Showing papers on "Coverage probability published in 1988"

Bayesian Confidence Intervals for Smoothing Splines

Abstract: The frequency properties of Wahba's Bayesian confidence intervals for smoothing splines are investigated by a large-sample approximation and by a simulation study. When the coverage probabilities for these pointwise confidence intervals are averaged across the observation points, the average coverage probability (ACP) should be close to the nominal level. From a frequency point of view, this agreement occurs because the average posterior variance for the spline is similar to a consistent estimate of the average squared error and because the average squared bias is a modest fraction of the total average squared error. These properties are independent of the Bayesian assumptions used to derive this confidence procedure, and they explain why the ACP is accurate for functions that are much smoother than the sample paths prescribed by the prior. This analysis accounts for the choice of the smoothing parameter (bandwidth) using cross-validation. In the case of natural splines an adaptive method for avo...

Estimating Animal Abundance with Capture Frequency Data

Abstract: I describe a new method for estimating the number of animals in a closed population with capture-recapture data under a heterogeneity model. The model is applied to capture frequency records of striped skunk (Mephitis mephitis), eastern chipmunk (Tamias striatus), eastern cottontail (Sylvilagus floridanus), and taxicab populations. I also report the results of a Monte Carlo simulation to assess the relative merits of the proposed moment estimator and the jackknife estimator. If many individuals are caught more than twice, the jackknife estimator is superior to the proposed moment estimator. However, when the mean capture probability is small, so that most captured animals are caught only 1-2 times in the samples, the moment estimator is usually less biased than the jackknife estimator. The mean coverage probability of the confidence interval associated with the moment estimator is also closer to the nominal level than that of the

Balanced Simultaneous Confidence Sets

Abstract: Suppose that T(θ) = {Tu (θ)} is a family of parametric functions indexed by the variable u. For each u, an approximate confidence set Cn,u for Tu (θ) may be obtained by referring a function of Tu (θ) and the sample to an estimated quantile dn,u of that function's sampling distribution. This approach is sometimes called the pivotal method for constructing a confidence set, even when it is not based on a true pivot. A simultaneous confidence set Cn for T(θ) is then obtained by simultaneously asserting the individual confidence sets {Cn,u }. The problem is to choose the critical values for the {Cn,u } in such a way that the overall coverage probability of Cn is correct and the coverage probabilities of the individual confidence sets {Cn,u } are equal. The second property is termed balance. It means that the simultaneous confidence set Cn treats each constituent confidence statement Cn,u fairly. Aside from a few special cases, the problem just described is too difficult for analytical approaches, whe...

Exact confidence intervals following a group sequential trial: A comparison of methods

Abstract: SUMMARY With interest in designing group sequential clinical trials increasing, methodology for analysing the data arising in such trials is needed. We discuss several approaches to constructing confidence intervals for the parameter of interest in a group sequential trial after the trial has ended. We compare these met1fods to one already in the literature and point out interesting differences.

Attributable risk estimation from matched case-control data.

Abstract: A methodology is proposed for obtaining summary estimators, variances, and confidence intervals for attributable risk measures from data obtained through a case-control study design where one or more controls have been matched to each case. The sampling design for obtaining these data is conceptualized as a simple random sample of cases being equivalent to a simple random sample of matched sets. By combining information across the strata determined by the matched sets, this approach provides all of the benefits associated with the Mantel-Haenszel procedure for the estimators of attributable risk among the exposed and population attributable risk. Asymptotic variances are derived under the assumption that the frequencies of the unique response patterns follow the multinomial distribution. Simulation results indicate that these methods fare very well with respect to bias and coverage probability.

On the use of concordant pairs in matched case-control studies

Abstract: A new estimator of the common odds ratio in one-to-one matched case-control studies is proposed. The connection between this estimator and the James-Stein estimating procedure is highlighted through the argument of estimating functions. Comparisons are made between this estimator, the conditional maximum likelihood estimator, and the estimator ignoring the matching in terms of finite sample bias, mean squared error, coverage probability, and length of confidence interval. In many situations, the new estimator is found to be more efficient than the conditional maximum likelihood estimator without being as biased as the estimator that ignores matching. The extension to multiple risk factors is also outlined.

Summary attributable risk estimation from unmatched case‐control data

Abstract: 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

Accuracy of approximate confidence bounds computed from interval censored weibull and lognormal data

Abstract: Interval censored data arise frequently in industrial life tests and other applications. Maximum likelihood estimation provides a convenient means for making inferences on important distribution properties like quantiles and failure probabilities. The asymptotic normal distribution of the maximum likelihood estimators provides a simple method of setting approximate confidence bounds on these quantiles. Inverting likelihood ratio tests is another alternative. This paper uses Monte Carlo Simulation to investigate the finite sample properties of maximum likelihood estimators of Weibull and lognormal parameters and quantiles from interval censored data. We evaluate the accuracy of large sample one- and two-sided confidence bounds based on asymptotic normal theory and compare their accuracy (with respect to coverage probability) to those obtained by inverting likelihood ratio tests. Even though these procedures are asymptotically equivalent, our results show that the intervals based on inverting a likelihood r...

Inferences on the lognormal mean for complete samples

Abstract: In many engineering problems it is necessary to draw statistical inferences on the mean of a lognormal distribution based on a complete sample of observations. Statistical demonstration of mean time to repair (MTTR) is one example. Although optimum confidence intervals and hypothesis tests for the lognormal mean have been developed, they are difficult to use, requiring extensive tables and/or a computer. In this paper, simplified conservative methods for calculating confidence intervals or hypothesis tests for the lognormal mean are presented. In this paper, “conservative” refers to confidence intervals (hypothesis tests) whose infimum coverage probability (supremum probability of rejecting the null hypothesis taken over parameter values under the null hypothesis) equals the nominal level. The term “conservative” has obvious implications to confidence intervals (they are “wider” in some sense than their optimum or exact counterparts). Applying the term “conservative” to hypothesis tests should not be conf...

Improved approximation for estimation following closed sequential tests

Abstract: SUMMARY Siegmund (1978) has developed a procedure for estimation following a closed sequential test, known as the repeated significance tests. Here a more elaborate approximation is proposed for a wider class of closed sequential tests. The major goal is to account for the excess over boundary in addition to the stopping time when the sequential test is terminated. Some hypothetical examples are used for comparing the two approximations. To investigate the properties of these procedures further, simulation results are presented in terms of estimated mean squared error, bias and variance for point estimators, and in terms of coverage probability and width for confidence intervals.

Convergence rates for two-stage confidence intervals based on U-statistics

Abstract: We consider a modified two-stage procedure for constructing a fixed-width confidence interval for the mean of a U-statistic First, we discuss a few asymptotic results with the associated rates of convergence The main result gives the rate of convergence for the coverage probability of our proposed confidence interval which is seen to be slower than that for the purely sequential procedure