We introduce the 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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {G}=\{ G(\varvec{\theta }): \varvec{\theta } \in \Theta \}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">G</mi> <mo>=</mo> <mo stretchy="false">{</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mrow> <mi mathvariant="bold-italic">θ</mi> </mrow> <mo stretchy="false">)</mo> <mo>:</mo> <mrow> <mi mathvariant="bold-italic">θ</mi> </mrow> <mo>∈</mo> <mi mathvariant="normal">Θ</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>, we consider the testing problem <Equation ID="Equ41"> <EquationSource Format="TEX">$ H_0: \Vert F - G(\varvec{\theta }_F) \Vert _p \ge \epsilon \quad \text {vs} \quad H_1: \Vert F - G(\varvec{\theta }_F) \Vert _p < \epsilon , $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi>H</mi> <mn>0</mn> </msub> <mrow> <mo>:</mo> <mo stretchy="false">‖</mo> <mi>F</mi> <mo>-</mo> <mi>G</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mrow> <mi mathvariant="bold-italic">θ</mi> </mrow> <mi>F</mi> </msub> <mo stretchy="false">)</mo> </mrow> <msub> <mrow> <mo stretchy="false">‖</mo> </mrow> <mi>p</mi> </msub> <mo>≥</mo> <mi>ϵ</mi> <mspace width="1em" /> <mtext>vs</mtext> <mspace width="1em" /> <msub> <mi>H</mi> <mn>1</mn> </msub> <mo>:</mo> <msub> <mrow> <mo stretchy="false">‖</mo> <mi>F</mi> <mo>-</mo> <mi>G</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mrow> <mi mathvariant="bold-italic">θ</mi> </mrow> <mi>F</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">‖</mo> </mrow> <mi>p</mi> </msub> <mo><</mo> <mi>ϵ</mi> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\epsilon >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϵ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> is a margin of error and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$G(\varvec{\theta }_F)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo stretchy="false">(</mo> <msub> <mrow> <mi mathvariant="bold-italic">θ</mi> </mrow> <mi>F</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> denotes a representative of F within the parametric class. The approximate model is determined via an M-estimator of the parameters. The methodology also quantifies the percentage improvement of the proposed model relative to a non-informative (constant) benchmark. The test statistic is the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\textrm{L}^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mtext>L</mtext> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-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

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2025年10月31日

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Amparo Baíllo,
Javier Cárcamo

摘要

We introduce the 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 $\mathcal {G}=\{ G(\varvec{\theta }): \varvec{\theta } \in \Theta \}$ $G = {G (θ) : θ \in Θ}$ , we consider the testing problem $ H_0: \Vert F - G(\varvec{\theta }_F) \Vert _p \ge \epsilon \quad \text {vs} \quad H_1: \Vert F - G(\varvec{\theta }_F) \Vert _p < \epsilon , $ $H_{0} : ‖ F - G (θ_{F}) ‖_{p} \geq ϵ vs H_{1} : {‖ F - G (θ_{F}) ‖}_{p} < ϵ,$ where $\epsilon >0$ $ϵ > 0$ is a margin of error and $G(\varvec{\theta }_F)$ $G (θ_{F})$ denotes a representative of F within the parametric class. The approximate model is determined via an M-estimator of the parameters. The methodology also quantifies the percentage improvement of the proposed model relative to a non-informative (constant) benchmark. The test statistic is the $\textrm{L}^p$ $L^{p}$ -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.