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Two new methods are suggested for estimating the dependence function of a bivariate extreme-value distribution. One is based on a multiplicative modi®cation of an earlier technique proposed by Pickands, and the other employs spline smoothing under constraints. Both produce estimators that satisfy all the conditions that de®ne a dependence function, including convexity and the restriction that its curve lie within a certain triangular region. The ®rst approach does not require selection of smoothing parameters; the second does, and for that purpose we suggest explicit tuning methods, one of them based on cross-validation.

Motivated by applications in high-dimensional settings, we suggest a test of the hypothesis H 0 that two sampled distributions are identical. It is assumed that two independent datasets are drawn from the respective populations, which may be very general. In particular, the distributions may be multivariate or infinite-dimensional, in the latter case representing, for example, the distributions of random functions from one Euclidean space to another. Our test uses a measure of distance between data. This measure should be symmetric but need not satisfy the triangle inequality, so it is not essential that it be a metric. The test is based on ranking the pooled dataset, with respect to the distance and relative to any fixed data value; and repeating this operation for each fixed datum. A permutation argument enables a critical point to be chosen such that the test has concisely known significance level, conditional on the set of all pairwise distances.

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This paper suggests using a mixture of parametric and non-parametric methods to construct prediction regions in bivariate extreme-value problems. The non-parametric part of the technique is used to estimate the dependence function, or copula, and the parametric part is employed to estimate the marginal distributions. A bootstrap calibration argument is suggested for reducing coverage error. This combined approach is compared with a more parametric one, relative to which it has the advantages of being more flexible and simpler to implement. It also enjoys these features relative to predictive likelihood methods. The paper shows how to construct both compact and semi-infinite bivariate prediction regions, and it treats the problem of predicting the value of one component conditional on the other. The methods are illustrated by application to Australian annual maximum temperature data.

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