Rudolf Beran

Prepivoting Test Statistics: A Bootstrap View of Asymptotic Refinements

Abstract Approximate tests for a composite null hypothesis about a parameter θ may be obtained by referring a test statistic to an estimated critical value. Either asymptotic theory or bootstrap methods can be used to estimate the desired quantile. The simple asymptotic test ϕA refers the test statistic to a quantile of its asymptotic null distribution after unknown parameters have been estimated. The bootstrap approach used here is based on the concept of prepivoting. Prepivoting is the transformation of a test statistic by the cdf of its bootstrap null distribution. The simple bootstrap test ϕB refers the prepivoted test statistic to a quantile of the uniform (0, 1) distribution. Under regularity conditions, the bootstrap test ϕB has a smaller asymptotic order of error in level than does the asymptotic test ϕA , provided that the asymptotic null distribution of the test statistic does not depend on unknown parameters. In the contrary case, both ϕA and ϕB have the same order of level error. Certain class...

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 functions 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...

Calibrating Prediction Regions

Abstract Suppose that the variable X to be predicted and the learning sample Yn that was observed have a joint distribution, which depends on an unknown parameter θ. The parameter θ can be finite- or infinite-dimensional. A prediction region Dn for X is a random set, depending on Yn , that contains X with prescribed probability α. This article studies methods for controlling simultaneously the conditional coverage proability of Dn , given Yn , and the overall (unconditional) coverage probability of Dn . The basic construction yields a prediction region Dn , which has the following properties in regular models: Both the conditional and overall coverage probabilities of Dn converge to α as the size n of the learning sample increases. The convergence of the former is in probability. Moreover, the asymptotic distribution of the conditional coverage probability about α is typically normal; and the overall coverage probability tends to α at rate n −1. Can one reduce the dispersion of the conditional coverage pr...

Adaptive estimates for autoregressive processes

Let {Xt:t=0, ±1, ±2, ...} be a stationaryrth order autoregressive process whose generating disturbances are independent identically distributed random variables with marginal distribution functionF. Adaptive estimates for the parameters of {Xt} are constructed from the observed portion of a sample path. The asymptotic efficiency of these estimates relative to the least squares estimates is greater than or equal to one for all regularF. The nature of the adaptive estimates encourages stable behavior for moderate sample sizes. A similar approach can be taken to estimation problems in the general linear model.

Diagnosing Bootstrap Success

We show that convergence of intuitive bootstrap distributions to the correct limit distribution is equivalent to a local asymptotic equivariance property of estimators and to an asymptotic independence property in the bootstrap world. The first equivalence implies that bootstrap convergence fails at superefficiency points in the parameter space. However, superefficiency is only a sufficient condition for bootstrap failure. The second equivalence suggests graphical diagnostics for detecting whether or not the intuitive bootstrap is trustworthy in a given data analysis. Both criteria for bootstrap convergence are related to Hájeks (1970, Zeit. Wahrscheinlichkeitsth., 14, 323-330) formulation of the convolution theorem and to Basus (1955, Sankhyā, 15, 377-380) theorem on the independence of an ancillary statistic and a complete sufficient statistic.

Refining Bootstrap Simultaneous Confidence Sets

Abstract Simultaneous confidence sets for a collection of parametric functions may be constructed in several different ways. These ways include: (a) the exact pivotal method that underlies Tukeys (1953) and Scheffes (1953) simultaneous confidence intervals for linear parametric functions in the normal linear model; (b) the method of asymptotic pivots, which is an approximate extension of the pivotal method; (c) the method of bootstrapped roots developed in Beran (1988). These three methods share several features. Each method simultaneously asserts a collection of confidence sets, one confidence set for every parametric function of interest. Each method obtains the uth constituent confidence set by referring a root to a critical value; the uth root is a function of the sample and of the uth parametric function. Each method has the same aim: to control the overall level of the simultaneous confidence set and to keep equal the marginal levels of the individual confidence statements that make up the simulta...

Convergence of stochastic empirical measures

Let Pn be a random probability measure on a metric space S. Let P^n be the empirical measure of kn iid random variables, each distributed according to Pn. Our main theorem asserts that if {Pn} converges in distribution, as random probability measures on S, then so does {P^n}. Applications of the result to the study of bootstrap and other stochastic procedures are given.

React Scatterplot Smoothers: Superefficiency through Basis Economy

Abstract REACT estimators for the mean of a linear model involve three steps: transforming the model to a canonical form that provides an economical representation of the unknown mean vector, estimating the risks of a class of candidate linear shrinkage estimators, and adaptively selecting the candidate estimator that minimizes estimated risk. Applied to one- or higher-way layouts, the REACT method generates automatic scatterplot smoothers that compete well on standard datasets with the best fits obtained by alternative techniques. Historical precursors to REACT include nested model selection, ridge regression, and nested principal component selection for the linear model. However, REACTs insistence on working with an economical basis greatly increases its superefficiency relative to the least squares fit. This reduction in risk and the possible economy of the discrete cosine basis, of the orthogonal polynomial basis, or of a smooth basis that generalizes the discrete cosine basis are illustrated by fitting scatterplots drawn from the literature. Flexible monotone shrinkage of components rather than nested 1–0 shrinkage achieves a secondary decrease in risk that is visible in these examples. Pinsker bounds on asymptotic minimax risk for the estimation problem express the remarkable role of basis economy in reducing risk.

Efficient robust estimates in parametric models

SummaryLet {Pθn:θ∈Θ},Θ an open subset ofRk, be a regular parametric model for a sample ofn independent, identically distributed observations. This paper describes estimates {Tn;n≧1} ofθ which are asymptotically efficient under the parametric model and are robust under small deviations from that model. In essence, the estimates are adaptively modified, one-step maximum likelihood estimates, which adjust themselves according to how well the parametric model appears to fit the data. When the fit seems poor,Tn discounts observations that would have large influence on the value of the usual one-step MLE. The estimates {Tn} are shown to be asymptotically minimax, in the Hájek-LeCam sense, for a Hellinger ball contamination model. An alternative construction of robust asymptotically minimax estimates, as modified MLEs, is described for canonical exponential families.

Confidence sets centered at C p -estimators

Suppose Xn is an observation, or average of observations, on a discretized signal ξn that is measured at n time points. The random vector Xn has a N(ξn, σ2nI) distribution, the mean and variance being unknown. Under squared error loss, the unbiased estimator Xn of ξn can be improved by variable-selection. Consider the candidate estimator ξn(A) whose i-th component equals the i-th component of Xn whenever i/(n+1) lies in A and vanishes otherwise. Allow the set A to range over a large collection of possibilities. A Cp-estimator is a candidate estimator that minimizes estimated quadratic loss over A. This paper constructs confidence sets that are centered at a Cp-estimator, have correct asymptotic coverage probabiligy for ξn, and are geometrically smaller than or equal to the competing confidence balls centered at Xn. The asymptotics are locally uniform in the parameters (ξn, σ2n). The results illustrate an approach to inference after variable-selection.