Minimum-volume confidence sets for normal linear regression models

In this article, we establish the minimum-volume confidence sets for normal linear regression models, extending the results in Zhang [Minimum volume confidence sets for parameters of normal distributions. Adv Stat Anal. 2017;101:309–320] on building the minimum-volume confidence sets for parameters of normal distributions. Compared with classical confidence sets, the proposed optimal confidence set is proved to have the smallest volume, for whatever confidence level, sample size and sample data.     

Improved set estimators for the coefficients of a linear model when the error distribution is spherically symmetric with unknown variances

The usual confidence set for p (p ≥ 3) coefficients of a linear model is known to be dominated by the James-Stein confidence sets under the assumption of spherical symmetric errors with known variance (Hwang and Chen 1986). For the same confidence-set problem but for the unknown-variance case, naturally one replaces the unknown variance by an estimator. For the normal case, many previous studies have shown numerically that the resultant James-Stein confidence sets dominate the resultant usual confidence sets, i.e., the F confidence sets. In this paper we provide a further asymptotic justification, and we discover the same advantage of the James-Stein confidence sets for normal error as well as spherically symmetric error.     

Bootstrap in functional linear regression

We have considered the functional linear model with scalar response and functional explanatory variable. One of the most popular methodologies for estimating the model parameter is based on functional principal components analysis (FPCA). In recent literature, weak convergence for a wide class of FPCA-type estimates has been proved, and consequently asymptotic confidence sets can be built. In this paper, we have proposed an alternative approach in order to obtain pointwise confidence intervals by means of a bootstrap procedure, for which we have obtained its asymptotic validity. Besides, a simulation study allows us to compare the practical behaviour of asymptotic and bootstrap confidence intervals in terms of coverage rates for different sample sizes.     

Simultaneous confidence sets and confidence intervals for multiple ratios

This paper deals with the problem of simultaneously estimating multiple ratios. In the simplest case of only one ratio parameter, Fieller's theorem (J. Roy. Statist. Soc. Ser. B 16 (1954) 175) provides a confidence interval for the single ratio. For multiple ratios, there is no method available to construct simultaneous confidence intervals that exactly satisfy a given familywise confidence level. Many of the methods in use are conservative since they are based on probability inequalities. In this paper, first we consider exact simultaneous confidence sets based on the multivariate t-distribution. Two approaches of determining the exact simultaneous confidence sets are outlined. Second, approximate simultaneous confidence intervals based on the multivariate t-distribution with estimated correlation matrix and a resampling approach are discussed. The methods are applied to ratios of linear combinations of the means in the one-way layout and ratios of parameter combinations in the general linear model. Extensive Monte Carlo simulation is carried out to compare the performance of the various methods with respect to the stability of the estimated critical points and of the coverage probabilities.     

Empirical Likelihood Inference for the Cox Model with Time-dependent Coefficients via Local Partial Likelihood

Abstract.  The Cox model with time-dependent coefficients has been studied by a number of authors recently. In this paper, we develop empirical likelihood (EL) pointwise confidence regions for the time-dependent regression coefficients via local partial likelihood smoothing. The EL simultaneous confidence bands for a linear combination of the coefficients are also derived based on the strong approximation methods. The EL ratio is formulated through the local partial log-likelihood for the regression coefficient functions. Our numerical studies indicate that the EL pointwise/simultaneous confidence regions/bands have satisfactory finite sample performances. Compared with the confidence regions derived directly based on the asymptotic normal distribution of the local constant estimator, the EL confidence regions are overall tighter and can better capture the curvature of the underlying regression coefficient functions. Two data sets, the gastric cancer data and the Mayo Clinic primary biliary cirrhosis data, are analysed using the proposed method.     

Empirical Likelihood for Linear Models Under Linear Process Errors

In this article, we study the construction of confidence intervals for regression parameters in a linear model under linear process errors by using the blockwise technique. It is shown that the blockwise empirical likelihood (EL) ratio statistic is asymptotically χ²-type distributed. The result is used to obtain EL based confidence regions for regression parameters. The finite-sample performance of the method is evaluated through a simulation study.     

Generalized simultaneous confidence regions for regression coefficients and their ratios

This study constructs a simultaneous confidence region for two combinations of coefficients of linear models and their ratios based on the concept of generalized pivotal quantities. Many biological studies, such as those on genetics, assessment of drug effectiveness, and health economics, are interested in a comparison of several dose groups with a placebo group and the group ratios. The Bonferroni correction and the plug-in method based on the multivariate-t distribution have been proposed for the simultaneous region estimation. However, the two methods are asymptotic procedures, and their performance in finite sample sizes has not been thoroughly investigated. Based on the concept of generalized pivotal quantity, we propose a Bonferroni correction procedure and a generalized variable (GV) procedure to construct the simultaneous confidence regions. To address a genetic concern of the dominance ratio, we conduct a simulation study to empirically investigate the probability coverage and expected length of the methods for various combinations of sample sizes and values of the dominance ratio. The simulation results demonstrate that the simultaneous confidence region based on the GV procedure provides sufficient coverage probability and reasonable expected length. Thus, it can be recommended in practice. Numerical examples using published data sets illustrate the proposed methods.     

Estimation of Balanced Simultaneous Confidence Sets for SIR Models

Stochastic compartmental (e.g., SIR) models have proven useful for studying the epidemics of childhood diseases while taking into account the variability of the epidemic dynamics. Here, we present a method for estimating balanced simultaneous confidence sets for the mean sample path of a stochastic SIR model, thus providing a simple representation of both the typical behavior and the variability of the epidemic. The confidence sets are estimated by a bootstrap procedure, using asymptotic properties of density dependent jump Markov processes. The method is applied to chickenpox epidemics in France and the coverage probability of the confidence sets is estimated in that context.     

A METHOD FOR ESTIMATING PARAMETER IN NONNEGATIVE MA(1) MODELS

ABSTRACT

A method for estimating parameter in nonnegative MA(1) models is proposed and investigated in the paper. The method also gives nontrivial confidence sets on confidence level 1. Small sample properties of new estimator are demonstrated in a simulation study.     

Confidence sets of fixed size with predetermined confidence level

In this paper a three stage method to obtain a confidence sets of given form and size, with a preassigned confidence coefficient, is suggested for the case of normally distributed observations from populations with different means and common unknown variance. The method is quite general, and it allows for most common forms of confidence sets. Asymptotic properties of the random sample sizes of the method are studied and compared to the same properties of other methods with the same aim.     

Using the bootstrap in correlation analysis,with application to a longitudinal data set

The problem of constructing simultaneous confidence intervals for various measures of association is considered. Alternative bootstrap algorithms are given for approximating the sampling distributions of the quantities generating the confidence sets. The small sample performance of the procedures is illustrated using simulated data from 3- and 6-variate normal populations. The results are applied to a large multidimensional longitudinal data set from a study of the relationship between drug use and several behavioral attributes.     

Optimal confidence intervals for the geometric parameter

Abstract

This article discusses optimal confidence estimation for the geometric parameter and shows how different criteria can be used for evaluating confidence sets within the framework of tail functions theory. The confidence interval obtained using a particular tail function is studied and shown to outperform others, in the sense of having smaller width or expected width under a specified weight function. It is also shown that it may not be possible to find the most powerful test regarding the parameter using the Neyman-Pearson lemma. The theory is illustrated by application to a fecundability study.     

Simultaneous Small Sample Inference for Linear Combinations of Generalized Linear Model Parameters

A method is proposed to construct simultaneous confidence intervals for multiple linear combinations of generalized linear model parameters, that uses a multivariate normal- or t-distribution together with the signed likelihood root statistic. In an application to a case study simultaneous confidence bands for logistic regression are calculated. A simulation study based on the example evaluation suggests superior performance compared to the common Wald-type approaches. The proposed methods are readily implemented in the R extension package mcprofile.     

Asymptotic normality of the lengths of a class of nonparametric confidence intervals for a regression parameter

In the linear regression model, the asymptotic distributions of certain functions of confidence bounds of a class of confidence intervals for the regression parameter arc investigated. The class of confidence intervals we consider in this paper are based on the usual linear rank statistics (signed as well as unsigned). Under suitable assumptions, if the confidence intervals are based on the signed linear rank statistics, it is established that the lengths, properly normalized, of the confidence intervals converge in law to the standard normal distributions; if the confidence intervals arc based on the unsigned linear rank statistics, it is then proved that a linear function of the confidence bounds converges in law to a normal distribution.     

Confidence sets and the stein effect

The problem of improving upon the usual set estimator of a multivariate normal mean has only recently seen significant advances. Improved sets that take advantage of the Stein effect have been constructed. It is shown here that the Stein effect is so powerful that one can construct improved confidence sets that can have zero radius on a set of positive probability. Other, somewhat more sensible, sets which attain arbitrarily small radius are also constructed, and it is argued that one way to eliminate unreasonable confidence sets is through a conditional evaluation.     

Minimum-area confidence sets for a normal distribution

When making inference on a normal distribution, one often seeks either a joint confidence region for the two parameters or a confidence band for the cumulative distribution function. A number of methods for constructing such confidence sets are available, but none of these methods guarantees a minimum-area confidence set. In this paper, we derive both a minimum-area joint confidence region for the two parameters and a minimum-area confidence band for the cumulative distribution function. The minimum-area joint confidence region is asymptotically equivalent to other confidence regions in the literature, but the minimum-area confidence band improves on existing confidence bands even asymptotically.     

Empirical Likelihood for Nonparametric Components in Additive Partially Linear Models

Empirical likelihood-based inference for the nonparametric components in additive partially linear models is investigated. An empirical likelihood approach to construct the confidence intervals of the nonparametric components is proposed when the linear covariate is measured with and without errors. We show that the proposed empirical log-likelihood ratio is asymptotically standard chi-squared without requiring the undersmoothing of the nonparametric components. Then, it can be directly used to construct the confidence intervals for the nonparametric functions. A simulation study indicates that, compared with a normal approximation-based approach, the proposed method works better in terms of coverage probabilities and widths of the pointwise confidence intervals.     

A Simultaneous Confidence Band for Dense Longitudinal Regression

We present a method of using local linear smoothing to construct simultaneous confidence bands for the mean function of densely spaced functional data. Our approach works well under mild conditions. In addition, the local linear estimator and its accompanying confidence band enjoy semiparametric efficiency in the sense that they are asymptotically equivalent to the counterparts obtained from the random trajectories entirely observed without errors. We illustrate the performance of the proposed confidence band through a simulation study. Furthermore, an application in food science is presented.     

A Note on the Use of Confidence Limits following Rejection of a Null Hypothesis

The equivalence of some tests of hypothesis and confidence limits is well known. When, however, the confidence limits are computed only after rejection of a null hypothesis, the usual unconditional confidence limits are no longer valid. This refers to a strict two-stage inference procedure: first test the hypothesis of interest and if the test results in a rejection decision, then proceed with estimating the relevant parameter. Under such a situation, confidence limits should be computed conditionally on the specified outcome of the test under which estimation proceeds. Conditional confidence sets will be longer than unconditional confidence sets and may even contain values of the parameter previously rejected by the test of hypothesis. Conditional confidence limits for the mean of a normal population with known variance are used to illustrate these results. In many applications, these results indicate that conditional estimation is probably not good practice.     

On multivariate confidence regions and simultaneous confidence limits for ratios

Convenient general linear model computational procedures are presented for constructing multivariate confidence regions and simultaneous confidence limits for ratios of linear combinations of the parameters. The practical consequence is that a single general linear model computer program, capable of validating the underlying model and estimating the parameters, can (after slight modification) also construct the confidence regions, and even determine their precise analytic form (ellipsoid, hyperboloid, etc.). The text is deliberately factual while the appendices extend and help clarify earlier work by Henry Scheffe. As an example, a confidence ellipse and simultaneous confidence limits are constructed for several relative potencies in a classical multiple parallel line bioassay.