[Paper Review] A test of goodness-of-fit for the copula densities

Ghislaine Gayraud, Karine Tribouley|ArXiv.org|Feb 27, 2009

Statistical Methods and Inference21 references3 citations

TL;DR

This paper proposes an adaptive goodness-of-fit test for copula densities using a smoothness-free test statistic that achieves the minimax rate of testing under $L_2$-norm separation, even when the true copula density belongs to a Besov ball. The method adapts to unknown smoothness and incurs only a $\log\log n$-term loss in the minimax rate, with theoretical guarantees for both lower and upper bounds, validated via simulations and real data.

ABSTRACT

We consider the problem of testing hypotheses on the copula density from $n$ bi-dimensional observations. We wish to test the null hypothesis characterized by a parametric class against a composite nonparametric alternative. Each density under the alternative is separated in the $L_2$-norm from any density lying in the null hypothesis. The copula densities under consideration are supposed to belong to a range of Besov balls. According to the minimax approach, the testing problem is solved in an adaptive framework: it leads to a $\log\log$ term loss in the minimax rate of testing in comparison with the non-adaptive case. A smoothness-free test statistic that achieves the minimax rate is proposed. The lower bound is also proved. Besides, the empirical performance of the test procedure is demonstrated with both simulated and real data.

Motivation & Objective

• To develop a minimax-optimal goodness-of-fit test for copula densities under unknown smoothness.
• To address the challenge of testing a parametric null hypothesis against a nonparametric composite alternative in the presence of unknown regularity.
• To establish a minimax rate of testing that accounts for the worst-case separation distance in $L_2$-norm between the null and alternative densities.
• To propose a test statistic that adapts to unknown smoothness without prior knowledge of the Besov ball parameters.
• To validate the theoretical performance through empirical studies on simulated and real data.

Proposed method

• Formulates the testing problem in a minimax framework, defining the alternative hypothesis as all densities separated from the null by at least $v_n$ in $L_2$-norm.
• Uses wavelet-based nonparametric estimation of the copula density to construct a test statistic that is smoothness-free and adaptive.
• Applies the minimax theory of testing (Ingster, 1982) to derive the optimal separation rate $v_n$ for the alternative hypothesis.
• Establishes a lower bound on the minimax rate using a testing risk criterion, proving that no test can achieve better separation than $v_n$.
• Derives an upper bound by constructing a test statistic based on wavelet coefficients of the empirical copula density, ensuring it achieves the minimax rate up to a $\log\log n$ factor.
• Employs concentration inequalities and moment bounds on wavelet projections to control the variance and bias of the test statistic under both null and alternative hypotheses.

Experimental results

Research questions

• RQ1 What is the optimal rate of testing for goodness-of-fit of copula densities when the true density belongs to a Besov ball with unknown smoothness?
• RQ2 Can a test statistic be constructed that adapts to unknown smoothness while achieving the minimax rate of testing?
• RQ3 What is the minimal separation distance $v_n$ in $L_2$-norm between the null and alternative hypotheses for which testing remains feasible?
• RQ4 How does the presence of unknown smoothness affect the minimax rate, and can the $\log\log n$-term loss be avoided in adaptive testing?
• RQ5 How does the proposed test perform empirically in finite samples compared to existing methods?

Key findings

• The minimax rate of testing for copula densities in the $L_2$-norm is characterized, with a $\log\log n$-term loss in the adaptive case compared to the non-adaptive setting.
• A smoothness-free test statistic based on wavelet coefficients of the empirical copula density achieves the minimax rate up to the $\log\log n$ factor.
• The lower bound for the minimax rate is proven using a testing risk criterion, showing that no test can achieve a faster separation rate than the derived $v_n$.
• The upper bound is established by constructing a test that controls both Type I and Type II errors uniformly over the alternative class.
• Empirical results on simulated and real data confirm the robustness and good performance of the test in finite samples.
• The method is valid under the assumption that the copula density lies in a Besov ball, allowing for a wide range of smoothness levels.