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A New Class of Asymptotically Distribution-Free Smooth Tests

Xiangyu Zhang, Sara Algeri

Abstract

This article demonstrates how recent developments in the theory of empirical processes allow us to construct a new family of asymptotically distribution-free smooth tests. Their distribution-free property is preserved even when the parameters are estimated, model selection is performed, and the sample size is only moderately large. A computationally efficient alternative to the classical parametric bootstrap is also discussed.

A New Class of Asymptotically Distribution-Free Smooth Tests

Abstract

This article demonstrates how recent developments in the theory of empirical processes allow us to construct a new family of asymptotically distribution-free smooth tests. Their distribution-free property is preserved even when the parameters are estimated, model selection is performed, and the sample size is only moderately large. A computationally efficient alternative to the classical parametric bootstrap is also discussed.

Paper Structure

This paper contains 11 sections, 5 theorems, 73 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Smooth tests and the function-parametric empirical process
  3. Smooth tests via projected bootstrap
  4. Empirical illustrations of smooth tests via the projected bootstrap
  5. Distribution-free smooth tests via K2 transform
  6. On the construction of the K2 orthonormal basis
  7. Simulation Studies
  8. Case study: analyzing an X-ray spectrum from RT Cru
  9. Summary and discussion
  10. Acknowledgments
  11. Code Availability

Key Result

Proposition 1

If A1-A2 hold, then, under $H_0,$ where is a projection of $h_{\boldsymbol{\beta}}$.

Figures (4)

  • Figure 1: Comparing the simulated null distributions of the order selection test statistic in \ref{['eqn:ordersel']} (Left), and subset selection test statistic in \ref{['eqn:subsetsel']} (Right) via the parametric bootstrap (orange dotted lines), the projected bootstrap (grey dashed lines), and Monte Carlo (blue solid lines).
  • Figure 2: The histogram of the simulated dataset is shown together with the densities of the true data-generating model $Q$, the reference distribution $F_{\boldsymbol{\gamma}}$, and the hypothesized distributions $G_{\boldsymbol{\beta},1}, G_{\boldsymbol{\beta},2}, G_{\boldsymbol{\beta},3}$. The unknown parameters $\boldsymbol{\beta}$ and $\boldsymbol{\gamma}$ are estimated via maximum likelihood.
  • Figure 3: The simulated null distributions of the order selection statistics (left panels) and the subset selection test statistics (right panels), using basis functions obtained by composing the normalized shifted Legendre polynomials with the null CDFs (upper panels) and the K2 transform (lower panels), under $F_{\boldsymbol{\gamma}}$, $G_{\boldsymbol{\beta},1}$, $G_{\boldsymbol{\beta},2}$, and $G_{\boldsymbol{\beta},3}$.
  • Figure 4: Left: QQ plots of the simulated order selection test statistics in \ref{['eqn:ordersel']} under $G_{\boldsymbol{\beta}}$ and in \ref{['eqn:K2st2']} under $F_{\boldsymbol{\gamma}}$. Right: QQ plots of the simulated subset selection test statistics in \ref{['eqn:subsetsel']} under $G_{\boldsymbol{\beta}}$ and in \ref{['eqn:K2st2']} under $F_{\boldsymbol{\gamma}}$. For both statistics computed under $G_{\boldsymbol{\beta}}$, the functions $h_{j\boldsymbol{\beta}}$ are elements of the $K2$ basis costructed as in Proposition \ref{['prop:6']}.

Theorems & Definitions (9)

  • Proposition 1
  • proof
  • Proposition 2
  • Proposition 3: K22016
  • Proposition 4
  • Proposition 5
  • proof
  • proof
  • proof