Showing papers on "Coverage probability published in 1987"
TL;DR: In this article, two approaches for dealing with the problem of poor coverage probabilities of certain standard confidence intervals are proposed: the first is a recommendation that the actual coverage be estimated directly from the data and its value reported in addition to the nominal level.
Abstract: Two approaches for dealing with the problem of poor coverage probabilities of certain standard confidence intervals are proposed. The first is a recommendation that the actual coverage be estimated directly from the data and its value reported in addition to the nominal level. This is achieved through a combination of computer simulation and density estimation. The asymptotic validity of the procedure is proved for a number of common situations. A classical example is the nonparametric estimation of the variance of a population using the normal-theory interval. Here it is shown that the estimated coverage probability consistently estimates the true coverage probability if the population distribution possesses a finite sixth moment. The second approach is more traditional. It is a procedure for modifying an interval to yield improved coverage properties. Given a confidence interval, its estimated coverage probability obtained in the first approach is used to alter the nominal level of the interval...
180 citations
TL;DR: In this paper, the authors give a lower bound for the simultaneous coverage probability when such confidence intervals are applied to all of the proportions in a poll, for small values of α.
Abstract: Opinion polls often give an indication of sampling error based on the standard conservative confidence intervals for a single binomial proportion. We give a lower bound for the simultaneous coverage probability when such intervals are applied to all of the proportions in a poll. This bound is equal to 1 – 2α. for small values of α. The same result is shown to apply to the standard confidence intervals for changes in proportions between surveys.
77 citations
TL;DR: In this paper, the problem of finding α-level confidence intervals for the ratio of two normally estimated means was studied, and it was shown that there is no procedure that with probability 1 gives bounded α-Level confidence intervals.
Abstract: We consider the problem of finding $\alpha$-level confidence intervals for the ratio of two normally estimated means We show that there is no procedure that with probability 1 gives bounded $\alpha$-level confidence intervals for the ratio, and we show that within a large class of sensible procedures the Fieller solution is the only one with exact coverage probability
41 citations
TL;DR: In this article, the pivotal quantities used by Sen (1967, Sankhya, Series A 29, 95-102) and Govindarajulu (1968, Annals of the Institute of Statistical Mathematics 20, 229-238) are modified to take into account that the variance of estimated p depends on p.
Abstract: SUMMARY We review the problem of obtaining a distribution-free confidence interval for the probability (p) that one random variable is less than another independent random variable based on the two-sample uncensored Wilcoxon-Mann-Whitney statistic. The pivotal quantities used by Sen (1967, Sankhya, Series A 29, 95-102) and Govindarajulu (1968, Annals of the Institute of Statistical Mathematics 20, 229-238) are modified to take into account that the variance of estimated p depends on p. A number of simulations were conducted to compare the various methods. The results suggest that the proposed modification is superior in the sense that over the entire range of p the modification generally yields values of coverage probability much closer to nominal coverage than the methods of Sen (S) and Govindarajulu (G). The coverage values for the S and G methods are very close to each other; both share the characteristics for one-sided lower confidence limits of being conservative for small values of p and nonconservative (sometimes extremely so) for large values of p. One-sided upper limits have coverages that are essentially the mirror image of lower-limit coverages. The contrasts between the modified method and the S and G methods diminish in magnitude as sample size increases but are still nontrivial for a sample size of 80. Our tentative conclusion is that the modified procedure is preferable to that of Sen and Govindarajulu and can be used when the sample size in both groups is equal to or greater than 20. Limitations of our simulations are briefly discussed. An example illustrating the use of our modified pivotal quantity and comparing it with the S and G results is presented.
38 citations
TL;DR: In this article, a modified empirical Bayes argument is used to construct confidence sets centered at improved estimators of the mean of a multivariate normal distribution, which have uniformly higher coverage probability than the usual confidence set (a sphere centered at the observations), with no increase in volume.
35 citations
TL;DR: In this article, the authors presented a method for approximating the coverage probability of simultaneous confidence bounds for multiple regression functions when the vector of predictor variables is constrained to lie in a polyhedral convex set.
Abstract: This article presents a method for approximating the coverage probability of simultaneous confidence bounds for multiple regression functions when the vector of predictor variables is constrained to lie in a polyhedral convex set. The method is useful because it allows one to construct simultaneous confidence intervals with prescribed coverage probability for the regression function evaluated at various settings of the predictor variables, which are narrower than bounds obtained without using the predictor variable constraints. For a family of two-sided simultaneous confidence bounds that includes Scheffe-type and constant-width bounds, the probability of coverage is related to the distribution of the maximum Euclidean norm of the projection onto a polyhedral cone for a pair of random vectors with known joint distribution. An analogous relation holds for one-sided bounds. If an algorithm for computing the projection onto the cone is available, then these results enable one to use the Monte Carlo ...
34 citations
TL;DR: In this article, three-stage sampling procedures have been developed for both the one-and two-sample situations and discussed with second-order expansions of various characteristics of the proposed procedures including those for the achieved coverage probability in either problem.
Abstract: Fixed-width confidence interval estimation problems for location parameters of negative exponential populations have been studied. Three-stage sampling procedures have been developed for both the one- and two-sample situations. Our discussions are primarily concerned with second-order expansions of various characteristics of the proposed procedures including those for the achieved coverage probability in either problem. Some simulated results are also presented to indicate the usefulness of our procedures for moderate sample sizes.
17 citations
TL;DR: In this paper, the authors gave a distribution-free prediction interval for X based on X 1, X 2, X n of the form [[Xbar] -A, [Xbar + A] with A 2 ≈ λ2(1 + 1/n)S 2.
Abstract: Saw, Yang, and Mo (1984) gave a distribution-free prediction interval for X based on X 1,…, Xn of the form [[Xbar] -A, [Xbar] + A] with A 2 ≈ λ2(1 + 1/n)S 2. As compared with the range [X (1), X (2)], which has length R(say) and size (minimum coverage probability) (n − 1)/(n + 1), their intervals can have size as high as n/(n +1), a value that is attained when λ2 = n +1. For n = 2, this interval (with λ2 = 3) becomes the “triple range” [X (1) - R, X (2)+ R] and has size 2/3; it coincides with the “normal interval” for n = 2 with coverage probability 2/3 under normality. For all n > 2, the size of their interval (with λ2 = n + 1) equals approximately the coverage probability of the normal interval based on three observations only. A table is given for the value of A required to guarantee a size of at least h' for the distribution-free interval for selected values of h' and for all n ≤ 100. It may also be used when applying a Chebyshev-type inequality for simple random sampling from a finite popula...
15 citations
TL;DR: In this paper, a criterion for conditional acceptability of a recentered confidence set is established using the theory of relevant betting procedures, which provides an objective means for assessing the conditional performance of a statistical procedure.
Abstract: The usual confidence sphere for a multivariate normal mean can be uniformly improved upon, in terms of coverage probability, by recentering it at a Stein-type estimator. However, these improved sets can have poor conditional performance. Using the theory of relevant betting procedures, which provides an objective means for assessing the conditional performance of a statistical procedure, a criterion for conditional acceptability can be established. A method of constructing such sets is outlined and applied to some recentered confidence sets. In particular, recentering at the positive-part James-Stein estimator yields a conditionally acceptable confidence set.
11 citations
TL;DR: In this article, the duality between confidence bands and (convex) set estimators in a simple linear regression is explored, including the nature of polygonal sets and the development of an algorithm that approximates the coverage probability of smooth confidence band functions.
9 citations
01 Jan 1987
TL;DR: In this paper, the various point and interval estimators of a common odds ratio from multiple 2 × 2 tables are reviewed, with particular emphasis on the point estimators, their asymptotic properties, and what is known about finite-sample properties.
Abstract: For comparing two proportions from stratified samples, one approach is via inference on the odds ratio. The various point and interval estimators of a common odds ratio from multiple 2 × 2 tables are reviewed in this paper, with particular emphasis on the point estimators, their asymptotic properties, and what is known about finite-sample properties. Based on research to date, the conditional maximum likelihood and Mantel-Haenszel estimators are recommended as the point estimators of choice. Confidence interval methods have not been studied as well, but there is a method associated with the Mantel-Haenszel estimator that is a good choice. There is the possibility of improvement in the finite-sample properties of these estimators. However, further work is needed before one of these modifications can be recommended for general use.
TL;DR: The confidence coefficient of a two-sided confidence interval for the binomial parameter p is the infimum of the coverage probability of the interval as p ranges between 0 and 1 as discussed by the authors.
Abstract: The confidence coefficient of a two-sided confidence interval for the binomial parameter p is the infimum of the coverage probability of the interval as p ranges between 0 and 1. The confidence coefficients for five different approximate confidence intervals are computed and compared to the confidence coefficient for the two-sided Clopper-Pearson confidence interval. Pratt's approximation method [10] yields virtually the same confidence coefficients as the Clopper-Pearson interval, and is easily computed without resorting to interative methods.
TL;DR: In this article, the influence of two kinds of contamination on sequential confidence intervals for the mean of a normal population is studied when the contamination probability tends to zero with the length d of the confidence interval.
Abstract: The influence of two kinds of contamination on sequential confidence intervals for the mean of a normal population is studied when the contamination probability tends to zero with the length d of the confidence interval. It is shown that if ∊/d → a>0 the mean of the asymptotic distribution of the stopping variable (properly normalized) of eachprocedure considered is of special interest.In the case of contamination by a point mass, the asymptotic mean is closely related to the influence function of the measure of spread in the definition of the stopping rule. For the Chow Robbins procedure the influence function of the standard deviation appears and this is unbounded. If the standard deviation in this procedure is replaced by a fractile difference estimator, the influence function, and hence the asymptotic mean, remains bounded. Similar results are obtained for a gross errors contamination model. The asymptotic coverage probability of both procedures is biased under the nonsymmetric point mass contaminatio...
TL;DR: In this paper, the authors examined confidence bands for the simple linear model to assess their degrees of robustness to departures from the model, taking the true model to be a quadratic polynomial, and departure from the linear case in terms of increasing magnitude of a curvature parameter.
Abstract: Confidence bands for the simple linear model are examined to assess their degrees of robustness to departures from the model. All calculations are made under an interval constraint on the range of interest for the predictor variable. The true model is taken to be a quadratic polynomial, and departure from the linear case is considered in terms of increasing magnitude of a curvature parameter. Proposed measures of robustness include the actual coverage probability under the true model and a measure of percentage coverage over the predictor variable axis when the band fails to cover the true quadratic model. The different band functions considered include hyperbolic bands constructed from Scheffd's 5 method, linear-segmented bands, and fixed-width (minimax) and minimax-regret bands. In terms of preserving coverage probability under quadratic mis-specification, the fixed-width and linear-segment bands perform best, the former being preferred when the constraint interval on the predictor variable is ...
01 Jan 1987
01 Jan 1987
TL;DR: In this article, the authors examined Fisher's early papers on Fiducial probability from the performance point of view, and showed that there are usually many possible FIDucial distributions, and that the problem is to find one that is in some sense optimal.
Abstract: This is an examination of Fisher’s early papers on Fiducial probability, from the performance point of view. A Fiducial distribution for a real- valued parameter θ is identified as a distribution-function-valued statistic (each observed value is a probability distribution on the parameter space) such that for each γ between 0 and 1 the γ-quantile of this distribution has probability γ of exceeding θ. The view taken is that there are usually many possible Fiducial distributions, and that the problem is to find one that is in some sense optimal. It is shown that Fiducial distributions can be found for discrete random variables using randomization: this is done for the Binomial (n,p) by inverting the uniformly most powerful one-sided tests. The resulting Fiducial distribution has the corresponding optimum property that its γ-quantile, subject to having probability γ of exceeding the truep, has uniformly minimum probability of exceeding each p 1 > p.
TL;DR: In this paper, the coverage probability of prediction regions when the parameter is estimated for dependent observations is investigated. And the asymptotic power of the predicted regions is also studied briefly, and both stationary and non-stationary type models are considered as applications.
Abstract: We obtain and asymptotic correction for the coverage probability of prediction regions when the parameter are estimated for dependent observations. Both stationary adn non-stationary type models are considered as applications. Asymptotic power of theses regions is also studied briefly.
01 Jan 1987
TL;DR: The duality between periodogram testing in time series analysis and coverage probability calculation in geometrical probability was discovered by Fisher and Siegel as mentioned in this paper, who proposed a test based on spacings, which generalizes this by summing the amount by which each strong period exceeds a threshold.
Abstract: Publisher Summary This chapter discusses periodogram testing based on spacings. The duality between periodogram testing in time series analysis and coverage probability calculation in geometrical probability was discovered by Fisher. Fisher's test for periodicity is based on the ratio of the largest value to the sum of all the periodogram values; its null distribution corresponds to the probability that a collection of randomly-placed arcs covers a circle. Siegel's test for periodicity generalizes this by summing the amount by which each strong period exceeds a threshold; the null distribution corresponds to the distribution of the vacancy, that is, that portion of a circle not covered by a collection of randomly-placed arcs.