$$D_s$$ D s -optimality in copula models

Abstract

Optimum experimental design theory has recently been extended for parameter estimation in copula models. The use of these models allows one to gain in flexibility by considering the model parameter set split into marginal and dependence parameters. However, this separation also leads to the natural issue of estimating only a subset of all model parameters. In this work, we treat this problem with the application of the -optimality to copula models. First, we provide an extension of the corresponding equivalence theory. Then, we analyze a wide range of flexible copula models to highlight the usefulness of -optimality in many possible scenarios. Finally, we discuss how the usage of the introduced design criterion also relates to the more general issue of copula selection and optimal design for model discrimination.Electronic supplementary materialThe online version of this article (doi:10.1007/s10260-016-0375-6) contains supplementary material, which is available to authorized users.

Copula‐based robust optimal block designs

“…35 The optimality of a block design ⋆ under the robust D A -criterion, regardless of how it was found, can be assessed via application of the following Kiefer-Wolfowitz-type equivalence theorem. The proof is similar to that for completely randomized experiments with multivariate response; see the work of Perrone et al 29 for the locally optimal design case. Theorem 1.…”

“…The pseudo‐Bayesian D A ‐optimality criterion was added to the R package docopulae version 0.4 (see the work of Rappold) with the functions wDsensitivity and wDefficiency, both relying on a prespecified quadrature scheme for evaluation of the integrals. In this paper, we have concentrated on finding designs to estimate the complete parameter vector but the implementation provides flexibility for checking for symmetry, model discrimination, etc, as investigated in the work of Perrone et al…”

“…The pseudo-Bayesian D A -optimality criterion was added to the R package docopulae version 0.4 (see the work of Rappold 35 ) with the functions wDsensitivity and wDefficiency, both relying on a prespecified quadrature scheme for evaluation of the integrals. In this paper, we have concentrated on finding designs to estimate the complete parameter vector but the implementation provides flexibility for checking for symmetry, model discrimination, etc, as investigated in the work of Perrone et al 29 Our examples are confined to the case k = 2. While there is no theoretical necessity for that, it is difficult to specify high-dimensional parametric copulas with a sufficient range of dependence, for details, see the excellent survey of Nikoloulopoulos.…”

“…Denman et al found D ‐optimal designs for a bivariate response ( k =2) that maximized , and Perrone and Müller developed a corresponding equivalence theorem. These methods were extended to the local D A ‐criterion and, as a special case, for the D s ‐criterion in the work of Perrone et al Other relevant uses of design of experiments in copula models are those by Deldossi et al and Durante and Perrone, but until now, all relied on the availability of a single “best guess” vector of parameter values. It is well recognized that, for many nonlinear models, optimal designs for particular values of the model parameters may be very inefficient under different values; see the work of Woods et al for the case of scalar response GLMs.…”

“…To overcome this dependence on assumed parameter values, here we adopt a pseudo‐Bayesian approach for constructing block designs. Adopting this approach provides a more robust approach to design than the localized methods provided by Perrone et al Furthermore, our primary interest is typically in s meaningful linear combination of the parameters. Such combinations can be defined as elements of the vector A T γ , where A T is an s ×( r + l ) matrix of rank s <( r + l ).…”