Correct Coverage Probability Research Articles

Estimating kappa from binocular data

A common error in statistical analysis of ophthalmic data is the lack of accounting for the positive correlation generally present between observations made in fellow eyes. The alternative of data analysis from only one eye in each patient may lead to loss of power and unrealistically large confidence intervals. This paper discusses a method to estimate kappa, a measure of agreement between two graders, when both graders rate the same set of pairs of eyes. The method assumes that the true left-eye and right-eye kappa values are equal and makes use of the correlated binocular data to estimate confidence intervals for the common kappa. Simulations show that the new estimators are better than the estimator based on only one eye; new confidence intervals had the correct coverage probability, but were usually only about 70 per cent as wide as single-eye intervals. The general methodology described here applies to analysis of grader agreement in rating other paired body structures.

Bootstrap Simultaneous Error Bars for Nonparametric Regression

Simultaneous error bars are constructed for nonparametric kernel estimates of regression functions. The method is based on the bootstrap, where resampling is done from a suitably estimated residual distribution. The error bars are seen to give asymptotically correct coverage probabilities uniformly over any number of gridpoints. Applications to an economic problem are given and comparison to both pointwise and Bonferroni-type bars is presented through a simulation study.

Sequential Confidence Regions in Inverse Regression Problems

In inverse regression problems (or more generally, the estimation of ratios of regression parameters) and errors-in-variables models, it has been shown by Gleser and Hwang that the length of any confidence interval with positive confidence level is infinite with positive probability. Therefore the confidence sets derived using asymptotic theory, although having correct asymptotic coverage probability, typically have zero confidence level when the sample size is fixed. Is it possible to construct a sequential confidence interval with finite length and $1 - \alpha > 0$ confidence level? The answer is no for any finite stage sequential sampling. The answer is, however, yes for a fully sequential scheme, as demonstrated by Hwang and Liu. For the inverse regression problem, and more generally the set estimation of a ratio of regression parameters, we construct a $(1 - \alpha)$ confidence sequence. Applying such a confidence sequence, we can construct a $(1 - \alpha)$ sequential confidence interval with the length less than a prespecified quantity.

Refining Bootstrap Simultaneous Confidence Sets

Abstract Simultaneous confidence sets for a collection of parametric functions may be constructed in several different ways. These ways include: (a) the exact pivotal method that underlies Tukey's (1953) and Scheffé's (1953) simultaneous confidence intervals for linear parametric functions in the normal linear model; (b) the method of asymptotic pivots, which is an approximate extension of the pivotal method; (c) the method of bootstrapped roots developed in Beran (1988). These three methods share several features. Each method simultaneously asserts a collection of confidence sets, one confidence set for every parametric function of interest. Each method obtains the uth constituent confidence set by referring a root to a critical value; the uth root is a function of the sample and of the uth parametric function. Each method has the same aim: to control the overall level of the simultaneous confidence set and to keep equal the marginal levels of the individual confidence statements that make up the simultaneous confidence set. The three methods differ in how they construct the necessary critical values. The pivotal method requires that the roots be identically distributed and that the supremum over all the roots be a pivot—a function of the sample and of the unknown parameter whose distribution is completely known. The method of asymptotic pivots relaxes these assumptions slightly by requiring them to hold only asymptotically. Nevertheless, the practical scope of both methods is very narrow. In contrast, the method of bootstrapped roots, or B method, is widely applicable. Under general conditions, it generates simultaneous confidence sets that, asymptotically, are balanced and have correct overall coverage probability. Moreover, the B method has a direct Monte Carlo approximation, which makes it easy to use. Under supplementary conditions, the pivotal method turns out to be a special case of the B method. At finite sample sizes, a B method simultaneous confidence set may suffer error in its overall coverage probability and lack of balance among the marginal coverage probabilities of its constituent confidence sets. Of what orders are these errors? What can be done to reduce them? We answer both questions in this article, by introducing and studying the B 2 method—a double iteration of the B method. Under regularity conditions, the B 2 method reduces the asymptotic order of imbalance in the B method simultaneous confidence set; and at the same time, it reduces the asymptotic order of error in overall coverage probability. The B 2 method has a direct Monte Carlo approximation that makes its use straightforward, though computer-intensive.

Confidence Bands for a Distribution Function Using the Bootstrap

Abstract We first discuss the construction of bootstrap confidence bands for the distribution function F of a population for simple random sampling but do not assume that F is continuous. The known alternative approach is to use the quantiles from the tabled Kolmogorov distribution. This approach is known to be conservative. It has already been shown in Bickel and Freedman (1981) that the bootstrap confidence band has the correct coverage probability asymptotically. We show by simulation that the bootstrap works well for small samples and outperforms the conservative approach, particularly for distributions that have small carriers. We also investigate the analogous problem of finding bootstrap confidence bands for the distribution function F of a population for a more complicated situation, stratified random sampling. The conservative approach in the previous situation is extended to this case when sampling is with replacement (we expect that it holds for sampling without replacement) and the supports of...