Subsampling Inference with K Populations and a Non‐standard Behrens–Fisher Problem

Resume  Nous passons en revue la methodologie et le developpement historique des methodes de sous-echantillonnage, et explorons en detail leur utilisation dans le cadre des problemes de tests d'hypotheses, ou ces methodes ont eteetonnamment peu prises en consideration. Nous explorons, en particulier, le cas tres general d'un modele aKechantillons faisant intervenir un parametre de grande dimension. Nous mettons en evidence le role important que joue le centrage de la loi de sous-echantillonnage, et montrons que ce centrage augmente la puissance des tests consideres. Nous montrons egalement comment les methodes de sous-echantillonnage permettent de traiter une forme non standard du probleme de Behrens-Fisher dans laquelle les populations dont on desire comparer les moyennes ne different pas seulement par leurs variances (qui peuvent eventuellement etre infinies), mais peuvent presenter des lois completement heterogenes. Notre formulation est tres generale, et s'applique meme a des observations de nature fonctionnelle. Enfin, nous developpons une theorie pour les statistiques-U permettant d'etablir, dans le cadre tres general de Kechantillons, la validite asymptotique des intervalles de confiance et des tests fondes sur le sous-echantillonnage. Summary We revisit the methodology and historical development of subsampling, and then explore in detail its use in hypothesis testing, an area which has received surprisingly modest attention. In particular, the general set-up of a possibly high-dimensional parameter with data fromKpopulations is explored. The role of centring the subsampling distribution is highlighted, and it is shown that hypothesis testing with a data-centred subsampling distribution is more powerful. In addition we demonstrate subsampling’s ability to handle a non-standard Behrens–Fisher problem, i.e., a comparison of the means of two or more populations which may possess not only different and possibly infinite variances, but may also possess different distributions. However, our formulation is general, permitting even functional data and/or statistics. Finally, we provide theory forK-sampleU-statistics that helps establish the asymptotic validity of subsampling confidence intervals and tests in this very general setting.

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