The Mann–Whitney U-statistic for α-dependent sequences
Tóm tắt
We give the asymptotic behavior of the Mann–Whitney U-statistic for two independent stationary sequences. The result applies to a large class of short-range dependent sequences, including many nonmixing processes in the sense of Rosenblatt [17]. We also give some partial results in the long-range dependent case, and we investigate other related questions. Based on the theoretical results, we propose some simple corrections of the usual tests for stochastic domination; next we simulate different (nonmixing) stationary processes to see that the corrected tests perform well.
Từ khóa
#Statistical Theory and MethodsTài liệu tham khảo
R. C. Bradley, “Basic Properties of Strong Mixing Conditions”, in Dependence in Probability and Statistics. A Survey of Recent Results. Oberwolfach, 1985, Ed. by E. Eberlein and M. S. Taqqu (Birkhäuser, 1986), pp. 165–192.
S. Dédé, Théorèmes limites fonctionnels et estimation de la densitéspectrale pour des suites stationnaires, PhD thesis (Univ. Pierre etMarie Curie, 2009). https://tel. archives-ouvertes. fr/tel-00440850
J. Dedecker, “An Empirical Central Limit Theorem for Intermittent Maps”, Probab. Theory Rel. Fields 148, 177–195 (2010).
J. Dedecker, H. Dehling, and M. S. Taqqu, “Weak Convergence of the Empirical Process of Intermittent Maps in L2 under Long-Range Dependence”, Stochastics and Dynamics 15 (2015), 29 pages.
J. Dedecker, S. Gouëzel and F. Merlevède, “Some Almost Sure Results for Unbounded Functions of Intermittent Maps and Their Associated Markov Chains”, Ann. Inst. Henri PoincaréProbab. Statist. 46, 796–821 (2010).
J. Dedecker and F. Merlevède, “A Deviation Bound for α-Dependent Sequences with Applications to IntermittentMaps”, Stochastics and Dynamics 17 (2017), 27 pages.
J. Dedecker and C. Prieur, “NewDependenceCoefficients. Examples and Applications to Statistics”, Probab. Theory Rel. Fields 132, 203–236 (2005).
J. Dedecker and C. Prieur, “An Empirical Central Limit Theorem for Dependent Sequences”, Stochastic Process. Appl. 117, 121–142 (2007).
J. Dedecker and C. Prieur, “Some Unbounded Functions of Intermittent Maps for Which the Central Limit Theorem Holds”, ALEA Lat. Am. J. Probab. Math. Statist. 5, 29–45 (2009).
J. Dedecker and E. Rio, “On Mean Central Limit Theorems for Stationary Sequences”, Ann. Inst. Henri PoincaréProbab. Statist. 44, 693–726 (2008).
H. Dehling and R. Fried, “Asymptotic Distribution of Two-Sample Empirical U-Quantiles with Applications to Robust Tests for Shifts in Location”, J. Multivar. Anal. 105, 124–170 (2012).
H. Dehling, A. Rooch, and M. S. Taqqu, “Nonparametric Change-Point Tests for Long-Range Dependent Data”, Scand. J. Statist. 40, 153–173 (2013).
I. Dewan and B. L. S. Prakasa Rao, “Mann–Whitney Test for Associated Sequences”, Ann. Inst. Statist. Math. 55, 111–119 (2003).
S. Gouëzel, “Central Limit Theorems and Stable Laws for Intermittent Maps”, Probab. Theory Rel. Fields 128, 82–122 (2004).
C. Liverani, B. Saussol and S. Vaienti, “A Probabilistic Approach to Intermittency”, Ergodic Theory Dynam. Systems 19, 671–685 (1999).
E. Rio, Théorie asymptotique des processus aléatoires faiblement dépendants, in Mathématiques et Applications (Springer-Verlag, 2000), Vol. 31.
M. Rosenblatt, “A Central Limit Theorem and a Strong Mixing Condition”, Proc. Nat. Acad. Sci. USA 42, 43–47 (1956).
R. J. Serfling, “The Wilcoxon Two-Sample Statistic on Strongly Mixing Processes”, Ann. Math. Statist. 39, 1202–1209 (1968).