×

Nonparametric two-sample tests for increasing convex order. (English) Zbl 1200.62046

Summary: Given two independent samples of non-negative random variables with unknown distribution functions \(F\) and \(G\), respectively, we introduce and discuss two tests for the hypothesis that \(F\) is less than or equal to \(G\) in increasing convex order. The test statistics are based on the empirical stop-loss transform, and critical values are obtained by a bootstrap procedure. It turns out that for the resampling a size switching is necessary. We show that the resulting tests are consistent against all alternatives and that they are asymptotically of the given size \(\alpha \). A specific feature of the problem is the behavior of the tests ‘inside’ the hypothesis, where \(F\neq G\). We also investigate and compare this aspect for the two tests.

MSC:

62G10 Nonparametric hypothesis testing
62G09 Nonparametric statistical resampling methods
62G20 Asymptotic properties of nonparametric inference

References:

[1] Beran, R. and Millar, P. (1986). Confidence sets for a multivariate distribution., Ann. Statist. 14 431-442. · Zbl 0599.62057 · doi:10.1214/aos/1176349931
[2] Billingsley, P. (1968)., Convergence of Probability Measures . New York: Wiley. · Zbl 0172.21201
[3] Conover, W. (1971)., Practical Nonparametric Statistics . New York: Wiley.
[4] Denuit, M., Dhaene, J., Goovaerts, M. and Kaas, R. (2005)., Actuarial Theory for Dependent Risks . Chichester: Wiley. · Zbl 1086.91035
[5] Kaas, R., van Heerwarden, A. and Goovaerts, M. (1994)., Ordering of Actuarial Risk. Caire Education Series 1 . Brussels: CAIRE.
[6] Liu, X. and Wang, J. (2003). Testing for increasing convex order in several populations., Ann. Inst. Statist. Math. 55 121-136. · Zbl 1052.62023
[7] Müller, A. and Stoyan, D. (2002)., Comparison Methods for Stochastic Models . New York: Wiley. · Zbl 0999.60002
[8] Rolski, T. Schmidli, H. Schmidt, V. and Teugels, J. (1999)., Stochastic Processes for Insurance and Finance . Chichester: Wiley. · Zbl 0940.60005
[9] Shorack, G. R. and Wellner, J. A. (1986)., Empirical Processes with Applications to Statistics . New York: Wiley. · Zbl 1170.62365
[10] Tsirelson, B. (1975). The density of the distribution function of the maximum of a Gaussian process., Theory. Probab. Appl. 20 847-856. · Zbl 0348.60050 · doi:10.1137/1120092
[11] van der Vaart, A. and Wellner, J. (1996)., Weak Convergence and Empirical Processes . New York: Springer. · Zbl 0862.60002
[12] Witting H. and Nölle, G. (1970)., Angewandte Mathematische Statistik . Stuttgart: Teubner. · Zbl 0212.20901
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.