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Parametric-rate inference for one-sided differentiable parameters.

ABSTRACT: Suppose one has a collection of parameters indexed by a (possibly infinite dimensional) set. Given data generated from some distribution, the objective is to estimate the maximal parameter in this collection evaluated at the distribution that generated the data. This estimation problem is typically non-regular when the maximizing parameter is non-unique, and as a result standard asymptotic techniques generally fail in this case. We present a technique for developing parametric-rate confidence intervals for the quantity of interest in these non-regular settings. We show that our estimator is asymptotically efficient when the maximizing parameter is unique so that regular estimation is possible. We apply our technique to a recent example from the literature in which one wishes to report the maximal absolute correlation between a prespecified outcome and one of p predictors. The simplicity of our technique enables an analysis of the previously open case where p grows with sample size. Specifically, we only require that log p grows slower than n , where n is the sample size. We show that, unlike earlier approaches, our method scales to massive data sets: the point estimate and confidence intervals can be constructed in O(np) time.

SUBMITTER: Luedtke AR

PROVIDER: S-EPMC6075853 | biostudies-literature | 2018

REPOSITORIES: biostudies-literature

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Parametric-rate inference for one-sided differentiable parameters.

Luedtke Alexander R AR van der Laan Mark J MJ

Journal of the American Statistical Association 20170228 522

Suppose one has a collection of parameters indexed by a (possibly infinite dimensional) set. Given data generated from some distribution, the objective is to estimate the maximal parameter in this collection evaluated at the distribution that generated the data. This estimation problem is typically non-regular when the maximizing parameter is non-unique, and as a result standard asymptotic techniques generally fail in this case. We present a technique for developing parametric-rate confidence in ...[more]

PMID: 30078921

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Project description:PurposeThe purpose of this tutorial is to provide visual scientists with various approaches for comparing two or more groups of data using parametric statistical tests, which require that the distribution of data within each group is normal (Gaussian). Non-parametric tests are used for inference when the sample data are not normally distributed or the sample is too small to assess its true distribution.MethodsMethods are reviewed using retinal thickness, as measured by optical coherence tomography (OCT), as an example for comparing two or more group means. The following parametric statistical approaches are presented for different situations: two-sample t-test, Analysis of Variance (ANOVA), paired t-test, and the analysis of repeated measures data using a linear mixed-effects model approach.ResultsAnalyzing differences between means using various approaches is demonstrated, and follow-up procedures to analyze pairwise differences between means when there are more than two comparison groups are discussed. The assumption of equal variance between groups and methods to test for equal variances are examined. Examples of repeated measures analysis for right and left eyes on subjects, across spatial segments within the same eye (e.g. quadrants of each retina), and over time are given.ConclusionsThis tutorial outlines parametric inference tests for comparing means of two or more groups and discusses how to interpret the output from statistical software packages. Critical assumptions made by the tests and ways of checking these assumptions are discussed. Efficient study designs increase the likelihood of detecting differences between groups if such differences exist. Situations commonly encountered by vision scientists involve repeated measures from the same subject over time, measurements on both right and left eyes from the same subject, and measurements from different locations within the same eye. Repeated measurements are usually correlated, and the statistical analysis needs to account for the correlation. Doing this the right way helps to ensure rigor so that the results can be repeated and validated.

Dataset Information

Parametric-rate inference for one-sided differentiable parameters.

Publications

Parametric-rate inference for one-sided differentiable parameters.

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