On Combining Wavelets Expansion and Sparse Linear Models for Regression on Metabolomic Data and Biomarker Selection

Wavelet thresholding of spectra has to be handled with care when the spectra are the predictors of a regression problem. Indeed, a blind thresholding of the signal followed by a regression method often leads to deteriorated predictions. The scope of this article is to show that sparse regression methods, applied in the wavelet domain, perform an automatic thresholding: the most relevant wavelet coefficients are selected to optimize the prediction of a given target of interest. This approach can be seen as a joint thresholding designed for a predictive purpose. The method is illustrated on a real world problem where metabolomic data are linked to poison ingestion. This example proves the usefulness of wavelet expansion and the good behavior of sparse and regularized methods. A comparison study is performed between the two-steps approach (wavelet thresholding and regression) and the one-step approach (selection of wavelet coefficients with a sparse regression). The comparison includes two types of wavelet bases, various thresholding methods, and various regression methods and is evaluated by calculating prediction performances. Information about the location of the most important features on the spectra was also obtained and used to identify the most relevant metabolites involved in the mice poisoning.     

A new approach to statistical efficiency of weighted least squares fitting algorithms for reparameterization of nonlinear regression models

We study nonlinear least-squares problem that can be transformed to linear problem by change of variables. We derive a general formula for the statistically optimal weights and prove that the resulting linear regression gives an optimal estimate (which satisfies an analogue of the Rao-Cramer lower bound) in the limit of small noise.     

Generalised Rank Regression Estimator with Standard Error Adjusted Lasso

One of the standard variable selection procedures in multiple linear regression is to use a penalisation technique in least‐squares (LS) analysis. In this setting, many different types of penalties have been introduced to achieve variable selection. It is well known that LS analysis is sensitive to outliers, and consequently outliers can present serious problems for the classical variable selection procedures. Since rank‐based procedures have desirable robustness properties compared to LS procedures, we propose a rank‐based adaptive lasso‐type penalised regression estimator and a corresponding variable selection procedure for linear regression models. The proposed estimator and variable selection procedure are robust against outliers in both response and predictor space. Furthermore, since rank regression can yield unstable estimators in the presence of multicollinearity, in order to provide inference that is robust against multicollinearity, we adjust the penalty term in the adaptive lasso function by incorporating the standard errors of the rank estimator. The theoretical properties of the proposed procedures are established and their performances are investigated by means of simulations. Finally, the estimator and variable selection procedure are applied to the Plasma Beta‐Carotene Level data set.     

Local composite quantile regression estimation of time-varying parameter vector for multidimensional time-inhomogeneous diffusion models

This paper is dedicated to the study of the composite quantile regression (CQR) estimations of time-varying parameter vectors for multidimensional diffusion models. Based on the local linear fitting for parameter vectors, we propose the local linear CQR estimations of the drift parameter vectors, and verify their asymptotic biases, asymptotic variances and asymptotic normality. Moreover, we discuss the asymptotic relative efficiency (ARE) of the local linear CQR estimations with respect to the local linear least-squares estimations. We obtain that the local estimations that we proposed are much more efficient than the local linear least-squares estimations. Simulation studies are constructed to show the performance of the estimations proposed.     

Practical procedures to deal with common support problems in matching estimation

This paper assesses the performance of common estimators adjusting for differences in covariates, such as matching and regression, when faced with the so-called common support problems. It also shows how different procedures suggested in the literature affect the properties of such estimators. Based on an empirical Monte Carlo simulation design, a lack of common support is found to increase the root-mean-squared error of all investigated parametric and semiparametric estimators. Dropping observations that are off support usually improves their performance, although the magnitude of the improvement depends on the particular method used.     

Robust block bootstrap panel predictability tests*

This article develops two block bootstrap-based panel predictability test procedures that are valid under very general conditions. Some of the allowable features include cross-sectional dependence, heterogeneous predictive slopes, persistent predictors, and complex error dynamics, including cross-unit endogeneity. While the first test procedure tests if there is any predictability at all, the second procedure determines the units for which predictability holds in case of a rejection by the first. A weak unit root framework is adopted to allow persistent predictors, and a novel theory is developed to establish asymptotic validity of the proposed bootstrap. Simulations are used to evaluate the performance of our tests in small samples, and their implementation is illustrated through an empirical application to stock returns.     

Discovering hidden statistical issues through individual-level models in ecological studies

ABSTRACT

In ecological studies, individual inference is made based on results from ecological models. Interpretation of the results requires caution since ecological analysis on group level may not hold in the individual level within the groups, leading to ecological fallacy. Using an ecological regression example for analyzing voting behaviors, we highlight that the explicit use of individual-level models is crucial in understanding the results of ecological studies. In particular, we clarify three relevant statistical issues for each individual-level models: assessment of the uncertainty of parameter estimates obtained from a wrong model, the use of shrinkage estimation method for simultaneous estimation of many parameters, and the necessity of sensitivity analysis rather than adhering to one seemingly most compelling assumption.     

Bias correction for parameter estimates of spatial point process models

When a spatial point process model is fitted to spatial point pattern data using standard software, the parameter estimates are typically biased. Contrary to folklore, the bias does not reflect weaknesses of the underlying mathematical methods, but is mainly due to the effects of discretization of the spatial domain. We investigate two approaches to correcting the bias: a Newton–Raphson-type correction and Richardson extrapolation. In simulation experiments, Richardson extrapolation performs best.