Showing papers on "Coverage probability published in 1997"
TL;DR: In this article, it was shown that a confidence set which does not satisfy this characterization has zero coverage probability (level) in the neighborhood of non-identification subsets and will have a nonzero probability of being unbounded under any distribution compatible with the model.
Abstract: General characterizations of valid confidence sets and tests in problems which involve locally almost unidentified (LAU) parameters are provided and applied to several econo- metric models. Two types of inference problems are studied: (i) inference about parame- ters which are not identifiable on certain subsets of the parameter space, and (ii) inference about parameter transformations with discontinuities. When a LAU parameter or parametric function has an unbounded range, it is shown under general regularity conditions that any valid confidence set with level 1 - a for this parameter must be unbounded with probability close to 1 - a in the neighborhood of nonidentification subsets and will have a nonzero probability of being unbounded under any distribution compatible with the model: no valid confidence set which is almost surely bounded does exist. These properties hold even if "identifying restrictions" are imposed. Similar results also obtain for parameters with bounded ranges. Consequently, a confidence set which does not satisfy this characterization has zero coverage probability (level). This will be the case in particular for Wald-type confidence intervals based on asymptotic standard errors. Furthermore, Wald-type statistics for testing given values of a LAU parameter cannot be pivotal functions (i.e., they have distributions which depend on unknown nuisance param- eters) and even cannot be usefully bounded over the space of the nuisance parameters. These results are applied to several econometric problems: inference in simultaneous equations (instrumental variables (IV) regressions), linear regressions with autoregressive errors, inference about long-run multipliers and cointegrating vectors. For example, it is shown that standard "asymptotically justified" confidence intervals based on IV estimators (such as two-stage least squares) and the associated "standard errors" have zero coverage probability, and the corresponding t statistics have distributions which cannot be bounded by any finite set of distribution functions, a result of interest for interpreting IV regressions with "weak instruments." Furthermore, expansion methods (e.g., Edgeworth expansions) and bootstrap techniques cannot solve these difficulties. Finally, in a number of cases where Wald-type methods are fundamentally flawed (e.g., IV regressions with poor instruments), it is observed that likelihood-based methods (e.g., likelihood-ratio tests and confidence sets) combined with projection techniques can easily yield valid tests and confidence sets.
567 citations
TL;DR: In this paper, an inequality is given for the expected length of a confidence interval given that a particular distribution generated the data and assuming that the confidence interval has a given coverage probability over a family of distributions.
Abstract: An inequality is given for the expected length of a confidence interval given that a particular distribution generated the data and assuming that the confidence interval has a given coverage probability over a family of distributions. As a corollary, attempts to adapt to the regularity of the true density within derivative smoothness classes cannot improve the rate of convergence of the length of the confidence interval over minimax fixed-length intervals and still maintain uniform coverage probability. However, adaptive confidence intervals can attain improved rates of convergence in some other classes of densities, such as those satisfying a shape restriction.
103 citations
TL;DR: A SAS macro is written using PROC IML that takes multinomial cell counts as input and returns simultaneous confidence intervals with the user-specified coverage probability, and allows the user to choose among six methods of constructing confidence intervals for multInomial proportions.
40 citations
TL;DR: Inversion of Pearson's chi-square statistic yields a confidence ellipsoid that can be used for simultaneous inference concerning multinomial proportions as mentioned in this paper, where the authors discuss the performance of these methods in terms of empirical coverage probabilities and enclosed volume.
Abstract: Inversion of Pearson's chi-square statistic yields a confidence ellipsoid that can be used for simultaneous inference concerning multinomial proportions. Because the ellipsoid is difficult to interpret, methods of simultaneous confidence interval construction have been proposed by Quesenberry and hurst,goodman,fitzpatrick and scott and sison and glaz . Based on simulation results, we discuss the performance of these methods in terms of empirical coverage probabilities and enclosed volume. None of the methods is uniformly better than all others, but the Goodman intervals control the empirical coverage probability with smaller volume than other methods when the sample size supports the large sample theory. If the expected cell counts are small and nearly equal across cells, we recommend the sison and glaz intervals.
30 citations
TL;DR: In this paper, the relationship between process potential index (Cp), process shift index (k) and percentage non-conforming (p) is depicted graphically and two-sided confidence limits for k and Cpk under two different scenarios are derived.
Abstract: The process capability index Cpk has been widely used as a process performance measure. In practice this index is estimated using sample data. Hence it is of great interest to obtain confidence limits for the actual index given a sample estimate. In this paper we depict graphically the relationship between process potential index (Cp), process shift index (k) and percentage non-conforming (p). Based on the monotone properties of the relationship, we derive two-sided confidence limits for k and Cpk under two different scenarios. These two limits are combined using the Bonferroni inequality to generate a third type of confidence limit. The performance of these limits of Cpk in terms of their coverage probability and average width is evaluated by simulation. The most suitable type of confidence limit for each specific range of k is then determined. The usage of these confidence limits is illustrated via examples. Finally a performance comparison is done between the proposed confidence limits and three non-parametric bootstrap confidence limits. The results show that the proposed method consistently gives the smallest width and yet provides the intended coverage probability. © 1997 John Wiley & Sons, Ltd.
28 citations
TL;DR: In this article, the problem of multivariate calibration is considered in the setup where a normally distributed response variable is related to an explanatory variable through a multivariate linear model, where the variance covariance matrix of the response variable was assumed to be a multiple of the identity matrix.
Abstract: The problem of multivariate calibration is considered in the setup where a normally distributed response variable is related to an explanatory variable through a multivariate linear model. The variance covariance matrix of the response variable is assumed to be a multiple of the identity matrix. The calibration data, that is, data obtained on the response variable corresponding to known values of the explanatory variable, are to be used for the construction of confidence regions for unknown values of the explanatory variable. The calibration problem addressed in this article deals with the construction of multiple use confidence regions; that is, the calibration data will be used repeatedly in order to construct a sequence of confidence regions for a sequence of unknown values of the explanatory variable. Such a procedure is characterized using two coverage probabilities, say 1 — α and 1 — β. Given that the confidence regions are constructed using the same calibration data, the proportion of conf...
26 citations
TL;DR: In this article, the authors considered the problem of finding an upper 1 −α confidence limit for a scalar parameter of interest o in the presence of a nuisance parameter vector θ when the data are discrete and proposed a method to shift the possible values t of T so that they are as small as possible subject both to the minimum coverage probability being greater than or equal to 1 -α, and to the shifted values being in the same order as the unshifted ts.
Abstract: summary
Consider the problem of finding an upper 1 –α confidence limit for a scalar parameter of interest o in the presence of a nuisance parameter vector θ when the data are discrete. Approximate upper limits T may be found by approximating the relevant unknown finite sample distribution by its limiting distribution. Such approximate upper limits typically have coverage probabilities below, sometimes far below, 1 –α for certain values of (θ, o). This paper remedies that defect by shifting the possible values t of T so that they are as small as possible subject both to the minimum coverage probability being greater than or equal to 1 –α, and to the shifted values being in the same order as the unshifted ts. The resulting upper limits are called ‘tight’. Under very weak and easily checked regularity conditions, a formula is developed for the tight upper limits.
25 citations
TL;DR: In this paper, a class of confidence sets with constant coverage probability for the mean of a p-variate normal distribution is proposed through a pseudo-empirical-Bayes construction.
Abstract: A class of confidence sets with constant coverage probability for the mean of a p-variate normal distribution is proposed through a pseudo-empirical-Bayes construction. When the dimension is greater than 2, by combining analytical results with some exact numerical calculations the proposed sets are proved to have a uniformly smaller volume than the usual confidence region. Sufficient conditions for the connectedness of the proposed confidence sets are also derived. In addition, our confidence sets could be used to construct tests for point null hypotheses. The resultant tests have convex acceptance regions and hence are admissible by Birnbaum. Tabular results of the comparison between the proposed region and other confidence sets are also given.
25 citations
TL;DR: In this article, the authors present robust inference procedures for linear and nonlinear models and derive robust analogues of likelihood ratio, Wald, and scores (Rao) tests for general parametric models.
Abstract: Publisher Summary This chapter discusses the concept of robust inference using an approach known as influence functions. It also describes the two key tools, the influence function and the breakdown point, and their application to the inference problem. They can be used to investigate the local stability and the global reliability of a test or confidence interval. Moreover, they provide the basis for constructing new robust tests and confidence intervals. The chapter presents robust tests for general parametric models. Robust analogues of likelihood ratio, Wald, and scores (Rao) tests are derived in the chapter. Robust inference procedures for linear and nonlinear models are discussed in the chapter. Some numerical results show the finite sample performance of these robust procedures. The purpose in robust testing is twofold. First, the level of a test should be stable under small, arbitrary departures from the null hypothesis (robustness of validity). Secondly, the test should still have a good power under small arbitrary departures from specified alternatives (robustness of efficiency). For confidence intervals, these criteria translate to coverage probability and length of the interval.
23 citations
TL;DR: In this article, the authors extended Woodroofe's technique to provide confidence intervals for the individual absolute success probabilities of the two treatments in the trial, for which no current methodology exists.
Abstract: Consider a sequential test applied to two streams of binary responses in a comparative clinical trial. After the completion of such a trial Woodroofe's (1992) technique for accurate confidence interval estimation for a treatment difference can be applied. In this paper the technique is extended to provide confidence intervals for the individual absolute success probabilities of the two treatments in the trial, for which no current methodology exists. Accuracy is explored by simulation of coverage probabilities and individual confidence limit probabilities. The simulations concern a triangular test and an O'Brien & Fleming test.
18 citations
TL;DR: In this article, the authors employ the method of empirical likelihood to construct confidence intervals for M-functionals in the presence of auxiliary information under a nonparametric setting and show that the modified empirical likelihood confidence intervals are asymptotically at least as narrow as the standard ones which do not utilize auxiliary information.
TL;DR: The construction of confidence sets when multivariate normality holds and in the general case where the usual spherical or elliptical structures may not occur is investigated in this article, where calibration is used to correct the coverage probability of the nonparametric sets.
Abstract: The construction of confidence sets when multivariate normality holds and in the general case where the usual spherical or elliptical structures may not occur is investigated. Calibration is used to correct the coverage probability of the nonparametric sets, and an example involving parameters from a chemical kinetics model in a biological system is used to demonstrate the techniques. Monte Carlo simulations validate the approach.
TL;DR: This article developed the conditional maximum likelihood (CMLE) estimator of the underlying common relative difference (RD) and its asymptotic conditional variance, and provided for the RD an exact interval calculation procedure, of which the coverage probability is always larger than or equal to the desired confidence level.
Abstract: On the basis of the conditional distribution, given the marginal totals of non-cases fixed for each of independent 2 × 2 tables under inverse sampling, this paper develops the conditional maximum likelihood (CMLE) estimator of the underlying common relative difference (RD) and its asymptotic conditional variance. This paper further provides for the RD an exact interval calculation procedure, of which the coverage probability is always larger than or equal to the desired confidence level and for investigating whether the underlying common RD equals any specified value an exact test procedure, of which Type I error is always less than or equal to the nominal α-level. These exact interval estimation and exact hypothesis testing procedures are especially useful for the situation in which the number of index subjects in a study is small and the asymptotically approximate methods may not be appropriate for use. This paper also notes the condition under which the CMLE of RD uniquely exists and includes a simple example to illustrate use of these techniques.
TL;DR: In this paper, the authors proposed two new methods for computing confidence intervals for the difference δ = p 1 - p 2 between two binomial proportions (p 1, P 2 ) for the two-sample case where the weighted tail probability is α/2 at each end on the δ scale.
Abstract: Two new methods for computing confidence intervals for the difference δ = p 1 - p 2 between two binomial proportions (p 1 , P 2 ) are proposed. Both the Mid-P and Max-P likelihood weighted intervals are constructed by mapping the tail probabilities from the two-dimensional (p 1 , p 2 )-space into a one-dimensional function of δ based on the likelihood weights. This procedure may be regarded as a natural extension of the CLOPPER-PEARSON (1934) interval to the two-sample case where the weighted tail probability is α/2 at each end on the δ scale. The probability computation is based on the exact distribution rather than a large sample approximation. Extensive computation was carried out to evaluate the coverage probability and expected width of the likelihood-weighted intervals, and of several other methods. The likelihood-weighted intervals compare very favorably with the standard asymptotic interval and with intervals proposed by HAUCK and ANDERSON (1986), Cox and SNELL (1989), SANTNER and SNELL (1980), SANTNER and YAMAGAMI (1993), and PESKUN (1993). In particular, the Mid-P likelihood-weighted interval provides a good balance between accurate coverage probability and short interval width in both small and large samples. The Mid-P interval is also comparable to COE and TAMHANE's (1993) interval, which has the best performance in small samples.
TL;DR: In this article, the authors address the problem of constructing confidence intervals for ordered population means of k independent normal populations using jackknife and bootstrap methodologies, where the goal is to achieve the nominal coverage probability with width of the interval no bigger than that of the standard confidence interval centered at the unrestricted maximum likelihood estimator (UMLE).
TL;DR: In this paper, a confidence interval for the scale parameter following a preliminary test concerning the equality of scale parameters is investigated, which is referred to as a pre-test confidence interval.
TL;DR: In this article, a modified version of the standard bootstrap procedure is proposed, which employs a data-based shrinkage towards the critical values β = ± 1, which converges to zero as the sample size grows, irrespective of the value of the autoregressive parameter.
TL;DR: In this paper, the authors investigated the interpretation of confidence interval following estimation of a Box-Cox transformation parameter and showed that, when model assumptions are satisfied and n is large, the nominal confidence level closely approximates the conditional coverage probability.
Abstract: What is the interpretation of a confidence interval following estimation of a Box-Cox transformation parameter λ? Several authors have argued that confidence intervals for linear model parameters ψ can be constructed as if λ. were known in advance, rather than estimated, provided the estimand is interpreted conditionally given . If the estimand is defined as , a function of the estimated transformation, can the nominal confidence level be regarded as a conditional coverage probability given , where the interval is random and the estimand is fixed? Or should it be regarded as an unconditional probability, where both the interval and the estimand are random? This article investigates these questions via large-n approximations, small- σ approximations, and simulations. It is shown that, when model assumptions are satisfied and n is large, the nominal confidence level closely approximates the conditional coverage probability. When n is small, this conditional approximation is still good for regression models with small error variance. The conditional approximation can be poor for regression models with moderate error variance and single-factor ANOVA models with small to moderate error variance. In these situations the nominal confidence level still provides a good approximation for the unconditional coverage probability. This suggests that, while the estimand may be interpreted conditionally, the confidence level should sometimes be interpreted unconditionally.
Quelle est l'interpretation d'un intervalle de confiance suivant l'estimation d'un parametre de transformation Box-Cox λ? Plusieurs auteurs ont pretendu que des intervalles de confiance pour des parametres ψ de modele lineaire peuvent etre construits comme si λ etait connu d'avance, plutot qu'estime, pourvu que l'estimateur soit interprete conditionnellement etant donne . Si l'estimateur est defini comme etant , une fonction de la transformation estimee, le niveau de confiance nominal peut-il etre considere comme une probabilite de couverture conditionnelle etant donne , ou l'intervalle est aleatoire et l'estimateur fixe? Ou devrait-il etre considere comme une probabilite inconditionnelle, ou l'intervalle et l'estimateur sont tous deux aleatoires? Cet article explore ces questions via des approximations pour de grands n, des approximations pour de petits σ, et des simulations. Il est demontre que, lorsque les hypotheses du modele sont satisfaites et que n est grand, le niveau de confiance nominal approxime etroitement la probabilite de couverture conditionnelle. Lorsque n est petit, cette approximation conditionnelle est encore bonne pour des modeles de regression ayant une petite variance d'erreur. L'approximation conditionnelle peut etre mediocre pour les modeles de regression avec variance d'erreur moderee et les modeles ANOVA a un seul facteur avec variance d'erreur petite a moderee. Dans ces situations, le niveau de confiance nominal procure tout de meme une bonne approximation de la probabilite de couverture inconditionnelle. Ceci suggere que, bien que l'estimateur peut etre interprete conditionnellement, le niveau de confiance devrait parfois etre interprete inconditionellement.
TL;DR: In this paper, the authors proposed an integrated approach for estimating a mean with a fixed width confidence interval through sampling in three stages to cover the additional problem of testing hypotheses concerning shifts in the mean with controlled Type II error.
Abstract: The rationale and methodology for estimating a mean with a fixed width confidence interval through sampling in three stages are extended to cover the additional problem of testing hypotheses concerning shifts in the mean with controlled Type II error. The coverage probability and operating characteristic function of the confidence interval based on the integrated approach are derived and compared with those of the usual triple sampling confidence interval. The extended methodology leads to better coverage probability and uniformly better Type II error probabilities. Achieving the additional objective of controlling Type II error inevitably implies a two- to threefold increase in the required optimal sample size. Some suggestions for dealing with this apparent limitation are discussed from a practical viewpoint. It is recommended that an integrated approach to estimation and testing based on confidence intervals be incorporated in the design stage for credible inferences.
TL;DR: In this article, the authors considered sequantial producers to construct fixed-width confidence intervals for some function θ of mean μ and variance σ 2 of normal distribution, and used nonlinear renewal theory to drive asymptotic expansion of expectation of the stopping time and estimate as the width of confidence interval decreases to zero.
Abstract: This paper considers sequantial producers to construct fixed-width confidence intervals for some function θ of mean μ and variance σ2 of normal distribution.Consideration is devoted to θ=exp( μ + σ2/2 ) and θ=μ/σ.Nonlinear renewal theory is used to drive asymptotic expansion of expectation of the stopping time and the estimate as the width of confidence interval decreases to zero.An improvement of the coverage probability is also discussed.
TL;DR: In this paper, both point and interval estimation of the survivor function S/sub 0/=Pr{X/spl ges/x/sub0/} for the geometric distribution is discussed.
Abstract: This paper discusses both point and interval estimation of the survivor function S/sub 0/=Pr{X/spl ges/x/sub 0/} for the geometric distribution. When the number of devices n/spl ges/50, the performance of the maximum likelihood estimator (MLE) and uniformly minimum variance unbiased estimator (UMVUE) of S/sub 0/ are essentially equivalent with respect to the relative mean-square-error (RMSE) to S/sub 0/. However, when the failure probability per time unit p/spl ges/0.50, and n/spl les/30, the UMVUE is preferable to the MLE with respect to the RMSE. For interval estimation of S/sub 0/ with no censoring, 4 asymptotic interval-estimators are derived from large-sample theory, and one from the exact distribution of the negative binomial. When p/spl les/0.2 and n/spl ges/30, all 5 interval-estimators perform reasonably well with respect to coverage probability. Since using the interval estimator derived from the exact distribution can assure "coverage probability"/spl ges/"desired confidence", this estimator is probably preferable to the other asymptotic ones when p/spl ges/0.50, and n/spl les/10. Finally, consider right-censoring, in which the failure-time that occurs after a fixed follow-up time period, is censored. We extend the interval estimator using the asymptotic properties of the MLE to account for right censoring. Monte Carlo simulation is used to evaluate the performance of this interval estimator; the censoring effect on efficiency is discussed for a variety of situations.
Posted Content•
TL;DR: In this article, the authors show that the bootstrap is not asymptotically correct to first order when the mean is restricted to be nonnegative, i.e., when the true value of the mean equals zero.
Abstract: The bootstrap of the maximum likelihood estimator of the mean of a sample of iid normal random variables with mean mu and variance one is not asymptotically correct to first order when the mean is restricted to be nonnegative. The problem occurs when the true value of the mean mu equals zero. This counterexample to the bootstrap generalizes to a wide variety of estimation problems in which the true parameter may be on the boundary of the parameter space. We provide some alternatives to the bootstrap that are asymptotically correct to first order. We consider two types of bootstrap percentile confidence intervals in the above example. We find that they both have asymptotic coverage probability that exceeds the nominal asymptotic level when the true value of the mean it equals zero.
TL;DR: In this article, a new purely sequential sampling strategy was proposed to construct a fixed-size confidence region for the mean vector of an unknown distribution function, and the coverage probability was shown to be at least (1??)?B?2d2+o(d2) asd?0.
TL;DR: In this article, the authors considered the interval estimation of the disturbance variance in a linear regression model with multivariate Student-t errors and derived the distribution function of the Stein type estimator under multivariate student-t error.
Abstract: This paper considers the interval estimation of the disturbance variance in a linear regression model with multivariate Student-t errors. The distribution function of the Stein type estimator under multivariate Student-t errors is derived, and the coverage probability of the Stein type confidence interval which is constructed under the normality assumption is numerically calculated under the multivariate Student-t distribution. It is shown that the coverage probability of the Stein type confidence interval is sometimes much smaller than the nominal level, and that it is larger than that of the usual confidence interval in almost all cases. For the case when it is known that errors have a multivariate Student-t distribution, sufficient conditions under which the Stein type confidence interval improves on the usual confidence interval are given, and the coverage probability of the stein type confidence interval is numerically evaluated.
TL;DR: In this paper, the authors investigated the rate of convergence of the coverage probability for fixed-width sequential confidence intervals of τ = aµ + bσ with a and b being given constants when the location parameter µ and the scale parameter σ of the negative exponential distribution are unknown.
Abstract: In this paper we consider sequential fixed-width confidence interval estimation for a parameter τ = aµ + bσ with a and b being given constants when the location parameter µ and the scale parameter σ of the negative exponential distribution are unknown. We investigate the rate of convergence of the coverage probability for fixed-width sequential confidence intervals of τ.
TL;DR: In this article, the authors compare higher order asymptotic properties of confidence intervals based on studentization and variance stabilizing transformations and show that neither of the methods outperforms the other in all situations, and hence the expansions given here can be used effectively to determine the better method in a specific application.
TL;DR: In this article, the coverage probability and the expected width of a traditional confidence region are derived as the threshold constant increases, and the nature of this bound is related to information numbers and first passage probabilities for random walks.
TL;DR: In this paper, the authors compared five interval estimators: the confidence interval using an idea similar to Fieller's theorem (CIFT), the CIMP using an exact parametric test (CIEP), the CIEP using the marginal likelihood ratio test (CILR), CILR assuming no matching effect (CINM), and the CIIP using a locally most powerful test(CIMP).
Abstract: This paper discusses interval estimation for the ratio of the mean failure times on the basis of paired exponential observations. This paper considers five interval estimators: the confidence interval using an idea similar to Fieller's theorem (CIFT), the confidence interval using an exact parametric test (CIEP), the confidence interval using the marginal likelihood ratio test (CILR), the confidence interval assuming no matching effect (CINM), and the confidence interval using a locally most powerful test (CIMP). To evaluate and compare the performance of these five interval estimators, this paper applies Monte Carlo simulation. This paper notes that with respect to the coverage probability, use of the CIFT, CILR, or CIMP, although which are all derived based on large sample theory, can perform well even when the number of pairs n is as small as 10. As compared with use of the CILR, this paper finds that use of the CIEP with equal tail probabilities is likely to lose efficiency. However, this loss can be reduced by using the optimal tail probabilities to minimize the average length when n is small (≤20). This paper further notes that use of the CIMP is preferable to the CIEP in a variety of situations considered here. In fact, the average length of the CIMP with use of the optimal tail probabilities can even be shorter than that of the CILR. When the intraclass correlation between failure times within pairs is 0 (i.e., the failure times within the same pair are independent), the CINM, which is derived for two independent samples, is certainly the best one among the five interval estimators considered here. When there is an intraclass correlation but which is small (≤0.10), the CIFT is recommended for obtaining a relatively short interval estimate without sacrificing the loss of the coverage probability. When the intraclass correlation is moderate or large, either the CILR or the CIMP with the optimal tail probabilities is preferable to the others. This paper also notes that if the intraclass correlation between failure times within pairs is large, use of the CINM can be misleading, especially when the number of pairs is large.
TL;DR: It is shown that there exist simulation estimates and confidence intervals for the expected first passage times and rewards as well as the expected average reward, with 100% coverage probability, in a finite state irreducible Markov reward chain.
Abstract: Consider a finite state irreducible Markov reward chain. It is shown that there exist simulation estimates and confidence intervals for the expected first passage times and rewards as well as the expected average reward, with 100% coverage probability. The length of the confidence intervals converges to zero with probability one as the sample size increases; it also satisfies a large deviations property.
Journal Article•
TL;DR: In this paper, the authors proposed several estimators of the reliability function R of the two-parameter exponential distribution, and compared those estimators in terms of the mean square error (MSE) through Monte Carlo method.
Abstract: We propose several estimators of the reliability function R of the two-parameter exponential distribution, and then compare those estimator in terms of the mean square error (MSE) through Monte Carlo method. We also consider the parametric bootstrap estimation. Using the parametric bootstrap estimator, we obtain the bootstrap confidence intervals for reliability function and compare the proposed bootstrap confidence intervals in terms of the length and the coverage probability through Monte Carlo method.