Usual Confidence Set Research Articles

Counting by weighing: construction of two-sided confidence intervals

ABSTRACTCounting by weighing is widely used in industry and often more efficient than counting manually which is time consuming and prone to human errors especially when the number of items is large. Lower confidence bounds on the numbers of items in infinitely many future bags based on the weights of the bags have been proposed recently in Liu et al. [Counting by weighing: Know your numbers with confidence, J. Roy. Statist. Soc. Ser. C 65(4) (2016), pp. 641–648]. These confidence bounds are constructed using the data from one calibration experiment and for different parameters (or numbers), but have the frequency interpretation similar to a usual confidence set for one parameter only. In this paper, the more challenging problem of constructing two-sided confidence intervals is studied. A simulation-based method for computing the critical constant is proposed. This method is proven to give the required critical constant when the number of simulations goes to infinity, and shown to be easily implemented on an ordinary computer to compute the critical constant accurately and quickly. The methodology is illustrated with a real data example.

Confidence distributions applied to propagating uncertainty to inference based on estimating the local false discovery rate: A fiducial continuum from confidence sets to empirical Bayes set estimates as the number of comparisons increases

ABSTRACTEmpirical Bayes estimates of the local false discovery rate can reflect uncertainty about the estimated prior by supplementing their Bayesian posterior probabilities with confidence levels as posterior probabilities. This use of coherent fiducial inference with hierarchical models generates set estimators that propagate uncertainty to varying degrees. Some of the set estimates approach estimates from plug-in empirical Bayes methods for high numbers of comparisons and can come close to the usual confidence sets given a sufficiently low number of comparisons.

Admissibility of the Usual Confidence Set for the Mean of a Univariate or Bivariate Normal Population: The Unknown Variance Case

Summary In the Gaussian linear regression model (with unknown mean and variance), we show that the standard confidence set for one or two regression coefficients is admissible in the sense of Joshi. This solves a long-standing open problem in mathematical statistics, and this has important implications on the performance of modern inference procedures post model selection or post shrinkage, particularly in situations where the number of parameters is larger than the sample size. As a technical contribution of independent interest, we introduce a new class of conjugate priors for the Gaussian location–scale model.

Counting by Weighing: Know Your Numbers with Confidence

SummaryCounting by weighing is often more efficient than counting manually, which is time consuming and prone to human errors, especially when the number of items (e.g. plant seeds, printed labels or coins) is large. Papers in the statistical literature have focused on how to count, by weighing, a random number of items that is close to a prespecified number in some sense. The paper considers the new problem, arising from a consultation with a company, of making inference about the number of 1p coins in a bag with known weight for infinitely many bags, by using the estimated distribution of coin weight from one calibration data set only. Specifically, a lower confidence bound has been constructed on the number of 1p coins for each of infinitely many future bags of 1p coins, as required by the company. As the same calibration data set is used repeatedly in the construction of all these lower confidence bounds, the interpretation of coverage frequency of the lower confidence bounds that is proposed is different from that of a usual confidence set.

Small Confidence Sets for the Mean of a Spherically Symmetric Distribution

Summary Suppose that X has a k-variate spherically symmetric distribution with mean vector θ and identity covariance matrix. We present two spherical confidence sets for θ, both centred at a positive part Stein estimator TS+(X). In the first, we obtain the radius by approximating the upper α-point of the sampling distribution of ‖TS+(X)−θ‖2 by the first two non-zero terms of its Taylor series about the origin. We can analyse some of the properties of this confidence set and see that it performs well in terms of coverage probability, volume and conditional behaviour. In the second method, we find the radius by using a parametric bootstrap procedure. Here, even greater improvement in terms of volume over the usual confidence set is possible, at the expense of having a less explicit radius function. A real data example is provided, and extensions to the unknown covariance matrix and elliptically symmetric cases are discussed.

Estimation améliorée explicite d'un degré de confiance conditionnel

We consider the problem of estimating the confidence statement of the usual confidence set, with confidence coefficient 1− α, of the mean of a p-variate normal distribution with identity covariance matrix. For p⩾5, we give an explicit sufficient condition for domination over the standard estimator 1− α by an estimator correcting it, that is, by 1− α+ s where s is a suitable function. That condition mainly relies on a partial differential inequality of the form kΔ s+ s 2⩽0 (for a certain constant k>0). It allows us to formally establish (with no recourse to simulations) this domination result. To cite this article: D. Fourdrinier, P. Lepelletier, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

Set-induced minimax estimators for a multivariate normal mean

Using a pseudo-empirical-Bayes approach, better confidence regions, C ∗(X) , for a multivariate normal mean are proposed in Tseng and Brown (Ann. Statist. 25 (1997) 2228). While C ∗(X) have a uniformly smaller volume than the classical confidence set and retaining the same constant coverage probability, they do not render straightforward point estimator as the usual confidence sets and recentered confidence sets because of their unfamiliar shapes. We propose here to use the centroid of C ∗(X) as its supplementary point estimator. With numerical aid, this estimator is shown to have a smaller mean squared errors than the maximum likelihood estimator, hence a minimax.

Asymptotic Improvement of the Usual Confidence Set in a Multivariate Normal Distribution with Unknown Variance

We consider confidence sets for the mean of a multivariate normal distribution with unknown covariance matrix of the formσ2I. The coverage probability of the usual confidence set is shown to be improved asymptotically by centering at a shrinkage estimator.

Estimated Confidence Under the Validity Constraint

We examine the decision theoretic estimated confidence approach proposed by Kiefer, Robinson and Berger, and focus on results under the frequentist validity constraint previously described by Brown and by Berger. Our main result is that the usual constant coverage probability estimator for the usual confidence set of a linear model is admissible under the frequentist validity constraint. Note that it is inadmissible without the frequentist validity constraint when the dimension is at least 5. The criterion of admissibility under the frequentist validity constraint is shown to be quite a reasonable one. Therefore the constant coverage probability estimator which has been widely used is justifiable from the post-data point of view.

Improved confidence sets for spherically symmetric distributions

The usual confidence set for a multivariate mean vector can be improved upon by recentering the set at a Stein-type estimator: this fact is known to be true under many different distributional assumptions. Thus far, however, the case of unknown variance has not been dealt with analytically. In this paper we prove that recentered set estimators dominate the usual set estimator when sampling is from any of a class of spherically symmetric distributions with unknown variance.

Improved confidence sets for the mean of a multivariate normal distribution

A new class of confidence sets for the mean of a p-variate normal distribution (p≥3) is introduced. They are neither spheres nor ellipsoids. We show that we can construct our confidence sets so that their coverage probabilities are equal to the specified confidence coefficient. Some of them are shown to dominate the usual confidence set, a sphere centered at the observations. Numerical results are also given which show how small their volumes are.

Employing vague prior information in the construction of confidence sets

In the problem of estimating the mean, θ, of a multivariate normal distribution, an experimenter will often be able to give some vague prior specifications about θ. This information is used to construct confidence sets centered at improved estimators of θ. These sets are shown to have uniformly (in θ) higher coverage probability than the usual confidence set (a sphere centered at the observations), with no increase in volume. Further, through the use of a modified empirical Bayes argument, a variable radius confidence set is constructed which provides a uniform reduction of volume. Strong numerical evidence is presented which shows that the empirical Bayes set also dominates the usual confidence set in coverage probability. All these improved sets provide substantial gains if the prior information is correct. Also considered are extensions to the unknown variance case, and a discussion of applications to the one-way analysis of variance. In particular, a procedure is presented which uniformly improves upon Scheffé's method of estimation of contrasts.

Improved Confidence Sets for the Coefficients of a Linear Model with Spherically Symmetric Errors

Under the assumption of normal errors, confidence spheres for $p(p \geq 3)$ coefficients of a linear model centered at the positive part James-Stein estimators were recently proved, by Hwang and Casella, to dominate the usual confidence set with the same radius. In this paper, the same domination results are established under various spherically symmetric distributions. These distributions include uniform distributions, double exponential distributions, and multivariate $t$ distributions.

Empirical Bayes Confidence Sets for the Mean of a Multivariate Normal Distribution

Abstract Through the use of an empirical Bayes argument, a confidence set for the mean of a multivariate normal distribution is derived. The set is a recentered sphere, is easy to compute, and has uniformly smaller volume than the usual confidence set. An exact formula for the coverage probability is derived, and numerical evidence is presented which shows that the empirical Bayes set uniformly dominates the usual set in coverage probability.

Admissibility of the Usual Confidence Sets for a Class of Bivariate Populations

For a bivariate population, with a probability density $p(|x - \theta|)$ where $p(t)$ is strictly decreasing for $t \geqq 0$, when a single observation is taken, the usual confidence sets are circles of constant radius centered at the observed value. Previous results of Kiefer imply that these circles have the minimax property of minimizing the maximum expected Lebesgue measure of the confidence sets for a given lower confidence level. It is now shown here that subject to mild conditions on $p(t)$, the confidence circles are also essentially unique in having the minimax property. The result generalizes the result proved previously for the bivariate normal case.

Admissibility of the Usual Confidence Sets for the Mean of a Univariate or Bivariate Normal Population

Admissibility of the Usual Confidence Sets for the Mean of a Univariate or Bivariate Normal Population

Inadmissibility of the Usual Confidence Sets for the Mean of a Multivariate Normal Population

Inadmissibility of the Usual Confidence Sets for the Mean of a Multivariate Normal Population