Using adaptive weighted least squares to reduce the lengths of confidence intervals

The author proposes an adaptive method which produces confidence intervals that are often narrower than those obtained by the traditional procedures. The proposed methods use both a weighted least squares approach to reduce the length of the confidence interval and a permutation technique to insure that its coverage probability is near the nominal level. The author reports simulations comparing the adaptive intervals to the traditional ones for the difference between two population means, for the slope in a simple linear regression, and for the slope in a multiple linear regression having two correlated exogenous variables. He is led to recommend adaptive intervals for sample sizes superior to 40 when the error distribution is not known to be Gaussian.     

Bias reduction for the ratio of means

Confidence intervals of third-order accuracy are given for the ratio of the means of two normal distributions. A simulation study is conducted to compare these intervals with ones known. A comparison is also made with non-parametric alternatives.     

Improved closed-form prediction intervals for binomial and Poisson distributions

The problems of constructing prediction intervals for the binomial and Poisson distributions are considered. Available approximate, exact and conditional methods for both distributions are reviewed and compared. Simple approximate prediction intervals based on the joint distribution of the past samples and the future sample are proposed. Exact coverage studies and expected widths of prediction intervals show that the new prediction intervals are comparable to or better than the available ones in most cases. The methods are illustrated using two practical examples.     

Approximate one-sided tolerance limits in random effects model and in some mixed models and comparisons

An approximate closed-form one-sided tolerance limit (TL) in a general mixed model is proposed. One-sided TLs for the distribution of observable random variable and for the distribution of unobservable random variable in one-way random model are obtained as special cases from the one for the general mixed model. Applications to a two-way nested random model are also given. The merits of the TLs are evaluated using Monte Carlo simulation and compared with the existing ones. Our comparison studies indicate that the approximate TLs are quite satisfactory for all parameter and sample size configurations, and better than the existing ones in some cases. Approximate confidence intervals for exceedance probabilities in one-way random effects model are also proposed. The procedures are illustrated using three examples.     

Robust adaptive estimators for binary regression models

This article introduces adaptive weighted maximum likelihood estimators for binary regression models. The asymptotic distribution under the model is established, and asymptotic confidence intervals are derived. Finite-sample properties are studied by simulation. For clean datasets, the proposed adaptive estimators are more efficient than the non-adaptive ones even for moderate sample sizes, and for outlier-contaminated datasets they show a comparable robustness. As for the asymptotic confidence intervals, the actual coverage levels under the model are very close to the nominal levels (even for moderate sample sizes), and they are reasonably stable under contamination.     

Weighted empirical likelihood inference for multiple samples

We propose a weighted empirical likelihood approach to inference with multiple samples, including stratified sampling, the estimation of a common mean using several independent and non-homogeneous samples and inference on a particular population using other related samples. The weighting scheme and the basic result are motivated and established under stratified sampling. We show that the proposed method can ideally be applied to the common mean problem and problems with related samples. The proposed weighted approach not only provides a unified framework for inference with multiple samples, including two-sample problems, but also facilitates asymptotic derivations and computational methods. A bootstrap procedure is also proposed in conjunction with the weighted approach to provide better coverage probabilities for the weighted empirical likelihood ratio confidence intervals. Simulation studies show that the weighted empirical likelihood confidence intervals perform better than existing ones.     

Approximate confidence intervals for a ratio of two variance components

New approximate confidence intervals for the ratio of two variance components in an unbalanced mixed .linear model with a single set of random effects are proposed. Contrary to the confidence intervals known in the literature the new intervals preserve the confidence coefficient and cover the exact confidence interval which, however, is not easy to establish as it requires the solution of complicated nonlinear equations.     

Confidence intervals for the mean of a normal distribution with restricted parameter space

For a normal distribution with known variance, the standard confidence interval of the location parameter is derived from the classical Neyman procedure. When the parameter space is known to be restricted, the standard confidence interval is arguably unsatisfactory. Recent articles have addressed this problem and proposed confidence intervals for the mean of a normal distribution where the parameter space is not less than zero. In this article, we propose a new confidence interval, rp interval, and derive the Bayesian credible interval and likelihood ratio interval for general restricted parameter space. We compare these intervals with the standard interval and the minimax interval. Simulation studies are undertaken to assess the performances of these confidence intervals.     

Non-parametric prediction intervals for the lifetime of coherent systems

In this paper, nonparametric methods are proposed to construct prediction intervals for the lifetime of a coherent system with known signatures. An explicit expression for the coverage probability of the prediction intervals is presented based on Samaniego’s signature. The existence and optimality of these intervals are discussed. In our derivation, we also obtain an exact expression for the marginal distribution of the \(i\) th order statistic from a pooled sample.     

Simultaneous Confidence Intervals for the Ratios of Marginal Means of a Multivariate Poisson Distribution

This article computes simultaneous confidence intervals for the ratios of marginal means of a multivariate Poisson distribution. For this, we propose a lognormal approximation technique and a bootstrap method. We demonstrate advantages of the proposed methods over existing ones through a simulation study. To illustrate their applicability to real-world problems, we apply the proposed methods to US data on infectious diseases.     

βExpectation and β-content tolerance intervals for dependent observations

Methods of constructing exact tolerance intervals (β-expectation and β-content) for independent observations are well known. For the case of dependent observations, obtaining exact results is not possible. In this article we provide an approximate method of constructing β-expectation tolerance intervals via a Taylor series expansion. Examples of independent observations are considered to compare the intervals constructed with those obtained by the exact method. For the case of non-stationary type processes we have proposed a method of constructing approximate β-content tolerance intervals. Once again an example is given to illustrate the results.     

Confidence sets based on sequential Tests

For given (small) a and β a sequential confidence set that covers the true parameter point with probability at least 1 - a and one or more specified false parameter points with probability at most β can be generated by a family of sequen-tial tests. Several situations are described where this approach would be a natural one. The following example is studied in some detail: obtain an upper (1 - α)-confidence interval for a normal mean μ (variance known) with β-protection at μ - δ(μ), where δ(.) is not bounded away from 0 so that a truly sequential procedure is mandatory. Some numerical results are presented for intervals generated by (1) sequential probability ratio tests (SPRT's), and (2) generalized sequential probability ratio tests (GSPRT's). These results indicate the superiority of the GSPRT-generated intervals over the SPRT-generated ones if expected sample size is taken as performance criterion     

Bayesian and Classical Approaches for Hypotheses Testing in Trisomies

We introduce classical approaches for testing hypotheses on the meiosis I non disjunction fraction in trisomies, such as the likelihood-ratio, bootstrap, and Monte Carlo procedures. To calculate the p-values for the bootstrap and Monte Carlo procedures, different transformations in the data are considered. Bootstrap confidence intervals are also used as a tool to perform hypotheses tests. A Monte Carlo study is carried out to compare the proposed test procedures with two Bayesian ones: Jeffreys and Pereira-Stern tests. The results show that the likelihood-ratio and the Bayesian tests present the best performance. Down syndrome data are analyzed to illustrate the procedures.     

Robust estimation and selection for single-index regression model

In this paper, we consider a single-index regression model for which we propose a robust estimation procedure for the model parameters and an efficient variable selection of relevant predictors. The proposed method is known as the penalized generalized signed-rank procedure. Asymptotic properties of the proposed estimator are established under mild regularity conditions. Extensive Monte Carlo simulation experiments are carried out to study the finite sample performance of the proposed approach. The simulation results demonstrate that the proposed method dominates many of the existing ones in terms of robustness of estimation and efficiency of variable selection. Finally, a real data example is given to illustrate the method.     

Constructing quantile confidence intervals using extended simple random sample in finite populations

This paper constructs quantile confidence intervals based on extended simple random sample (SRS) from a finite population, where ranks of population units are all known. Extended simple random sample borrows additional information from unmeasured observations in the population by conditioning on the population ranks of the measured units in SRS. The confidence intervals are improved using Rao-Blackwell theorem over the conditional distribution of sample ranks given the measured sample units. Empirical evidence shows that the proposed confidence intervals have shorter lengths than confidence intervals constructed from an SRS sample.     

Errors-in-Variables Regression Using Stein Estimates

A method is proposed for estimating regression parameters from data containing covariate measurement errors by using Stein estimates of the unobserved true covariates. The method produces consistent estimates for the slope parameter in the classical linear errors-in-variables model and applies to a broad range of nonlinear regression problems, provided the measurement error is Gaussian with known variance. Simulations are used to examine the performance of the estimates in a nonlinear regression problem and to compare them with the usual naive ones obtained by ignoring error and with other estimates proposed recently in the literature.     

Censored median regression and profile empirical likelihood

We implement profile empirical likelihood-based inference for censored median regression models. Inference for any specified subvector is carried out by profiling out the nuisance parameters from the “plug-in” empirical likelihood ratio function proposed by Qin and Tsao. To obtain the critical value of the profile empirical likelihood ratio statistic, we first investigate its asymptotic distribution. The limiting distribution is a sum of weighted chi square distributions. Unlike for the full empirical likelihood, however, the derived asymptotic distribution has intractable covariance structure. Therefore, we employ the bootstrap to obtain the critical value, and compare the resulting confidence intervals with the ones obtained through Basawa and Koul’s minimum dispersion statistic. Furthermore, we obtain confidence intervals for the age and treatment effects in a lung cancer data set.     

Statistical aspects of linkage analysis in interlaboratory studies

This paper investigates statistical issues that arise in interlaboratory studies known as Key Comparisons when one has to link several comparisons to or through existing studies. An approach to the analysis of such a data is proposed using Gaussian distributions with heterogeneous variances. We develop conditions for the set of sufficient statistics to be complete and for the uniqueness of uniformly minimum variance unbiased estimators (UMVUE) of the contrast parametric functions. New procedures are derived for estimating these functions with estimates of their uncertainty. These estimates lead to associated confidence intervals for the laboratories (or studies) contrasts. Several examples demonstrate statistical inference for contrasts based on linkage through the pilot laboratories. Monte Carlo simulation results on performance of approximate confidence intervals are also reported.     

Transformed Logit Confidence Intervals for Small Populations in Single Capture–Recapture Estimation

The good performance of logit confidence intervals for the odds ratio with small samples is well known. This is true unless the actual odds ratio is very large. In single capture–recapture estimation the odds ratio is equal to 1 because of the assumption of independence of the samples. Consequently, a transformation of the logit confidence intervals for the odds ratio is proposed in order to estimate the size of a closed population under single capture–recapture estimation. It is found that the transformed logit interval, after adding .5 to each observed count before computation, has actual coverage probabilities near to the nominal level even for small populations and even for capture probabilities near to 0 or 1, which is not guaranteed for the other capture–recapture confidence intervals proposed in statistical literature. Thus, given that the .5 transformed logit interval is very simple to compute and has a good performance, it is appropriate to be implemented by most users of the single capture–recapture method.     

Improved Prediction Intervals and Distribution Functions

Abstract.  The plug-in solution is usually not entirely adequate for computing prediction intervals, as their coverage probability may differ substantially from the nominal value. Prediction intervals with improved coverage probability can be defined by adjusting the plug-in ones, using rather complicated asymptotic procedures or suitable simulation techniques. Other approaches are based on the concept of predictive likelihood for a future random variable. The contribution of this paper is the definition of a relatively simple predictive distribution function giving improved prediction intervals. This distribution function is specified as a first-order unbiased modification of the plug-in predictive distribution function based on the constrained maximum likelihood estimator. Applications of the results to the Gaussian and the generalized extreme-value distributions are presented.