Consistency of bootstrap approximation to the null distributions of local spatial statistics with application to house price analysis

With the increasing availability of spatially extensive geo-referenced data, much attention has been paid to the use of local statistics to identify local patterns of spatial association, in which the null distributions of local statistics play an essential role in the related statistical inference. As a powerful tool to approximate the distribution of a statistic, the bootstrap method is used in this paper to derive null distributions of the commonly used local spatial statistics including local Getis and Ord’s $G_{i}$ G i , Moran’s $I_{i}$ I i and Geary’s $c_{i}$ c i . Strong consistency of the bootstrap approximation to the null distributions of the statistics is proved under some mild conditions, and the Boston housing price data are analyzed to demonstrate the application of the theoretical results.

Inference Under Random Limit Bootstrap Measures

Asymptotic bootstrap validity is usually understood as consistency of the distribution of a bootstrap statistic, conditional on the data, for the unconditional limit distribution of a statistic of interest. From this perspective, randomness of the limit bootstrap measure is regarded as a failure of the bootstrap. We show that such limiting randomness does not necessarily invalidate bootstrap inference if validity is understood as control over the frequency of correct inferences in large samples. We first establish sufficient conditions for asymptotic bootstrap validity in cases where the unconditional limit distribution of a statistic can be obtained by averaging a (random) limiting bootstrap distribution. Further, we provide results ensuring the asymptotic validity of the bootstrap as a tool for conditional inference, the leading case being that where a bootstrap distribution estimates consistently a conditional (and thus, random) limit distribution of a statistic. We apply our framework to several inference problems in econometrics, including linear models with possibly nonstationary regressors, CUSUM statistics, conditional Kolmogorov–Smirnov specification tests and tests for constancy of parameters in dynamic econometric models.

Statistical attribute filtering to detect faint extended astronomical sources

In astronomy, sky surveys contain a large number of light-emitting sources, often with intensities close to the noise level. Automatic extraction of astronomical objects is therefore needed. SExtractor is a widely used program for automated source extraction and cataloguing, but it is not optimal with faint extended sources. Using SExtractor as a reference, the paper describes an improvement of a previous method proposed by the authors. It is a Max-Tree-based method for extraction of faint extended sources without using a stronger image smoothing. The Max-Tree structure is a hierarchical representation of an image, in which attributes can be computed in every node. Object detection is performed on the nodes of the tree and it relies on the distribution of a statistic calculated using the power attribute, compared to the expected distribution in case of noise. Statistical tests are presented, a comparison with the object extraction of SExtractor is shown and results are discussed.

Resampling Methods in Paleontology

This chapter reviews major types of statistical resampling approaches used in paleontology. They are an increasingly popular alternative to the classic parametric approach because they can approximate behaviors of parameters that are not understood theoretically. The primary goal of most resampling methods is an empirical approximation of a sampling distribution of a statistic of interest, whether simple (mean or standard error) or more complicated (median, kurtosis, or eigenvalue). This chapter focuses on the conceptual and practical aspects of resampling methods that a user is likely to face when designing them, rather than the relevant mathematical derivations and intricate details of the statistical theory. The chapter reviews the concept of sampling distributions, outlines a generalized methodology for designing resampling methods, summarizes major types of resampling strategies, highlights some commonly used resampling protocols, and addresses various practical decisions involved in designing algorithm details. A particular emphasis has been placed here on bootstrapping, a resampling strategy used extensively in quantitative paleontological analyses, but other resampling techniques are also reviewed in detail. In addition,ad hocand literature-based case examples are provided to illustrate virtues, limitations, and potential pitfalls of resampling methods.