Constructing Confidence Sets Research Articles

Monte Carlo sampling in dual space for approximating the empirical halfspace distance

The Kolmogorov-Smirnov distance is an important tool for constructing confidence sets and tests in univariate problems. In multivariate settings, an analogous role is played by the halfspace distance, which has the merit of being invariant under linear transformations. However, the evaluation of the halfspace distance between two samples is a computationally very intensive combinatorial problem even in moderate dimensions, which severely restricts the use of the halfspace distance, especially in resampling procedures. To approximate this distance in a fast and data-dependent way, the notion of a dual measure is introduced. Based on geometric concepts, it will be shown how the above problem can be put as a density estimation problem using Monte Carlo sampling in a certain dual space. A central limit theorem for the empirical halfspace distance is derived and used as a gauge to compare the new procedure with a traditional random search.

Calibrated nonparametric confidence sets

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.

Data reduction and statistical inconsistency in linear inverse problems

An estimator or confidence set is statistically consistent if, in a well-defined sense, it converges in probability to the truth as the number of data grows. We give sufficient conditions for it to be impossible to find consistent estimators or confidence sets in some linear inverse problems. Several common approaches to statistical inference in geophysical inverse problems use the set of models that satisfy the data within a χ 2 measure of misfit to construct confidence sets and estimates. For example, the minimum-norm estimate of the unknown model is the model of smallest norm among those that map into a χ 2 ball around the data. We give weaker conditions under which the χ 2 misfit approach yields inconsistent estimators and confidence sets. Both sets of conditions depend on a measure of the redundancy of the observations, with respect to an a priori constraint on the model. When the observations are sufficiently redundant, using a χ 2 measure of misfit to selected averages of the data yields consistent confidence sets and minimum-norm estimates. Under still weaker conditions, one can find consistent estimates and confidence intervals for finite collections of linear functionals of the model. In an idealization of the problem of estimating the Gauss coefficients of the magnetic field at the core from satellite data, using a constraint on the energy stored in the field, suitable data averaging leads to consistent confidence intervals for finite collections of the Gauss coefficients.

Asymptotically honest confidence sets for structural errors-in-variables models

The problem of constructing confidence sets for the structural errors-in-variables model is considered under the assumption that the variance of the error associated with the covariate is known. Previously proposed confidence sets for this model suffer from the problem that they all have zero confidence levels for any sample size, where the confidence level of a confidence set is defined to be the infimum of coverage probability over the parameter space. In this paper we construct some asymptotically honest confidence sets; that is, the limiting values of their confidence levels are at least as large as the nominal probabilities when the sample size goes to $\infty$. A desirable property of the proposed confidence set for the slope is also established.

Direct use of regression quantiles to construct confidence sets in linear models

Direct use of the empirical quantile function provides a standard distribution-free approach to constructing confidence intervals and confidence bands for population quantiles. We apply this method to construct confidence intervals and confidence bands for regression quantiles and to construct prediction intervals based on sample regression quantiles. Comparison of the direct method with the studentization and the bootstrap methods are discussed. Simulation results show that the direct method has the advantage of robustness against departure from the normality assumption of the error terms.

A New General Method for Constructing Confidence Sets in Arbitrary Dimensions: With Applications

Let $\mathbf{X}$ have a star unimodal distribution $P_0$ on $\mathbb{R}^p$. We describe a general method for constructing a star-shaped set $S$ with the property $P_0(\mathbf{X} \in S) \geq 1 - \alpha$, where $0 < \alpha < 1$ is fixed. This is done by using the Camp-Meidell inequality on the Minkowski functional of an arbitrary star-shaped set $S$ and then minimizing Lebesgue measure in order to obtain size-efficient sets. Conditions are obtained under which this method reproduces a level (high density) set. The general theory is then applied to two specific examples: set estimation of a multivariate normal mean using a multivariate $t$ prior and classical invariant estimation of a location vector $\mathbf{\theta}$ for a mixture model. In the Bayesian example, a number of shape properties of the posterior distribution are established in the process. These results are of independent interest as well. A computer code is available from the authors for automated application. The methods presented here permit construction of explicit confidence sets under very limited assumptions when the underlying distributions are calculationally too complex to obtain level sets.

On nondifferentiable functions and the bootstrap

We investigate a class of statistical problems, where usual bootstrap methods fail, and discuss two alternative solutions. In particular, a stochastic procedure for constructing confidence sets is proposed. Special applications are the eigenvalues of a covariance matrix and minimum distance functionals.

A James-Stein Type Estimator for Combining Unbiased and Possibly Biased Estimators

Abstract We present a method for combining unbiased sample data with possibly biased auxiliary information. The estimator we derive is similar in spirit to the James–Stein estimator. We prove that the estimator dominates the sample mean under quadratic loss. When the auxiliary information is unbiased, our estimator has risk slightly greater than the usual combined estimator. As the bias increases, however, the risk of the usual estimator is unbounded, while the risk of our estimator is bounded by the risk of the sample mean. We show how our estimator can be considered an approximation to the best linear combination of the sample data and the auxiliary information, allude to how it can be derived as an empirical Bayes estimator, and suggest a method for constructing confidence sets. Finally, the performance of our estimator is compared to that of the sample mean and the usual combined estimator using real forestry data.

A James-Stein Type Estimator for Combining Unbiased and Possibly Biased Estimators

Abstract We present a method for combining unbiased sample data with possibly biased auxiliary information. The estimator we derive is similar in spirit to the James–Stein estimator. We prove that the estimator dominates the sample mean under quadratic loss. When the auxiliary information is unbiased, our estimator has risk slightly greater than the usual combined estimator. As the bias increases, however, the risk of the usual estimator is unbounded, while the risk of our estimator is bounded by the risk of the sample mean. We show how our estimator can be considered an approximation to the best linear combination of the sample data and the auxiliary information, allude to how it can be derived as an empirical Bayes estimator, and suggest a method for constructing confidence sets. Finally, the performance of our estimator is compared to that of the sample mean and the usual combined estimator using real forestry data.

Hierarchical Statistical Models and a Generalized Likelihood Ratio Test

SUMMARY This paper deals with statistical inference for statistical models which have a nested structure. The emphasis here is on the use of a classical test of significance and on a confidence set construction approach for model identification, and the paper proposes a generalized likelihood ratio test with equal marginal error rate as a basic tool. The power performance of this test is characterized and some applications are illustrated.

Evaluation of Normal Probabilities of Symmetric Regions

Let ${\bf X} = (X_1 , \cdots ,X_n )'$ be a standardized multivariate normal random vector and let $A \in \mathbb{R}^n $ be a permutation-symmetric region. We provide and justify an approximation to $P({\bf X} \in A)$ which is easy to compute. We then consider the special case of exceedance probabilities, ${\bf P}$ (k of the n components of ${\bf X}$ exceed a constant c), and use it to show how our approximation can be used to improve upon procedures for constructing confidence sets for the mean of a multivariate normal. We also assess the performance of our approximation by using numerical examples.

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.