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Bootstrap tests for almost goodness-of-fit

Amparo Baíllo, Javier Cárcamo

TL;DR

This paper develops Almost Goodness-of-Fit (AGoF) tests to determine whether a parametric model approximates the true distribution within a pre-specified margin $\epsilon$, using the $L^p$ distance between the empirical distribution and a model representative $G(\boldsymbol{\theta}_F)$. It builds two bootstrap-consistent procedures to approximate the critical region for the test, grounded in a rigorous asymptotic theory via Hadamard differentiability and empirical process convergence in $L^p$. The authors derive the limit distributions, provide practical implementation steps, and demonstrate the method through simulations and real-data applications (Haiti IgG serosurvey and carbon-fiber failure stress), including a model-improvement metric $G(F,\mathcal{G})$ that quantifies the relative gain over a non-informative benchmark. The framework enables flexible, margin-based model validation and robust comparison across competing parametric families, with clear guidance on choosing the margin and interpreting results for equivalence-type hypotheses.

Abstract

We introduce the \textit{almost goodness-of-fit} test, a procedure to assess whether a (parametric) model provides a good representation of the probability distribution generating the observed sample. Specifically, given a distribution function $F$ and a parametric family $\mathcal{G}=\{ G(\boldsymbolθ) : \boldsymbolθ \in Θ\}$, we consider the testing problem \[ H_0: \| F - G(\boldsymbolθ_F) \|_p \geq ε\quad \text{vs} \quad H_1: \| F - G(\boldsymbolθ_F) \|_p < ε, \] where $ε>0$ is a margin of error and $G(\boldsymbolθ_F)$ denotes a representative of $F$ within the parametric class. The approximate model is determined via an M-estimator of the parameters. %The objective is the approximate validation of a distribution or an entire parametric family up to a pre-specified threshold value. The methodology also quantifies the percentage improvement of the proposed model relative to a non-informative (constant) benchmark. The test statistic is the $\mathrm{L}^p$-distance between the empirical distribution function and that of the estimated model. We present two consistent, easy-to-implement, and flexible bootstrap schemes to carry out the test. The performance of the proposal is illustrated through simulation studies and analysis and real-data applications.

Bootstrap tests for almost goodness-of-fit

TL;DR

This paper develops Almost Goodness-of-Fit (AGoF) tests to determine whether a parametric model approximates the true distribution within a pre-specified margin , using the distance between the empirical distribution and a model representative . It builds two bootstrap-consistent procedures to approximate the critical region for the test, grounded in a rigorous asymptotic theory via Hadamard differentiability and empirical process convergence in . The authors derive the limit distributions, provide practical implementation steps, and demonstrate the method through simulations and real-data applications (Haiti IgG serosurvey and carbon-fiber failure stress), including a model-improvement metric that quantifies the relative gain over a non-informative benchmark. The framework enables flexible, margin-based model validation and robust comparison across competing parametric families, with clear guidance on choosing the margin and interpreting results for equivalence-type hypotheses.

Abstract

We introduce the \textit{almost goodness-of-fit} test, a procedure to assess whether a (parametric) model provides a good representation of the probability distribution generating the observed sample. Specifically, given a distribution function and a parametric family , we consider the testing problem where is a margin of error and denotes a representative of within the parametric class. The approximate model is determined via an M-estimator of the parameters. %The objective is the approximate validation of a distribution or an entire parametric family up to a pre-specified threshold value. The methodology also quantifies the percentage improvement of the proposed model relative to a non-informative (constant) benchmark. The test statistic is the -distance between the empirical distribution function and that of the estimated model. We present two consistent, easy-to-implement, and flexible bootstrap schemes to carry out the test. The performance of the proposal is illustrated through simulation studies and analysis and real-data applications.

Paper Structure

This paper contains 15 sections, 6 theorems, 71 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction and motivation
  2. Almost Goodness of Fit Tests
  3. The choice of the proximity measure
  4. The model framework
  5. The AGoF test
  6. The margin of error and a measure of AGoF
  7. Processes with estimated parameters
  8. Asymptotic behaviour in $\mathrm{L}^p$
  9. Bootstrap consistency
  10. Practical implementation
  11. A simulation study
  12. Application to two real data sets
  13. IgG antibodies in Haiti serosurvey
  14. Failure stress of carbon fibers
  15. Discussion

Key Result

Proposition 1

Let $\epsilon>0$ be fixed. For the testing problem AGoFspecific, the rejection region in c_alpha fulfills the following properties:

Figures (4)

  • Figure 1: Power function for (a) the Weibull(2,1) and the exponential model; (b) a normal mixture and the normal model and (c) a negative binomial and a Poisson model. The vertical red line is located at $\|F-G(\boldsymbol\theta_F)\|_p$.
  • Figure 2: Power function for (a) the Kumaraswamy(2,2) and the beta model; (b) the Student $t_4$ and the normal model; (c) the lognormal(0.5,0.5) and the gamma model. The vertical red line is located at $\|F-G(\boldsymbol\theta_F)\|_p$.
  • Figure 3: Histogram of log(MFI-bg) for antigen Bm33, normal fit and 2-component normal mixture fit.
  • Figure 4: For antigen Bm33, values of $\epsilon_k^*(0.05)$ when (a) $p=1$ and (b) $p=2$. Black points are the empirical $\mathrm{L}^p$-distances.

Theorems & Definitions (7)

  • Proposition 1
  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Theorem 3
  • Corollary 2
  • proof