Construction Of Confidence Regions Research Articles

Bootstrap confidence regions for correspondence analysis

A bootstrap-based method for constructing confidence regions (CRs) around row or column points projected onto a pair of axes from the correspondence analysis (CA) of a two-way contingency table is presented. These regions deal with the specific question of the sampling variation of sample row and column profile points around population row and column profile points when both are projected onto the observed axes, rather than the decomposition of the χ2-test of independence or the general question of the stability of the observed CA display which has been considered in previous work. The method therefore constructs the regions in a different way to what has been proposed before. A simulation experiment shows that the method performs well in most of the situations in which it might be used, with a few exceptions being noted. An example illustrates that the method produces conclusions which are consistent with those from detailed parametric modelling.

Empirical likelihood analysis of longitudinal data involving within-subject correlation

In this paper we use profile empirical likelihood to construct confidence regions for regression coefficients in partially linear model with longitudinal data. The main contribution is that the within-subject correlation is considered to improve estimation efficiency. We suppose a semi-parametric structure for the covariances of observation errors in each subject and employ both the first order and the second order moment conditions of the observation errors to construct the estimating equations. Although there are nonparametric estimators, the empirical log-likelihood ratio statistic still tends to a standard χ 2 variable in distribution after the nuisance parameters are profiled away. A data simulation is also conducted.

On optimal choice of order statistics in large samples for the construction of confidence regions for the location and scale

Given a large sample from a location-scale population we estimate the unknown parameters by means of confidence regions constructed on the basis of two order statistics. The problem of the best choice of those statistics to obtain good estimates, as \(n\rightarrow \infty ,\) is considered.

Statistical inference for a single-index varying-coefficient model

We investigate the estimators of parameters of interest for a single-index varying-coefficient model. To estimate the unknown parameter efficiently, we first estimate the nonparametric component using local linear smoothing, then construct an estimator of parametric component by using estimating equations. Our estimator for the parametric component is asymptotically efficient, and the estimator of nonparametric component has asymptotic normality and optimal uniform convergence rate. Our results provide ways to construct confidence regions for the involved unknown parameters. The finite-sample behavior of the new estimators is evaluated through simulation studies, and applications to two real data are illustrated.

Construction of confidence regions in problem on probabilistic study into motion of minor bodies of the solar system

Different approaches to constructing regions of possible motions of minor bodies are examined. An economical approach to the minimization of the number of points of the initial region by means of defining this region with its boundary surface is offered, and the estimations of its efficiency are given. The methods for estimating the admissibility of the linear approach are considered. For this purpose, simple methods for calculating the nonlinear factors are offered, which makes it possible to classify a problem to be solved as either strongly or weakly nonlinear. Recommendations are given on the possibility of reducing the concrete estimation problem to a weakly nonlinear one, where the more economical linear approach can be used. The combined method of mapping the initial region in time is also offered which unites the linear and nonlinear approaches. By example of two asteroids, the area of applicability of linear mappings is estimated.

Simultaneous confidence regions for closed tests, includingHolm‐,Hochberg‐, andHommel‐related procedures

The derivation of simultaneous confidence regions for some multiple-testing procedures (MTPs) of practical interest has remained an unsolved problem. This is the case, for example, for Hochberg's step-up MTP and Hommel's more powerful MTP that is neither a step-up nor a step-down procedure. It is shown in this article how the direct approach used previously by the author to construct confidence regions for certain closed-testing procedures (CTPs) can be extended to a rather general setup. The general results are then applied to a situation with one-sided inferences and CTPs belonging to a class studied by Wei Liu. This class consists of CTPs based on ordered marginal p-values. It includes Holm's, Hochberg's, and Hommel's MTPs. A property of the confidence regions derived for these three MTPs is that no confidence assertions sharper than rejection assertions can be made unless all null hypotheses are rejected. Briefly, this is related to the fact that these MTPs are quite powerful. The class of CTPs considered includes, however, also MTPs related to Holm's, Hochberg's, and Hommel's MTPs that are less powerful but are such that confidence assertions sharper than rejection assertions are possible even if not all null hypotheses are rejected. One may thus choose and prespecify such an MTP, though this is at the cost of less rejection power.

Random weighting [formula omitted]-estimation for linear errors-in-variables models

In this paper, we extend the random weighting method to linear errors-in-variables models and propose random weighting M-estimators (RWME) for parameters. Its large sample properties are studied and the consistency and asymptotic normality are proved under mild conditions. In addition, the results facilitate the construction of confidence regions and hypothesis testing for the unknown parameters. Extensive simulations are reported, showing that the proposed method works well in practical settings. The proposed methods are also applied to a data set from an AIDS clinical trial group study.

Treatment effect bounds: An application to Swan–Ganz catheterization

We reanalyze data from the observational study by Connors et al. (1996) on the impact of Swan–Ganz catheterization on mortality outcomes. The study by Connors et al. (1996) assumes that there are no unobserved differences between patients who are catheterized and patients who are not catheterized and finds that catheterization increases patient mortality. We instead allow for such differences between patients by implementing both the instrumental variable bounds of Manski (1990), which only exploits an instrumental variable, and the bounds of Shaikh and Vytlacil (2011), which exploit mild nonparametric, structural assumptions in addition to an instrumental variable. We propose and justify the use of indicators of weekday admission as an instrument for catheterization in this context. We find that in our application, the Manski (1990) bounds do not indicate whether catheterization increases or decreases mortality, where as the Shaikh and Vytlacil (2011) bounds reveal that at least for some diagnoses, Swan–Ganz catheterization reduces mortality at 7 days after catheterization. We show that the bounds of Shaikh and Vytlacil (2011) remain valid under even weaker assumptions than those described in Shaikh and Vytlacil (2011). We also extend the analysis to exploit a further nonparametric, structural assumption–that doctors catheterize individuals with systematically worse latent health–and find that this assumption further narrows these bounds and strengthens our conclusions. In our analysis, we construct confidence regions using the methodology developed in Romano and Shaikh (2008). We show in particular that the confidence regions are uniformly consistent in level over a large class of possible distributions for the observed data that include distributions where the instrument is arbitrarily “weak”.

Empirical Likelihood for a Heteroscedastic Partial Linear Errors-in-Variables Model

The purpose of this article is to use the empirical likelihood method to study construction of the confidence region for the parameter of interest in heteroscedastic partially linear errors-in-variables model with martingale difference errors. When the variance functions of the errors are known or unknown, we propose the empirical log-likelihood ratio statistics for the parameter of interest. For each case, a nonparametric version of Wilks’ theorem is derived. The results are then used to construct confidence regions of the parameter. A simulation study is carried out to assess the performance of the empirical likelihood method.

Testing cosmology with extreme galaxy clusters

ABSTRACT Motivated by recent suggestions that a number of observed galaxy clusters have masses which are too high for their given redshift to occur naturally in a standard model cosmology, we use extreme value statistics to construct confidence regions in the mass–redshift plane for the most extreme objects expected in the universe. We show how such a diagram not only provides a way of potentially ruling out the concordance cosmology, but also allows us to differentiate between alternative models of enhanced structure formation. We compare our theoretical prediction with observations, placing currently observed high- and low-redshift clusters on a mass–redshift diagram, and find – provided we consider the full sky to avoid a posteriori selection effects – that none is in significant tension with concordance cosmology.

Estimation for a partial-linear single-index model

In this paper, we study the estimation for a partial-linear single-index model. A two-stage estimation procedure is proposed to estimate the link function for the single index and the parameters in the single index, as well as the parameters in the linear component of the model. Asymptotic normality is established for both parametric components. For the index, a constrained estimating equation leads to an asymptotically more efficient estimator than existing estimators in the sense that it is of a smaller limiting variance. The estimator of the nonparametric link function achieves optimal convergence rates; and the structural error variance is obtained. In addition, the results facilitate the construction of confidence regions and hypothesis testing for the unknown parameters. A simulation study is performed and an application to a real dataset is illustrated. The extension to multiple indices is briefly sketched.

Two-step estimators in partial linear models with missing response variables and error-prone covariates

A partial linear model with missing response variables and error-prone covariates is considered. The imputation approach is developed to estimate the regression coefficients and the nonparametric function. The proposed parametric estimators are shown to be asymptotically normal, and the estimators for the nonparametric part are proved to converge at an optimal rate. To construct confidence regions for the regression coefficients and the nonparametric function, respectively, the authors also propose the empirical-likelihood-based statistics and investigate the limit distributions of the empirical likelihood ratios. The simulation study is conducted to compare the finite sample behavior for the proposed estimators. An application to an AIDS dataset is illustrated.

Estimation for the single-index models with random effects

In this paper, we generalize the single-index models to the scenarios with random effects. The introduction of the random effects raises interesting inferential challenges. Instead of treating the variance matrix as the tuning parameters in the nonparametric model of Gu and Ma (2005), we propose root-n consistent estimators for the variance components. Furthermore, the single-index part in our model avoids the curse of dimensionality and makes our model simpler. The variance components also cannot be treated as nuisance parameters and are canceled in the estimation procedure like Wang et al. (2010). A new set of estimating equations modified for the boundary effects is proposed to estimate the index coefficients. The link function is estimated by using the local linear smoother. Asymptotic normality is established for the proposed estimators. Also, the estimator of the link function achieves optimal convergence rate. These results facilitate the construction of confidence regions and hypothesis testing for the parameters of interest. Simulations show that our methods work well for high-dimensional p.

EUCLIDEAN REVEALED PREFERENCES: TESTING THE SPATIAL VOTING MODEL

SUMMARYIn the spatial model of voting, voters choose the candidate closest to them in the ideological space. Recent work by Degan and Merlo in 2009 shows that it is falsifiable on the basis of individual voting data in multiple elections. We show how to tackle the fact that the model only partially identifies the distribution of voting profiles and we give a formal revealed preference test of the spatial voting model in three national elections in the USA, and strongly reject the spatial model in all cases. We also construct confidence regions for partially identified voter characteristics in an augmented model with unobserved valence dimension, and identify the amount of voter heterogeneity necessary to reconcile the data with spatial preferences. Copyright © 2011 John Wiley & Sons, Ltd.

Hybrid Confidence Regions Based on Data Depth

Summary We consider the general problem of constructing confidence regions for, possibly multi-dimensional, parameters when we have available more than one approach for the construction. These approaches may be motivated by different model assumptions, different levels of approximation, different settings of tuning parameters or different Monte Carlo algorithms. Their effectiveness is often governed by different sets of conditions which are difficult to vindicate in practice. We propose two procedures for constructing hybrid confidence regions which endeavour to integrate all such individual approaches. The procedures employ the concept of data depth to calibrate the confidence region in two different ways, the first rendering its coverage error minimax and the second rendering its coverage error conservative. The resulting region reconciles in many important aspects the discrepancies between the various approaches, and is robust against misspecification of their governing conditions. Theoretical and empirical properties of our procedures are investigated in comparison with those of the constituent individual approaches.

Empirical likelihood for the parametric part in partially linear errors-in-function models

Partially linear errors-in-function models were proposed by Liang (2000), but their inferences have not been systematically studied. This article proposes an empirical likelihood method to construct confidence regions of the parametric components. Under mild regularity conditions, the nonparametric version of the Wilk’s theorem is derived. Simulation studies show that the proposed empirical likelihood method provides narrower confidence regions, as well as higher coverage probabilities than those based on the traditional normal approximation method.

A new statistical depth function with applications to multimodal data

We propose a new statistical depth function based on interpoint distances, which has the distinct property of respecting multimodality in data configurations. This property proves to be especially relevant to many inference problems including confidence region construction, classification, tests for equality of populations, p-value computation, etc. With specification of an appropriate interpoint distance, our depth function also applies to infinite-dimensional data. A number of examples are used to illustrate the diverse applicability of our proposed depth function in different problem settings, where the conventional centre-outward ordering depth functions are found to be inadequate.

Jackknife–blockwise empirical likelihood methods under dependence

Empirical likelihood for general estimating equations is a method for testing hypothesis or constructing confidence regions on parameters of interest. If the number of parameters of interest is smaller than that of estimating equations, a profile empirical likelihood has to be employed. In case of dependent data, a profile blockwise empirical likelihood method can be used. However, if too many nuisance parameters are involved, a computational difficulty in optimizing the profile empirical likelihood arises. Recently, Li et al. (2011) [9] proposed a jackknife empirical likelihood method to reduce the computation in the profile empirical likelihood methods for independent data. In this paper, we propose a jackknife–blockwise empirical likelihood method to overcome the computational burden in the profile blockwise empirical likelihood method for weakly dependent data.

Reduce computation in profile empirical likelihood method

Since its introduction by Owen in [29, 30], the empirical likelihood method has been extensively investigated and widely used to construct confidence regions and to test hypotheses in the literature. For a large class of statistics that can be obtained via solving estimating equations, the empirical likelihood function can be formulated from these estimating equations as proposed by [35]. If only a small part of parameters is of interest, a profile empirical likelihood method has to be employed to construct confidence regions, which could be computationally costly. In this paper we propose a jackknife empirical likelihood method to overcome this computational burden. This proposed method is easy to implement and works well in practice.

Empirical Likelihood for a Heteroscedastic Partial Linear Model

Consider the heteroscedastic regression model , where , β is a p × 1 column vector of unknown parameter, (X i , T i , Z i ) are random design points, Y i are the response variables, g(·) is an unknown function defined on the closed interval [0, 1], {e i , ℱ i } is a sequence of martingale differences. When f is known and unknown cases, we propose the empirical log-likelihood ratio statistics for the parameter β. For each case, a nonparametric version of Wilks' theorem is derived. The results are then used to construct confidence regions of the parameter. Simulation study shows that the empirical likelihood method performs better than a normal approximation-based approach.