Ranking ranks: a ranking algorithm for bootstrapping from the empirical copula

Nonparametric copula models are based on observations whose distributions are generally unknown. Estimation of these copula models is based on pseudo-observations consisting of the ranked data. To determine distributional properties (e.g., the variance) of the models and their estimators, resampling methods such as bootstrapping are involved. These methods require drawing samples with replacement from the ranked data. The newly generated samples have to be reranked and existing ties have to be solved by mid-ranks. Since a large number of samples has to be generated in order to attain a suitable accuracy of the estimate, the speed of the algorithm for reranking the samples highly affects the overall computation time. However, commonly used ranking procedures are computationally expensive and their running time is of order O(n* log(n*) + n*). We discuss a faster, more feasible approach using the specific copula setting with a running time that is only of order O(n + n*), where n denotes the sample size and n* the size of the bootstrap sample. In a simulation study, the algorithm performs up to 9 times faster than Matlab’s tiedrank.m-procedure. Copyright Springer-Verlag 2013