Showing papers by "Joseph P. Romano published in 1988"

A Review of Bootstrap Confidence Intervals

Abstract: On passe en revue plusieurs methodes distinctes basees sur le bootstrap pour construire des intervalles de confiance

On weak convergence and optimality of kernel density estimates of the mode

Abstract: A mode of a probability density $f(t)$ is a value $\theta$ that maximizes $f$. The problem of estimating the location of the mode is considered here. Estimates of the mode are considered via kernel density estimates. Previous results on this problem have several serious drawbacks. Conditions on the underlying density $f$ are imposed globally (rather than locally in a neighborhood of $\theta$). Moreover, fixed bandwidth sequences are considered, resulting in an estimate of the location of the mode that is not scale-equivariant. In addition, an optimal choice of bandwidth depends on the underlying density, and so cannot be realized by a fixed bandwidth sequence. Here, fixed and random bandwidths are considered, while relatively weak assumptions are imposed on the underlying density. Asymptotic minimax risk lower bounds are obtained for estimators of the mode and kernel density estimates of the mode are shown to possess a certain optimal local asymptotic minimax risk property. Bootstrapping the sampling distribution of the estimates is also discussed.

A Bootstrap Revival of Some Nonparametric Distance Tests

Abstract: Several tests based on the empirical measure have been proposed to test independence of variables, goodness of fit, equality of distributions, rotational invariance, and so forth. These tests have excellent power properties, but critical values are difficult, if not impossible, to obtain. Furthermore, these tests usually assume that the data are real-valued with continuous distributions. Here, critical values are determined by bootstrapping and the resulting tests are shown to have the correct asymptotic level under minimal assumptions. For example, given data Xi = (X i,1, …, Xi,d ), i = 1, …, n, it may be desired to test independence of the d components. The proposed test compares the empirical measure and the product of its marginals by taking a supremum over an appropriate Vapnik-Cervonenkis class of sets. No assumptions are made on the probability distribution of the data or on the space in which it lives; indeed, some components may be discrete, some continuous, and others categorical. Simil...

Bootstrapping the mode

Abstract: The problem of constructing bootstrap confidence intervals for the mode of a density is considered Estimates of the mode are derived from kernel density estimates based on fixed and data-dependent bandwidths The asymptotic validity of bootstrap techniques to estimate the sampling distribution of the estimates is investigated In summary, the results are negative in the sense that a straightforward application of a naive bootstrap yields invalid inferences In particular, the bootstrap fails if resampling is done from the kernel density estimate On the other hand, if one resamples from a smoother kernel density estimate (which is necessarily different from the one which yields the original estimate of the mode), the bootstrap is consistent The bootstrap also fails if resampling is done from the empirical distribution, unless the choice of bandwidth is suboptimal Similar results hold when applying bootstrap techniques to other functionals of a density