Goodness-of-fit tests for copulas: A review and a power study

Many proposals have been made recently for goodness-of-fit testing of copula models. After reviewing them briefly, the authors concentrate on “blanket tests”, i.e., those whose implementation requires neither an arbitrary categorization of the data nor any strategic choice of smoothing parameter, weight function, kernel, window, etc. The authors present a critical review of these procedures and suggest new ones. They describe and interpret the results of a large Monte Carlo experiment designed to assess the effect of the sample size and the strength of dependence on the level and power of the blanket tests for various combinations of copula models under the null hypothesis and the alternative. To circumvent problems in the determination of the limiting distribution of the test statistics under composite null hypotheses, they recommend the use of a double parametric bootstrap procedure, whose implementation is detailed. They conclude with a number of practical recommendations.     

An extension of the Wang transform derived from Bühlmann’s economic premium principle for insurance risk

It is well known that the Wang transform [Wang, S.S., 2002. A universal framework for pricing financial and insurance risks. Astin Bull. 32, 213–234] for the pricing of financial and insurance risks is derived from Bühlmann’s economic premium principle [Bühlmann, H., 1980. An economic premium principle. Astin Bull. 11, 52–60]. The transform is extended to the multivariate setting by [Kijima M., 2006. A multivariate extension of equilibrium pricing transforms: The multivariate Esscher and Wang transforms for pricing financial and insurance risks, Astin Bull. 36, 269–283]. This paper further extends the results to derive a class of probability transforms that are consistent with Bühlmann’s pricing formula. The class of transforms is extended to the multivariate setting by using a Gaussian copula, while the multiperiod extension is also possible within the equilibrium pricing framework.     

Shape preserving approximation on the real line with exponential weights

The well-known “Bernstein’s weighted problem” deals with the possibility of weighted approximation on the whole real line. In this paper, we show the possibility of k-monotone approximation on the real line with Freud’s weight .     

Third order accurate variable mesh discretization and application of TAGE iterative method for the non-linear two-point boundary value problems with homogeneous functions in integral form

In this paper, we discuss the application of two-parameter alternating group explicit (TAGE) iterative method to an efficient third order variable mesh method for the solution of non-linear differential equations with integral homogeneous functions subject to natural boundary conditions. The proposed method is applicable only when the internal grid points of the solution space are odd in number. The proposed iterative method is also applicable to the integro-differential equations with singular coefficients. Comparative numerical results are given to demonstrate the usefulness of the proposed method.     

Short proofs of summation and transformation formulas for basic hypergeometric series

We show that several terminating summation and transformation formulas for basic hypergeometric series can be proved in a straightforward way. Along the same line, new finite forms of Jacobi's triple product identity and Watson's quintuple product identity are also proved.