Coverage Probabilities Of Confidence Intervals Research Articles

Upper bounds for coverage probabilities of confidence intervals for nonmonotone parametric functions

Abstract Consider a statistical model F∈ F and let θ = θ ( F ) be a structural parameter which admits a (1− α )-level two-sided confidence interval based on a random sample taken from F . Let g ( θ ) be some parametric function of interest. The problem of deriving a confidence interval for g ( θ ) directly from that given on θ is considered. If g is one-to-one then a (1− α )-level two-sided confidence interval is immediately available. If however, g is not one-to-one the problem becomes more complex. In this paper the situation where g is a nonmonotone function is considered. Under the assumption that g has a unique minimum γ at x = δ and that g ( x ) is strictly decreasing (increasing) for x δ ( x > δ ) a two-sided confidence interval for g ( θ ) can be obtained from the (1− α )-level confidence interval on θ whose confidence level, while being at least 1− α , is not greater than 1− α /2. Moreover, if in addition g is symmetric then an improved upper bound, smaller than 1− α /2, can be achieved when F is a location or location and scale distribution.

A Degrees-of-freedom Approximation for a t-statistic with Heterogeneous Variance

Ordinary least squares is a commonly used method for obtaining parameter estimates for a linear model. When the variance is heterogeneous, the ordinary least squares estimate is unbiased, but the ordinary least squares variance estimate may be biased. Various jackknife estimators of variance have been shown to be consistent for the asymptotic variance of the ordinary least squares estimator of the parameters. In small samples, simulations have shown that the coverage probabilities of confidence intervals using the jackknife variance estimators are often low. We propose a finite sample degrees-of-freedom approximation to use with the t-distribution, which, in simulations, appears to raise the coverage probabilities closer to the nominal level. We also discuss an ‘estimated generalized least squares estimator’, in which we model and estimate the variance, and use the estimated variance in a weighted regression. Even when the variance is modelled correctly, the coverage can be low in small samples, and, in terms of the coverage probability, one appears better off using ordinary least squares with a robust variance and our degrees-of-freedom approximation. An example with real data is given to demonstrate how potentially misleading conclusions may arise when the degrees-of-freedom approximation is not used.

Recursive mean adjustment in time-series inferences

When time-series data are positively autocorrelated, mean adjustment using the overall sample mean causes biases for sample autocorrelations and parameter estimates, which decreases the coverage probabilities of confidence intervals. A new method for mean adjustment is proposed, in which a datum at a time is adjusted for the mean through the partial sample mean, the average of data up to the time point. The method is simple and reduces the biases of the parameter estimators and the sample autocorrelations when data are positively autocorrelated. The empirical coverage probabilities of the confidence intervals of the autoregressive coefficient become quite close to the nominal level.

Multiple comparison procedures with the average for scale families: normal and exponential populations

In this article, multiple comparison procedures with the average of scale parameters for normal and exponential populations are proposed. A subset selection procedure and a confidence interval procedure are investigated. These procedures will have broad applicability in identifying groups of treatments with smaller than the average, larger than the average and not different from the average variability in various fields. Bonferroni inequality is used to bound the probability of a correct selection and the coverage probability of confidence intervals. The Pearson’s approximation to the sampling distribution of the log chi-square ratio utilized in both subset selection and confidence intervals is shown to be satisfactory compared with the simulation results. Statistical tables to implement these procedures for the case of equal sample sizes are provided for use in practice.

Conditional coverage probability of confidence intervals in errors-in-variables and related models

Gleser and Hwang (1987) show that there exists no nontrivial confidence interval with finite expected length for many errors-in-variables and related models. Nonetheless, the validity of using these confidence intervals only when their realizations are of finite length remains uninvestigated. We scrutinize this practice from a conditional frequentist viewpoint and find it not justifiable. It is shown that under some conditions, the minimum coverage probability of the confidence interval is zero if conditioning on the event that the confidence interval has finite length. The result is then applied to confirm Neyman's conjecture for Fieller's confidence sets.

Bootstrap Methods for Covariance Structures

The optimal minimum distance (OMD) estimator for models of covariance structures is asymptotically efficient but has much worse finite-sample properties than does the equally weighted minimum distance (EWMD) estimator. This paper shows how the bootstrap can be used to improve the finite-sample performance of the OMD estimator. The theory underlying the bootstrap's ability to reduce the bias of estimators and errors in the coverage probabilities of confidence intervals is summarized. The results of numerical experiments and an empirical example show that the bootstrap often essentially eliminates the bias of the OMD estimator. The finite-sample estimation efficiency of the bias-corrected OMD estimator often exceeds that of the EWMD estimator. Moreover, the true coverage probabilities of confidence intervals based on the OMD estimator with bootstrap-critical values are very close to the nominal coverage probabilities.

Iterated bootstrap with applications to frontier models

The iterated bootstrap may be used to estimate errors which arise from a single pass of the bootstrap and thereby to correct for them. Here the iteration is employed to correct for coverage probability of confidence intervals obtained by a percentile method in the context of production frontier estimation with panel data. The parameter of interest is the maximum of the intercepts in a fixed firm effect model. The bootstrap distribution estimators are consistent if and only if there are no ties for this maximum. In the regular case (no ties), poor distribution estimators can result when the second largest intercept is close to the maximum. The iterated bootstrap is thus suggested to improve the accuracy of the obtained distributions. The result is illustrated in the analysis of labor efficiency of railway companies.

Use of Estimating Functions for Estimation from Complex Surveys

Abstract We describe a method for point and interval estimation of population parameters from complex surveys using estimating functions. The theory was originally developed for infinite populations and has recently been applied to finite populations. With estimating functions, a unifying framework can be given for point and interval estimation of both finite and infinite population parameters. We discuss test inversion methods to derive confidence intervals for one-dimensional parameters and propose a method for eliminating nuisance parameters in the multidimensional setting. We show that special cases of our proposal result in conditional and orthogonal methods proposed in the literature. We describe a simulation study using real data to compare the coverage probabilities of confidence intervals obtained under various approaches.

One‐step M‐estimators in the linear model, with dependent errors

AbstractWe consider the problem of robust M‐estimation of a vector of regression parameters, when the errors are dependent. We assume a weakly stationary, but otherwise quite general dependence structure. Our model allows for the representation of the correlations of any time series of finite length.We first construct initial estimates of the regression, scale, and autocorrelation parameters. The initial autocorrelation estimates are used to transform the model to one of approximate independence. In this transformed model, final one‐step M‐estimates are calculated.Under appropriate assumptions, the regression estimates so obtained are asymptotically normal, with a variance‐covariance structure identical to that in the case in which the autocorrelations are known a priori. The results of a simulation study are given. Two versions of our estimator are compared with the L1 ‐estimator and several Huber‐type M‐estimators. In terms of bias and mean squared error, the estimators are generally very close. In terms of the coverage probabilities of confidence intervals, our estimators appear to be quite superior to both the L1‐estimator and the other estimators. The simulations also indicate that the approach to normality is quite fast.

A comparison of two methods of estimating a common risk difference in a stratified analysis of a multicenter clinical trial

In this paper, we compare two methods of estimating the difference between the proportion of adverse events in a test treatment group and the proportion of adverse events in a control treatment group in a multicenter clinical trial. We used simulated data to compare the bias and mean squared error of the weighted least squares estimator to the bias and mean squared error of the Cochran-Mantel-Haenszel estimator. We also computed the coverage probabilities for confidence intervals derived from these estimators. We found that the weighted least squares method was often seriously biased. The coverage probabilities for the Cochran-Mantel-Haenszel estimators were often closer to their nominal values than were the coverage probabilities for the weighted least squares estimators. It also was found that these methods require a larger sample size to maintain coverage probabilities near their nominal values when unequal numbers of persons are assigned to the test and control treatments.

Analysis of Binary Data from a Multicentre Clinical Trial

We develop several methods for estimating the treatment effect difference defined as the overall log-odds ratio of favourable response in a multicentre clinical trial comparing two treatments with binary response. A simulation study compares the bias and mean squared error of the point estimates and the exact coverage probabilities of confidence intervals obtained distributions.

Analysis of binary data from a multicentre clinical trial

We develop several methods for estimating the treatment effect difference defined as the overall log-odds ratio of favourable response in a multicentre clinical trial comparing two treatments with binary response. A simulation study compares the bias and mean squared error of the point estimates and the exact coverage probabilities of confidence intervals obtained distributions.

Minimum Distance Estimators of Scale with Censored Data

Abstract This article examines how the level of censoring affects the performance of estimators of scale that minimize a weighted Cramervon Mises distance between the Kaplan-Meier product-limit estimator of the survival distribution function and an assumed model. The article shows that under certain conditions these estimators are asymptotically normal if the survival function is a member of the assumed scale family. Weights that minimize the asymptotic variance under these conditions also are found. Under stronger conditions, the minimum distance estimators (MDE's) are asymptotically normal even if the survival function is not a member of the assumed scale family. Asymptotic coverage probabilities of confidence intervals are found for percentiles constructed from exponential MDE's of scale under departures from the exponential distribution, and what happens in the limiting cases of no censoring and complete censoring is examined. Using the asymptotic results and Monte Carlo experiments, several exponenti...

An Improved Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimator

This paper considers a new class of heteroskedasticity and autocorrelation consistent (HAC) covariance matrix estimators. The estimators considered are prewhitened kernel estimators with vector autoregressions employed in the prewhitening stage. The paper establishes consistency, rate of convergence, and asymptotic truncated mean squared error (MSE) results for the estimators when a fixed or automatic bandwidth procedure is employed. Conditions are obtained under which prewhitening improves asymptotic truncated MSE. Monte Carlo results show that prewhitening is very effective in reducing bias, improving confidence interval coverage probabilities, and rescuing over-rejection of t-statistics constructed using kernel-HAC estimators. On the other hand, prewhitening is found to inflate variance and MSE of the kernel estimators. Since confidence interval coverage probabilities and over-rejection of t-statistics are usually of primary concern, prewhitened kernel estimators provide a significant improvement over the standard non-prewhitened kernel estimators.

Coverage Probabilities of Bootstrap-Confidence Intervals for Quantiles

An asymptotic expansion of length 2 is established for the coverage probabilities of confidence intervals for the underlying $q$-quantile which are derived by bootstrapping the sample $q$-quantile. The corresponding level error turns out to be of order $O(n^{-1/2})$ which is unexpectedly low. A confidence interval of even more practical use is derived by using backward critical points. The resulting confidence interval is of the same length as the one derived by ordinary bootstrap but it is distribution free and has higher coverage probability.

The performance of the two-stage analysis of two-treatment, two-period crossover trials.

In the two-treatment, two-period crossover trial, patients are randomly allocated either to one group that receives treatment A followed by treatment B, or to another group that receives the treatments in the reverse order. Grizzle first proposed a two-stage procedure for analysing the data from such a trial. This paper examines the long-run sampling properties of this procedure, in terms of mean square error of point estimates, coverage probability of confidence intervals and actual significance level of hypothesis tests for the differences between the effects of the two treatments. The advantages of incorporating baseline observations into the analysis are also explored. Because the preliminary test for carryover is highly correlated with the analysis of data from the first period only, actual significance levels are higher than nominal levels even when there is no differential carryover. When carryover is present, the nominal level very seriously understates the actual level, and this becomes even worse when baseline observations are ignored. Increasing sample size only exacerbates the problem since this adverse behaviour then occurs at smaller values of the carryover effect. It is concluded that the two-stage analysis is too potentially misleading to be of practical use.

Summary attributable risk estimation from unmatched case-control data.

We propose an alternative method to obtain summary estimators, variances and confidence intervals for attributable risk measures. This method combines weighted exposure prevalences for cases and controls across strata formed by the cross-classification of relevant covariates to form estimates of attributable risk among the exposed and attributable risk in the target population. The major benefit of this approach over those previously proposed in the literature is that it operates on data summed across strata rather than on estimation of statistics within each stratum. This alternative method for attributable risk measures utilizes the Mantel-Haenszel estimate of an average odds ratio, and can be implemented using the matrix procedure in SAS. This method is appropriate even when the within-stratum sample sizes are too small for other methods to be valid. Simulation results indicate that this method is superior to others with respect to bias and coverage probability for confidence intervals.

Confidence interval estimation and hypothesis testing for a common correlation coefficient

Several procedures have been proposed for testing the hypothesis that all off-diagonal elements of the correlation matrix of a multivariate normal distribution are equal. If the hypothesis of equal correlation can be accepted, it is then of interest to estimate and perhaps test hypotheses for the common correlation. In this paper, two versions of five different test statistics are compared via simulation in terms of adequacy of the normal approximation, coverage probabilities of confidence intervals, control of Type I error, and power. The results indicate that two test statistics based on the average of the Fisher z-transforms of the sample correlations should be used in most cases. A statistic based on the sample eigenvalues also gives reasonable results for confidence intervals and lower-tailed tests.

Robustness of Stein's Two-Stage Procedure for Mixtures of Normal Populations

Abstract Explicit expressions for the level of significance and power of Stein's two-stage hypothesis testing procedure are derived when the normal population is mixed with another normal population differing in the mean. Approximate expansions for these expressions have been obtained, and numerical studies of the size have been carried out using the expansions. Some numerical indication of the accuracy of the approximations has also been obtained. Stein's two-stage procedure is quite robust against mixtures of normal populations differing in location parameters. Similar conclusions can be drawn regarding the coverage probabilities of confidence intervals generated by this procedure.

Robustness of Stein's Two-Stage Procedure for Mixtures of Normal Populations

Abstract Explicit expressions for the level of significance and power of Stein's two-stage hypothesis testing procedure are derived when the normal population is mixed with another normal population differing in the mean. Approximate expansions for these expressions have been obtained, and numerical studies of the size have been carried out using the expansions. Some numerical indication of the accuracy of the approximations has also been obtained. Stein's two-stage procedure is quite robust against mixtures of normal populations differing in location parameters. Similar conclusions can be drawn regarding the coverage probabilities of confidence intervals generated by this procedure.