method by Luo and Huang (2011, Statistics in Medicine 30, 301-311), we extend the martingale-based estimating equation method considered by Peng and Huang (2008, Journal of the American Statistical Association 103, 637-649) for univariate survival data to analyze recurrent gap time data. The proposed estimation procedure can be easily implemented in existing software for univariate censored quantile regression. Uniform consistency and weak convergence of the proposed estimators are established. Monte Carlo studies demonstrate the effectiveness of the proposed method. An application to data from the Danish Psychiatric Central Register is presented to illustrate the methods developed in this article.

Semi-Parametric Analysis of Shape-Invariant Engel Curves with Control Function Approach

An extended generalised partially linear single-index (EGPLSI) model provides flexibility of a partially linear model and a single-index model. Furthermore, it also allows for the analysis of the shape-invariant specification. Especially, since it does not only provide the flexibility of a partially linear (PL) and a single-index (SI) models, but also enables an analysis of the shape-invariant specification. Nonetheless, the model's practicality in the empirical studies has been hampered by lack of appropriate estimation procedure and method to deal with endogeneity. In the current paper, we establish an alternative control function approach to address the endogeneity issue in the estimation of the EGPLSI model. We also show that all attractive features of the EGPLSI model discussed in the literature are still available under the proposed estimation procedure. Economic literature suggests that semiparametric technique is an important tool for an empirical analysis of Engel curves, which often involves endogeneity of the total expenditure. We show that our newly developed method is applicable and able to address the endogeneity problems in semiparametric analysis of the empirical Engel curves.

Effective Rotation Patterns in Successive Sampling Over Two Occasions

The problem of estimation of finite population mean on the current occasion in two-occasion successive sampling has been considered. Utilizing all the readily available information from first and second occasions, an efficient estimation procedure has been suggested. Optimum replacement policy relevant to the proposed estimation procedure has been discussed. A proposed estimator is compared with natural estimators and suitable recommendations have been made.

A universally consistent modification of maximum likelihood

In some models, both parametric and not, maximum likelihood estimation fails to be consistent. We investigate why the maximum likelihood method breaks down with some examples and notice the paradox that, in those same models, maximum likelihood estimation would have been consistent if the data had been measured with error. With this motivation we define doubly-smoothed maximum likelihood as a natural mechanism for adding measurement error without bias. We show the proposed estimation procedure gives universal consistency in independent and identically distributed data. Our method of proof is new. The same arguments can show maximum likelihood itself is universally consistent in a discrete sample space. It is shown that the asymptotic efficiency can be quite high even when the bandwidth parameters are held fixed. Practical guidelines for the choice of kernel and tuning parameter are also given.

Estimating Archimedean Copulas in High Dimensions

Abstract. This article presents a novel estimation procedure for high‐dimensional Archimedean copulas. In contrast to maximum likelihood estimation, the method presented here does not require derivatives of the Archimedean generator. This is computationally advantageous for high‐dimensional Archimedean copulas in which higher‐order derivatives are needed but are often difficult to obtain. Our procedure is based on a parameter‐dependent transformation of the underlying random variables to a one‐dimensional distribution where a minimum‐distance method is applied. We show strong consistency of the resulting minimum‐distance estimators to the case of known margins as well as to the case of unknown margins when pseudo‐observations are used. Moreover, we conduct a simulation comparing the performance of the proposed estimation procedure with the well‐known maximum likelihood approach according to bias and standard deviation.

Estimating the Period of a Cyclic Non‐Homogeneous Poisson Process

Abstract. Motivated by applications of Poisson processes for modelling periodic time‐varying phenomena, we study a semi‐parametric estimator of the period of cyclic intensity function of a non‐homogeneous Poisson process. There are no parametric assumptions on the intensity function which is treated as an infinite dimensional nuisance parameter. We propose a new family of estimators for the period of the intensity function, address the identifiability and consistency issues and present simulations which demonstrate good performance of the proposed estimation procedure in practice. We compare our method to competing methods on synthetic data and apply it to a real data set from a call center.

Local modal regression

A local modal estimation procedure is proposed for the regression function in a nonparametric regression model. A distinguishing characteristic of the proposed procedure is that it introduces an additional tuning parameter that is automatically selected using the observed data in order to achieve both robustness and efficiency of the resulting estimate. We demonstrate both theoretically and empirically that the resulting estimator is more efficient than the ordinary local polynomial regression (LPR) estimator in the presence of outliers or heavy-tail error distribution (such as t-distribution). Furthermore, we show that the proposed procedure is as asymptotically efficient as the LPR estimator when there are no outliers and the error distribution is a Gaussian distribution. We propose an expectation–maximisation-type algorithm for the proposed estimation procedure. A Monte Carlo simulation study is conducted to examine the finite sample performance of the proposed method. The simulation results confirm the theoretical findings. The proposed methodology is further illustrated via an analysis of a real data example.

Quantile regression for competing risks data with missing cause of failure

This paper considers generalized linear quantile regression for competing risks data when the failure type may be missing. Two estimation procedures for the regression co-efficients, including an inverse probability weighted complete-case estimator and an augmented inverse probability weighted estimator, are discussed under the assumption that the failure type is missing at random. The proposed estimation procedures utilize supplemental auxiliary variables for predicting the missing failure type and for informing its distribution. The asymptotic properties of the two estimators are derived and their asymptotic efficiencies are compared. We show that the augmented estimator is more efficient and possesses a double robustness property against misspecification of either the model for missingness or for the failure type. The asymptotic covariances are estimated using the local functional linearity of the estimating functions. The finite sample performance of the proposed estimation procedures are evaluated through a simulation study. The methods are applied to analyze the 'Mashi' trial data for investigating the effect of formula-versus breast-feeding plus extended infant zidovudine prophylaxis on HIV-related death of infants born to HIV-infected mothers in Botswana.