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Functional delta residuals and applications to simultaneous confidence bands of moment based statistics

Fabian J. E. Telschow, Samuel Davenport, Armin Schwartzman

Abstract

Given a functional central limit (fCLT) for an estimator and a parameter transformation, we construct random processes, called functional delta residuals, which asymptotically have the same covariance structure as the limit process of the functional delta method. An explicit construction of these residuals for transformations of moment-based estimators and a multiplier bootstrap fCLT for the resulting functional delta residuals are proven. The latter is used to consistently estimate the quantiles of the maximum of the limit process of the functional delta method in order to construct asymptotically valid simultaneous confidence bands for the transformed functional parameters. Performance of the coverage rate of the developed construction, applied to functional versions of Cohen's d, skewness and kurtosis, is illustrated in simulations and their application to test Gaussianity is discussed.

Functional delta residuals and applications to simultaneous confidence bands of moment based statistics

Abstract

Given a functional central limit (fCLT) for an estimator and a parameter transformation, we construct random processes, called functional delta residuals, which asymptotically have the same covariance structure as the limit process of the functional delta method. An explicit construction of these residuals for transformations of moment-based estimators and a multiplier bootstrap fCLT for the resulting functional delta residuals are proven. The latter is used to consistently estimate the quantiles of the maximum of the limit process of the functional delta method in order to construct asymptotically valid simultaneous confidence bands for the transformed functional parameters. Performance of the coverage rate of the developed construction, applied to functional versions of Cohen's d, skewness and kurtosis, is illustrated in simulations and their application to test Gaussianity is discussed.

Paper Structure

This paper contains 20 sections, 12 theorems, 72 equations, 23 figures.

Table of Contents

  1. Introduction
  2. Functional Delta Residuals
  3. Delta Residuals for Moment-based Statistics
  4. A Functional Central Limit Theorem for Moments
  5. Delta Residuals
  6. Examples of Delta Residuals Based on Moments
  7. A Multiplier Bootstrap Functional Limit Theorem
  8. Simultaneous Confidence Bands
  9. Construction of the SCBs
  10. Estimation of the quantile $q_\alpha$
  11. Testing Gaussianity using Skewness and Kurtosis
  12. Simulations of Coverage for Simultaneous Confidence Bands
  13. Functional Models of the Simulations
  14. Coverage Rates for Cohen's $d$
  15. Coverage Rates for Skewness and Kurtosis
  16. ...and 5 more sections

Key Result

Theorem \oldthetheorem

Let $N \in \mathbb{N}$ and $\hat{ \boldsymbol{\theta} }_N \in C(\, S, \mathbb{R}^P \,)$ be an estimator of a parameter $\boldsymbol{\theta} \in C(\, S, \mathbb{R}^P \,)$ such that as $N \rightarrow \infty$ weakly in $C(\, S, \mathbb{R}^P \,)$, where $\boldsymbol{G}$ denotes a zero-mean Gaussian process on $C(\, S, \mathbb{R}^P \,)$ with covariance function $\boldsymbol{\mathfrak{c}}$. Let $H \in

Figures (23)

  • Figure 1: Left: samples of a Gaussian process with square exponential covariance function with a linear combination of Gaussian densities as mean (bold black line). Middle: the untransformed residuals of this process. Right: the functional delta residuals of Cohen's $d$ of this process.
  • Figure 2: Left: the true asymptotic correlation function of Cohen's $d$ as given in Corollary \ref{['cor:SNRfCLT']} of the same process. Middle: correlation structure estimated from the sample correlation of a sample of untransformed residuals of size $100$. Right: correlation structure of Cohen's $d$ estimated from the sample correlation of a sample of delta residuals of size $100$.
  • Figure 3: Examples of ten sample paths of the considered models. The bold black line is the true population mean.
  • Figure 4: Examples of SCBs for moment-based statistics for two sample sizes. The bold black line is the true population parameter.
  • Figure 5: Simulations of coverage rates of SCBs for Model A. In the first two panels no bias correction is used in the construction of the SCBs, while in the third and fourth the coverage rate of bias-corrected SCBs is reported. The black dashed lines are 95% confidence intervals for the nominal level $0.95$.
  • ...and 18 more figures

Theorems & Definitions (31)

  • Theorem \oldthetheorem
  • proof
  • Remark 1
  • Definition 1
  • Remark 2
  • Remark 3
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 21 more