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Likelihood inference for Archimedean copulas

Marius Hofert, Martin Mächler, Alexander J. McNeil

arXiv:1108.6032v1math.ST

TL;DR

The paper addresses the difficulty of obtaining tractable, efficiently evaluable generator derivatives for Archimedean copula likelihood inference. It derives explicit formulas and applies them to likelihood estimation, confidence intervals, and multi-parameter families, showing feasibility even in large dimensions and implementing the methods in nacopula.

  • Problem

    Generator derivatives needed for Archimedean copula densities are difficult to derive and evaluate reliably, particularly in large dimensions.

  • Method

    The paper derives explicit generator-derivative formulas, develops likelihood and confidence-interval procedures, extends them to multi-parameter families, and implements the methods in nacopula.

  • Results

    Maximum-likelihood estimation is feasible for Archimedean copulas in large dimensions and for multi-parameter families, with confidence intervals and simulation results reported.

  • Takeaways & Limitations

    Explicit derivatives make likelihood inference and related quantities accessible for one- and multi-parameter Archimedean copulas, including asymmetric extensions.

  • Takeaways & Limitations

    Computer-algebra-derived derivative formulas can be numerically problematic and too slow for large-scale simulations.

Abstract

from arXiv · show

Explicit functional forms for the generator derivatives of well-known one-parameter Archimedean copulas are derived. These derivatives are essential for likelihood inference as they appear in the copula density, conditional distribution functions, or the Kendall distribution function. They are also required for several asymmetric extensions of Archimedean copulas such as Khoudraji-transformed Archimedean copulas. Access to the generator derivatives makes maximum-likelihood estimation for Archimedean copulas feasible in terms of both precision and run time, even in large dimensions. It is shown by simulation that the root mean squared error is decreasing in the dimension. This decrease is of the same order as the decrease in sample size. Furthermore, confidence intervals for the parameter vector are derived. Moreover, extensions to multi-parameter Archimedean families are given. All presented methods are implemented in the open-source R package nacopula and can thus easily be accessed and studied.

1 Introduction

The paper addresses the difficulty of deriving and evaluating Archimedean generator derivatives for likelihood inference, especially in large dimensions. It provides explicit formulas and applies them to maximum-likelihood estimation, asymmetric extensions, and multi-parameter families.

  • Motivation: Deriving tractable generator derivatives and evaluating them efficiently are the paper’s two central obstacles for likelihood-based estimation.These obstacles affect the copula density and become especially important in large dimensions.
  • Motivation: Computer algebra systems can fail in low dimensions and produce lengthy formulas that are numerically error-prone or slow to evaluate.This limits their suitability for large-scale simulation and practical likelihood calculations.
  • Contribution: Explicit generator-derivative formulas are derived for well-known Archimedean families in any dimension.The derivatives support densities, conditional distributions, Kendall distribution functions, and Khoudraji-transformed extensions.
  • Scope: The paper studies maximum-likelihood estimation in dimensions from about 10 to 100 and extends the framework to multi-parameter Archimedean families.The algorithms are implemented in the open-source R package nacopula.

2 Archimedean copulas

Archimedean copulas are represented through generator functions and their inverses, with density existence governed by derivative regularity. The paper focuses on completely monotone generators and several widely used one-parameter families, while also noting multi-parameter extensions and dependence summaries.

  • Definition: An Archimedean copula has the representation C(u)=ψ(ψ^-1(u_1)+···+ψ^-1(u_d)) for a suitable generator ψ.The generator maps [0,∞] to [0,1] and has an inverse on the relevant range.
  • Definition: A generator defines an Archimedean copula in dimension d when it satisfies the required d-monotonicity conditions.These conditions impose alternating-sign derivative inequalities through order d−2, plus monotonicity and convexity of the final derivative.
  • Density: A copula density exists when ψ^(d−1) exists and is absolutely continuous on (0,∞).The density therefore depends directly on generator derivative regularity.
  • Generator classes: Completely monotone generators satisfy alternating-sign derivative inequalities at every order and generate Archimedean copulas in all dimensions.They are Laplace–Stieltjes transforms of distributions on the positive real line.
  • Families: Ali-Mikhail-Haq, Clayton, Frank, Gumbel, and Joe are treated as widely used one-parameter Archimedean families.Outer-power transformations provide examples of extensions with more parameters.
  • Dependence summaries: Kendall’s tau and tail-dependence coefficients are summarized as model properties used in choosing and estimating Archimedean families.The paper uses Kendall’s tau in constructing initial likelihood-optimization intervals.

3 Maximum-likelihood estimation for Archimedean copulas

The likelihood procedure first converts observations into pseudo-observations using empirical margins, then numerically optimizes the copula likelihood. Standard likelihood theory supplies score, information, consistency, and asymptotic-normality results under regularity conditions.

  • Pseudo maximum-likelihood: The two-step procedure estimates marginal distributions first and then estimates the copula parameter vector.This is typically easier than estimating marginal and copula parameters simultaneously.
  • Pseudo maximum-likelihood: Pseudo maximum-likelihood estimation replaces each marginal distribution by its empirical distribution function and forms rank-based pseudo-observations.The scaling n/(n+1) keeps pseudo-observations inside the open unit hypercube for density evaluation.
  • Likelihood optimization: Maximum-likelihood estimation defines a likelihood and log-likelihood for copula observations and finds the estimator by numerical optimization.In practice, the observations are usually the pseudo-observations.
  • Likelihood theory: The score function and Fisher information provide derivative-based quantities for likelihood inference.The paper notes that the score and information require derivatives whose tractability can differ substantially.
  • Likelihood theory: Under regularity conditions, maximum-likelihood estimators are strongly consistent and asymptotically normal.The asymptotic-normality statement uses the p-dimensional identity matrix in its formulation.

Theorem 3.1

Theorem 3.1 states the two principal asymptotic properties of maximum-likelihood estimators under regularity conditions: consistency and asymptotic normality.

  • Consistency: Maximum-likelihood estimators are strongly consistent, converging almost surely to the true parameter vector as n→∞.The theorem states convergence in probability and almost sure convergence.
  • Asymptotic normality: Maximum-likelihood estimators are asymptotically normal under the theorem’s regularity conditions.The result is expressed using the p-dimensional identity matrix.

3.3 Generator derivatives and copula density

Efficient likelihood inference requires tractable generator derivatives and numerical evaluation, but computer algebra systems can fail for high-order derivatives or produce unstable, slow formulas. The section introduces explicit derivative formulas for the Archimedean families in Table 1.

  • Maximum-likelihood estimation requires efficient evaluation of the copula log-density through tractable generator derivatives.Both algebraic tractability and numerical precision and run time are required.
  • Computer algebra systems may fail to obtain higher-order derivatives and may return formulas that are unstable or slow to evaluate.The reported problems include the 50th derivative of a Gumbel generator.
  • Mathematica was aborted after ten minutes while computing the 50th Gumbel derivative at θ = 1.25 and t = 15.Maple returned values including 10 628 and -29 800 without warning.
  • Such computational failures can compromise numerical results and make large-scale simulation studies unsuitable when based on computer algebra outputs.The paper specifically mentions goodness-of-fit tests and parametric bootstrap simulations.
  • Theorem 3.2 derives explicit formulas for generator derivatives of all Archimedean families listed in Table 1.

Theorem 3.2

Theorem 3.2 derives explicit generator-derivative representations for the well-known Ali-Mikhail-Haq, Clayton, Frank, Gumbel, and Joe Archimedean families. The derivations use family-specific algebraic transformations and series identities, including polylogarithms and Stirling numbers.

  • The Ali-Mikhail-Haq derivation uses the polylogarithm representation and its defining series.
  • The Clayton result follows directly by taking derivatives of its generator expression.
  • The Frank and Joe cases are obtained through analogous family-specific transformations and algebraic steps.
  • The Gumbel derivation rewrites the generator using the exponential and applies Stirling-number identities.
  • Theorem 3.2 provides generator-derivative formulas for the Ali-Mikhail-Haq, Clayton, Frank, Gumbel, and Joe families.The representations cover the five well-known Archimedean families considered in the theorem.

Corollary 3.3

Corollary 3.3 converts the generator-derivative representations into density formulas for the Ali-Mikhail-Haq, Clayton, Frank, Gumbel, and Joe Archimedean families. Its proof follows from Formula (2) and Theorem 3.2.

  • Corollary 3.3 gives density representations for the Ali-Mikhail-Haq, Clayton, Frank, Gumbel, and Joe families.
  • The proof is obtained from Formula (2) together with the generator-derivative results in Theorem 3.2.
  • The corollary is presented as an important consequence of Theorem 3.2 for computing Archimedean copula densities.

Remark 3.4

Remark 3.4 explains why explicit generator derivatives matter beyond the five direct density formulas: they support densities, conditional distributions, Kendall distribution functions, and asymmetric extensions. The formulas can also replace slower Monte Carlo derivative approximations.

  • Explicit derivative formulas allow densities, conditional distribution functions, and Kendall distribution functions to be computed for the corresponding Archimedean families.The Kendall distribution function is relevant, for example, to goodness-of-fit testing.
  • Theorem 3.2 provides explicit formulas where earlier work provided recursive forms, enabling direct computation of the corresponding densities.
  • The formulas make density evaluation available even in large dimensions, including the Gumbel extreme-value family.Densities for extreme-value copulas are generally rarely available in explicit form.
  • Generator derivatives support density formulas for asymmetric Khoudraji-transformed Archimedean copulas.The transformed-copula density is given once the generator derivatives are available.
  • Simple generator transformations may extend the theorem’s results to other Archimedean families, including outer power transformations.
  • Monte Carlo approximation can be accurate and straightforward to implement, but it is slower than using direct derivative formulas.

4 Sample size n vs dimension d

Across the considered Archimedean families, dependence levels, and sample sizes, RMSE decreases as dimension increases. Its decrease with dimension has the same order as its decrease with sample size.

  • Dimension effects: RMSE decreases with dimension for all considered Archimedean families, dependence levels, and sample sizes.The study covers AMH, Clayton, Frank, Gumbel, and Joe families; τ ∈ {0.25, 0.5, 0.75}; and n ∈ {20, 50, 100, 200}.
  • Dimension effects: The decrease in RMSE with dimension is of the same order as its decrease with sample size.The log-log analysis relates both effects to the mean squared error.
  • Interpretation: The dimension-dependent behavior contrasts with findings based on simulations involving only small dimensions.The authors note that numerical problems may be underestimated or mishandled in small dimensions, while larger dimensions are typically more numerically involved.

5 Constructing confidence intervals

The paper develops several asymptotic and likelihood-based approaches for confidence intervals and regions for Archimedean copula parameters. A simulation study finds comparable performance overall, while highlighting computational trade-offs among the methods.

  • Methods: Confidence regions and intervals are constructed using Fisher-information estimators, observed information, and likelihood-based methods.The likelihood-ratio approach can construct confidence regions without requiring derivatives.
  • Asymptotic intervals: Under regularity conditions, asymptotic confidence regions use chi-square quantiles, while one-parameter intervals use standard-normal quantiles.The corresponding constructions are given for parameter vectors and scalar parameters.
  • Fisher information: The expected information is difficult to obtain, motivating alternative estimators based on first- and second-order derivatives.The first-order estimator can be preferable when second-order derivatives are much more difficult to access.
  • Simulation study: The simulation study evaluates coverage across n ∈ {100, 400}, d ∈ {5, 20}, τ ∈ {0.25, 0.5, 0.75}, and N = 1000 replications.It compares I(θ̂_n), Î^(1)(θ̂_n), Î^(2)(θ̂_n), and the likelihood-based approach W.
  • Simulation study: All confidence-interval methods work comparably well overall, but their computational costs differ.Î^(1)(θ̂_n) is preferred to simulated expected information, while Î^(2)(θ̂_n) is typically difficult to evaluate because of complicated second-order derivatives.
  • Likelihood-based intervals: The likelihood-based approach W is typically simpler to evaluate, can yield asymmetric intervals, and supports intervals for Kendall’s tau or tail-dependence coefficients through re-parameterization.These quantities are otherwise often obtained with the Delta Method based on an approximate normal distribution.

6 Multi-parameter families

The paper extends likelihood-based inference to multi-parameter Archimedean families by deriving usable generator derivatives and constructing initial parameter intervals from Kendall’s tau. It demonstrates these ideas for outer power Clayton and GIG copulas, including explicit density calculations and sampling procedures.

  • Multi-parameter extensions: Multi-parameter extensions include outer power transformations and the Archimedean GIG family, both estimated by maximum likelihood.These families are available in the R package nacopula.
  • Initial intervals: Kendall’s tau provides an intuitive concordance scale for constructing initial likelihood-optimization intervals.For one-parameter families, it helps define initial intervals; for multi-parameter families, it reduces the search region before optimization.
  • Outer power Clayton: Outer power generator derivatives can be obtained from a composition-derivative formula, enabling explicit densities for the transformed copulas.The resulting transformed generator remains completely monotone for β ∈[1, ∞).
  • Outer power Clayton: For the two-parameter outer power Clayton copula, Kendall’s tau and both tail-dependence coefficients are available explicitly.Unlike the Clayton copula, this extension can exhibit upper tail dependence.
  • Initial intervals: Algorithm 6.1 constructs an initial rectangle for outer power Clayton parameters by exploiting that τ(θ, β) increases in both parameters.The rectangle is determined from three points using concordance offsets h− and h+ and an admissibility truncation ε.
  • The GIG family: GIG generator derivatives have a comparatively simple form, and its initial-interval algorithm accounts for Kendall’s tau decreasing in both ν and θ.The GIG family approaches a Clayton copula with parameter 1/ν as θ decreases to zero.

7 Conclusion

The paper concludes that explicit generator derivatives make likelihood inference feasible for Archimedean copulas, including multi-parameter families and dimensions as large as d = 100. Simulations show that MSE decreases with dimension at the same order as with sample size, while likelihood-based confidence intervals and initial optimization intervals are also developed.

  • GIG results: Under weak concordance, the GIG family requires a larger sample size n for the bias to be small.This finding is reported alongside the summary results for two-parameter GIG copulas.
  • Main conclusions: Explicit generator derivatives enable maximum-likelihood estimation for various Archimedean copulas, including multi-parameter families in dimensions up to d = 100.They also support explicit conditional distributions and Kendall distribution functions.
  • Main conclusions: MSE decreases with dimension d at the same order as with sample size n, summarized as MSE ∝1/(nd).The paper also constructs initial intervals for likelihood optimization and derives likelihood-based confidence intervals for parameter vectors.
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