Bootstrap tests for almost goodness-of-fit
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arXiv
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| Format: | Preprint |
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2024
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| author | Baíllo, Amparo Cárcamo, Javier |
| author_facet | Baíllo, Amparo Cárcamo, Javier |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_20918 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bootstrap tests for almost goodness-of-fit Baíllo, Amparo Cárcamo, Javier Methodology Statistics Theory Applications 62G10, 62G20, 62F40 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. |
| title | Bootstrap tests for almost goodness-of-fit |
| topic | Methodology Statistics Theory Applications 62G10, 62G20, 62F40 |
| url | https://arxiv.org/abs/2410.20918 |