[Paper Review] Some copula inference procedures adapted to the presence of ties

Ivan Kojadinovic|arXiv (Cornell University)|Sep 18, 2016

Financial Risk and Volatility ModelingEconomics, Econometrics and Finance3 citations

TL;DR

This paper proposes tie-adapted statistical inference procedures for copulas in the presence of tied data due to rounding or measurement precision, using modified test statistics and parametric bootstrap resampling. The key contribution is that these adapted tests maintain valid Type I error rates and provide reliable p-values even with ties, as demonstrated through Monte Carlo simulations and real-world insurance data analysis.

ABSTRACT

When modeling the distribution of a multivariate continuous random vector using the so-called \emph{copula approach}, it is not uncommon to have ties in the coordinate samples of the available data because of rounding or lack of measurement precision. Yet, the vast majority of existing inference procedures on the underlying copula were both theoretically derived and practically implemented under the assumption of no ties. Applying them nonetheless can lead to strongly biased results. Some of the existing statistical tests can however be adapted to provide meaningful results in the presence of ties. It is the case of some tests of exchangeability, radial symmetry, extreme-value dependence and goodness of fit. Detailed algorithms for computing approximate p-values for the modified tests are provided and their finite-sample behaviors are empirically investigated through extensive Monte Carlo experiments. An illustration on a real-world insurance data set concludes the work.

Motivation & Objective

• Address the problem of biased inference in copula modeling when data contain ties due to rounding or measurement limitations.
• Develop modified statistical tests for exchangeability, radial symmetry, extreme-value dependence, and goodness-of-fit that remain valid despite ties.
• Ensure practical usability by providing algorithms for approximate p-value computation using parametric bootstrap.
• Validate the performance of the adapted tests through extensive Monte Carlo simulations and real data application.
• Enable reproducibility by implementing all tests in the R package 'copula'.

Proposed method

• Modify existing test statistics (e.g., R_n,C, Q_n, T_n, S_n) to account for ties in the component samples of multivariate data.
• Use a parametric bootstrap approach with multiplier resampling to compute approximate p-values for the adapted test statistics.
• Apply a data discretization mechanism in simulations: binning uniform variates into k intervals based on power transformation (a_i = i^t / k^t) to induce controlled ties.
• Implement the bootstrap procedure with N = 10,000 replicates to estimate p-values for each test under the null hypothesis.
• Adapt the test statistics by replacing empirical ranks with average ranks in tied groups to preserve asymptotic properties under dependence.
• Use the R package 'copula' to implement and reproduce all numerical experiments, ensuring full reproducibility.

Experimental results

Research questions

• RQ1 How do ties in multivariate data affect the validity of standard copula inference tests?
• RQ2 Can existing tests for exchangeability, radial symmetry, extreme-value dependence, and goodness-of-fit be modified to remain valid under the presence of ties?
• RQ3 What is the finite-sample performance of the proposed tie-adapted tests compared to their non-adapted counterparts?
• RQ4 How robust are the adapted tests to varying levels of tie intensity and dependence structures?
• RQ5 Can the parametric bootstrap procedure be effectively adapted to compute reliable p-values when ties are present?

Key findings

• The non-adapted test for exchangeability on the LOSS/ALAE data yielded a p-value of 0.000, while the tie-adapted version gave a p-value of 0.049, indicating reduced Type I error inflation.
• Both the non-adapted and tie-adapted radial symmetry tests rejected the null hypothesis with p-values of 0.000, showing strong evidence against radial symmetry in the data.
• The test for extreme-value dependence showed p-values of 0.602 (non-adapted) and 0.87 (adapted), indicating no evidence against extreme-value dependence.
• For goodness-of-fit testing, the Gumbel–Hougaard copula was the only family not rejected at the 1% significance level, with p-values of 0.230 (non-adapted) and 0.221 (adapted).
• The adapted tests maintained consistent p-values across different estimation methods (Kendall’s tau inversion and maximum pseudo-likelihood), confirming robustness.
• Monte Carlo experiments showed that the adapted tests control Type I error rates under ties, whereas non-adapted tests become overly liberal.