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For given α (0\u003Cα\u003C1) andd>0 we define a certain class of stopping timesN=N(α,d, x) and takeI\n                  N,d\n                (x)=[m\n                  N\n                (x)−d, m\n                  N\n                (x)+d] as a 2d-width confidence interval form(x) at a given pointx. In this paper it is shown that the probability P{m(x)∈I\n                  N,d\n                (x)} converges to α asd tends to zero.",{"EN":103,"VI":104},"Asymptotic consistency of fixed-width sequential confidence intervals for a multiple regression function","Tính vững tiệm cận của các khoảng tin cậy tuần tự có độ rộng cố định cho hàm hồi quy bội",{"VOID":106},"Ahmad, I. A., and Lin, P. (1976). Nonparametric sequential estimation of a multiple regression function,Bull. Math. Statist.,17, 63–75.\nBillingsley, P. (1968).Convergence of Probability Measures, John Wiley and Sons, New York.\nChow, Y. S. and Robbins, H. (1965). On the asymptotic theory of fixed-width sequential confidence intervals for the mean,Ann. Math. Statist.,36, 457–462.\nDevroye, L. P., and Wagner, T. J. (1980). On theL t convergence of kernel estimators of regression functions with applications in discrimination, Zeit. Wahrscheinlichkeitsth.,51, 15–25.\nIsogai, E. (1981). Stopping rules for sequential density estimation,Bull. Math. Statist.,19, 53–67.\nIsogai, E. (1983). A class of nonparametric recursive estimators of a multiple regression function,Bull. Inform. Cybernetics,20, 33–44.\nLoève, M. (1963).Probability Theory, 3rd edition, D. Van Nostrand, Princeton.\nNadaraya, E. A. (1964). On estimating regression,Theor Prob. Appl.,9, 141–142.\nPetrov, V. V. (1975).Sums of Independent Random Variables, Springer-Verlag.\nPrakasa Rao, B. L. S. (1983).Nonparametric Functional Estimation, Academic Press.\nRichter, W. (1965). Limit theorems for sequeuce of random variables with sequences of random indices,Theor. Prob. Appl.,10, 74–84.\nSamanta, M. (1984). On sequential estimation of the regression function,Bull. Inform. Cybernetics,21, 19–27.\nStone, C. J. (1977). Consistent nonparametric regression,Ann. Statist.,5, 595–645.\nWatson, G. S. (1964). Smooth regression analysis,Sankhyā, A.,26, 359–372.",{"VOID":108},"10.1007\u002FBF02482501","PUBLICATION","VERIFIED","2025-01-20T07:38:29.701+00:00","Auto Verify",[114],"VI","http:\u002F\u002Flink.springer.com\u002F10.1007\u002FBF02482501",[117],{"id":118,"sortIndex":23,"researcher":22,"roles":119,"affiliations":121,"properties":130,"displayName":132,"givenName":22,"familyName":22},"bea02536-0518-4ae3-af31-7ba3d2639367",[120],"AUTHOR",[122],{"id":123,"sortIndex":23,"affiliation":124,"properties":22},"7251a3e3-d312-44a0-a2f0-217d5cfcdbae",{"id":123,"createTime":22,"updateTime":22,"relativeEntities":125,"slug":22,"properties":126,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":129,"statistic":22},[],{"title":127},{"VI":128},"Niigata University, Niigata, Japan",[],{"title":131},{"VI":132},"Eiichi Isogai","ARTICLE",{"url":115,"publisher":135,"properties":177},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":136,"slug":10,"properties":137,"entityType":20,"verifyStatus":21,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":23,"subjectFields":141,"manageAffiliations":146,"indexDatabases":157,"url":22,"thumbnailPath":22,"statistic":172,"gsStatistic":22,"type":88,"analyzePriority":22},[],{"issn":138,"title":139,"eissn":140},{"VOID":15},{"EN":17},{"VOID":13},[142],{"id":26,"createTime":22,"updateTime":22,"relativeEntities":143,"label":144,"description":145,"parentId":22,"standard":22,"scholarHubFieldId":22},[],{"EN":29},{},[147,152],{"id":33,"createTime":22,"updateTime":22,"relativeEntities":148,"slug":22,"properties":149,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":151,"statistic":22},[],{"title":150},{"EN":37},[39],{"id":41,"createTime":22,"updateTime":22,"relativeEntities":153,"slug":22,"properties":154,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":156,"statistic":22},[],{"title":155},{"EN":45},[39],[158,165],{"id":49,"indexDatabase":159,"url":62,"indexYears":22,"academicFieldIds":164,"indexDatabaseRanking":22},{"id":51,"createTime":22,"updateTime":22,"relativeEntities":160,"label":161,"description":162,"key":58,"publicationTags":163,"standard":22},[],{"EN":54,"VI":54},{"EN":56,"VI":57},[60,61],[64],{"id":66,"indexDatabase":166,"url":77,"indexYears":78,"academicFieldIds":171,"indexDatabaseRanking":81},{"id":68,"createTime":22,"updateTime":22,"relativeEntities":167,"label":168,"description":169,"key":74,"publicationTags":170,"standard":22},[],{"EN":71,"VI":71},{"EN":71,"VI":73},[76],[80],{"impactFactor":23,"impactFactorByYear":173,"i10Index":23,"i10IndexLast5Year":23,"totalPublication":84,"totalPublicationByYear":174,"totalCitation":23,"totalCitationByYear":175,"totalCitationPerPublication":23,"totalCitationPerPublicationByYear":176,"hindexLast5Year":23,"hindex":23},{},{"1979":84},{},{},{"pages":178,"volume":180},{"VOID":179},"69-83",{"VOID":181},"38","1986-12-01",1986,[81,60],false,{"id":187,"createTime":188,"updateTime":189,"relativeEntities":190,"slug":191,"properties":192,"entityType":109,"verifyStatus":110,"verifyTime":203,"verifyNote":112,"languages":22,"translateLanguages":22,"viewCount":23,"primaryUrl":204,"fullTextUrl":22,"authors":205,"publicationType":133,"publisherRelationship":221,"citationCount":22,"citationInfo":22,"publishDate":269,"publishYear":270,"citationAnalyzeStatus":271,"lastCitationAnalyze":272,"indexDatabases":273,"openAccess":22,"references":22,"isForceReanalyzing":185},"4c8e7f72-6b9d-4cc0-8170-ccff5da6d880","2024-02-13T14:09:34.511+00:00","2026-08-24T11:21:55.037+00:00",[],"A-note-on-uniform-asymptotic-normality-of-intermediate-order-statistics",{"abstract":193,"title":195,"gsPaper":197,"references":199,"doi":201},{"EN":194},"It is proved that under fairly general von Mises-type conditions on the underlying distribution, the intermediate order statistics, properly standardized, converge uniformly over all Borel sets to the standard normal distribution. This closes the gap between central order statistics and extremes, where uniform convergence under mild conditions is well-known.",{"EN":196},"A note on uniform asymptotic normality of intermediate order statistics",{"VOID":198},"[]",{"VOID":200},"Balkema, A. A. and de, Haan, L. (1972). On R. von Mises' condition for the domain of attraction of exp (−e -x), Ann. Math. Statist., 43, 1352–1354.\nBalkema, A. A. and de, Haan, L. (1978a). Limit distributions for order statistics I, Theory Probab. Appl. 23, 77–92.\nBalkema, A. A. and de, Haan, L. (1978b). Limit distributions for order statistics II, Theory Probab. Appl., 23, 341–358.\nChibisov, D. M. (1964). On limit distributions for order statistics, Theory Probab. Appl., 9, 142–148.\nCooil, B. (1985). Limiting multivariate distributions of intermediate order statistics, Ann. Probab., 13, 469–477.\nde Haan, L. (1975). On Regular Variation and Its Application to the Weak Convergence of Sample Extremes, 3rd ed., Mathematical Centre Tracts, Vol. 32, Amsterdam.\nde, Haan, L. and Resnick, S. I. (1982). Local limit theorems for sample extremes, Ann. Probab., 10, 396–413.\nFalk, M. (1985). Uniform Convergence of Extreme Order Statistics, Habilitationsschrift, University of Siegen.\nFalk, M. (1986). Rates of uniform convergence of extreme order statistics, Ann. Inst. Statist. Math., 38, 245–262.\nGalambos, J. (1987). The Asymptotic Theory of Extreme Order Statistics, 2nd ed., Krieger, Melbourne, Florida.\nGnedenko, B. (1943). Sur la distribution limite du terme maximum d'une série aléatoire, Ann. Math., 44, 423–453.\nHall, P. (1979). On the rate of convergence of normal extremes, J. Appl. Probab., 16, 433–439.\nIkeda, S. and Matsunawa, T. (1972). On the uniform asymptotic joint normality of sample quantiles, Ann. Inst. Statist. Math., 24, 33–52.\nIkeda, S. and Matsunawa, T. (1976). Uniform asymptotic distribution of extremes, Essays in Probability and Statistics, (eds. S., Ikeda et al.), 419–432, Shinko Tsusho, Tokyo.\nPickands, J.III (1967). Sample sequences of maxima, Ann. Math. Statist., 38, 1570–1574.\nReiss, R.-D. (1976). Asymptotic expansions for sample quantiles, Ann. Probab., 4, 249–258.\nReiss, R.-D. (1981). Uniform approximation to distributions of extreme order statistics, Adv. in Appl. Probab., 13, 533–547.\nSmirnov, N. V. (1952). Limit distributions for the terms of a variational series, Amer. Math. Soc. Transl., 67, 82–143.\nSmirnov, N. V. (1967). Some remarks on limit laws for order statistics, Theory Probab. Appl., 12, 337–339.\nSweeting, T. J. (1985). On domains of uniform local attraction in extreme value theory, Ann. Probab., 13, 196–205.\nvon Mises, R. (1936). La distribution de la plus grande de n valeurs, reprinted in Selected Papers II, Amer. Math. Soc., Providence, Rhode Island, 1954, 271–294.\nWeiss, L. (1969). The asymptotic joint distribution of an increasing number of sample quantiles, Ann. Inst. Statist. Math., 21, 257–263.\nWeiss, L. (1971). Asymptotic inference about a density function at an end of its range, Naval Res. Logist. Quart., 18, 111–114.",{"VOID":202},"10.1007\u002FBF00049107","2024-09-05T06:47:25.247+00:00","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002FBF00049107",[206],{"id":207,"sortIndex":23,"researcher":22,"roles":208,"affiliations":209,"properties":218,"displayName":220,"givenName":22,"familyName":22},"a1373b09-8bad-45f6-b17f-2eaefa158273",[120],[210],{"id":211,"sortIndex":23,"affiliation":212,"properties":22},"53345ab4-fc60-44f1-b9cb-c27e27fdfbbc",{"id":211,"createTime":22,"updateTime":22,"relativeEntities":213,"slug":22,"properties":214,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":217,"statistic":22},[],{"title":215},{"VI":216},"Department of Mathematics, University of Siegen, Siegen 21, West Germany",[],{"title":219},{"VI":220},"Michael Falk",{"url":204,"publisher":222,"properties":264},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":223,"slug":10,"properties":224,"entityType":20,"verifyStatus":21,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":23,"subjectFields":228,"manageAffiliations":233,"indexDatabases":244,"url":22,"thumbnailPath":22,"statistic":259,"gsStatistic":22,"type":88,"analyzePriority":22},[],{"issn":225,"title":226,"eissn":227},{"VOID":15},{"EN":17},{"VOID":13},[229],{"id":26,"createTime":22,"updateTime":22,"relativeEntities":230,"label":231,"description":232,"parentId":22,"standard":22,"scholarHubFieldId":22},[],{"EN":29},{},[234,239],{"id":33,"createTime":22,"updateTime":22,"relativeEntities":235,"slug":22,"properties":236,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":238,"statistic":22},[],{"title":237},{"EN":37},[39],{"id":41,"createTime":22,"updateTime":22,"relativeEntities":240,"slug":22,"properties":241,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":243,"statistic":22},[],{"title":242},{"EN":45},[39],[245,252],{"id":49,"indexDatabase":246,"url":62,"indexYears":22,"academicFieldIds":251,"indexDatabaseRanking":22},{"id":51,"createTime":22,"updateTime":22,"relativeEntities":247,"label":248,"description":249,"key":58,"publicationTags":250,"standard":22},[],{"EN":54,"VI":54},{"EN":56,"VI":57},[60,61],[64],{"id":66,"indexDatabase":253,"url":77,"indexYears":78,"academicFieldIds":258,"indexDatabaseRanking":81},{"id":68,"createTime":22,"updateTime":22,"relativeEntities":254,"label":255,"description":256,"key":74,"publicationTags":257,"standard":22},[],{"EN":71,"VI":71},{"EN":71,"VI":73},[76],[80],{"impactFactor":23,"impactFactorByYear":260,"i10Index":23,"i10IndexLast5Year":23,"totalPublication":84,"totalPublicationByYear":261,"totalCitation":23,"totalCitationByYear":262,"totalCitationPerPublication":23,"totalCitationPerPublicationByYear":263,"hindexLast5Year":23,"hindex":23},{},{"1979":84},{},{},{"pages":265,"volume":267},{"VOID":266},"19-29",{"VOID":268},"41","1989-03-01",1989,"ERROR_IN_GET_PLATFORM_ID","2026-08-24T11:21:55.036+00:00",[81,60],{"id":275,"createTime":276,"updateTime":277,"relativeEntities":278,"slug":279,"properties":280,"entityType":109,"verifyStatus":110,"verifyTime":292,"verifyNote":112,"languages":22,"translateLanguages":22,"viewCount":23,"primaryUrl":293,"fullTextUrl":22,"authors":294,"publicationType":133,"publisherRelationship":325,"citationCount":22,"citationInfo":22,"publishDate":368,"publishYear":369,"citationAnalyzeStatus":271,"lastCitationAnalyze":277,"indexDatabases":370,"openAccess":22,"references":22,"isForceReanalyzing":185},"2e62c3a5-c3e8-4044-8173-330dd9177215","2024-04-08T03:56:39.429+00:00","2026-08-19T18:29:45.485+00:00",[],"On-the-Simes-inequality-in-elliptical-models",{"abstract":281,"title":283,"gsPaper":285,"keywords":286,"references":288,"doi":290},{"EN":282},"We provide some necessary and some sufficient conditions for the validity of the inequality of Simes in models with elliptical dependencies. Necessary conditions are presented in terms of sufficient conditions for the reverse Simes inequality. One application of our main results concerns the problem of model misspecification, in particular the case that the assumption of Gaussianity of test statistics is violated. Since our sufficient conditions require non-negativity of correlation coefficients between test statistics, we also develop two exact tests for vectors of correlation coefficients and compare their powers in computer simulations.",{"EN":284},"On the Simes inequality in elliptical models",{"VOID":198},{"EN":287},"",{"VOID":289},"Aitchison, J. (1964). Confidence-region tests. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 26, 462–476.\nBenjamini, Y., Hochberg, Y. (1995). Controlling the false discovery rate: a practical and powerful approach to multiple testing. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 57(1), 289–300.\nBenjamini, Y., Yekutieli, D. (2001). The control of the false discovery rate in multiple testing under dependency. Annals of Statistics, 29(4), 1165–1188.\nBlock, H. W., Savits, T. H., Shaked, M. (1985). A concept of negative dependence using stochastic ordering. Statistics & Probability Letters, 3, 81–86.\nBlock, H. W., Savits, T. H., Wang, J., Sarkar, S. K. (2013). The multivariate-\\(t\\) distribution and the Simes inequality. Statistics & Probability Letters, 83(1), 227–232. doi:10.1016\u002Fj.spl.2012.08.013.\nDickhaus, T. (2014). Simultaneous Statistical Inference with Applications in the Life Sciences. Berlin: Springer.\nFinner, H., Strassburger, K. (2014). On the Simes test under dependence. Talk at the 60. Biometrisches Kolloquium. Bremen, 12 March 2014.\nGupta, A. K., Varga, T., Bodnar, T. (2013). Elliptically contoured models in statistics and portfolio theory (2nd ed.). New York: Springer. doi:10.1007\u002F978-1-4614-8154-6.\nHommel, G. (1988). A stagewise rejective multiple test procedure based on a modified Bonferroni test. Biometrika, 75(2), 383–386. doi:10.1093\u002Fbiomet\u002F75.2.383.\nHothorn, T., Bretz, F., Westfall, P. (2008). Simultaneous inference in general parametric models. Biometrical Journal, 50(3), 346–363.\nLäuter, J. (2013). Simes’ theorem is generally valid for dependent normally distributed variables. Invited talk at the international conference on simultaneous inference 2013, Hannover, 24 September 2013.\nMuirhead, R. (1982). Aspects of multivariate statistical theory. Wiley, New York.\nRüschendorf, L. (2009). On the distributional transform, Sklar’s theorem, and the empirical copula process. Journal of Statistical Planning and Inference, 139(11), 3921–3927. doi:10.1016\u002Fj.jspi.2009.05.030.\nSarkar, S. K. (1998). Some probability inequalities for ordered MTP\\(_2\\) random variables: a proof of the Simes conjecture. Annals of Statistics, 26(2), 494–504. doi:10.1214\u002Faos\u002F1028144846.\nSarkar, S. K. (2002). Some results on false discovery rate in stepwise multiple testing procedures. Annals of Statistics, 30(1), 239–257.\nSarkar, S. K. (2008). On the Simes inequality and its generalization. IMS Collections Beyond Parametrics in Interdisciplinary Research: Festschrift in Honor of Professor Pranab K. Sen, 1, 231–242.\nSimes, R. J. (1986). An improved Bonferroni procedure for multiple tests of significance. Biometrika, 73, 751–754.\nWestfall, P. H., Young, S. S. (1993). Resampling-based multiple testing: examples and methods for p-value adjustment. Wiley, New York: Wiley Series in Probability and Mathematical Statistics. 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Different types of canonical correlations are considered and their connection with connectedness and orthogonality are examined.",{"EN":381},"Canonical correlations in multi-way layout",{"VOID":383},"15177169369754542205",{"VOID":385},"10.1007\u002FBF02506481","2024-04-25T20:27:13.069+00:00","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002FBF02506481",[389,406],{"id":390,"sortIndex":23,"researcher":22,"roles":391,"affiliations":392,"properties":401,"displayName":403,"givenName":22,"familyName":22},"0abbda4e-c786-4811-a18f-74cab27099ee",[120],[393],{"id":394,"sortIndex":23,"affiliation":395,"properties":22},"4165e12f-6693-42d0-84e9-de036c8b2670",{"id":394,"createTime":22,"updateTime":22,"relativeEntities":396,"slug":22,"properties":397,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":400,"statistic":22},[],{"title":398},{"VI":399},"Department of Mathematics, National Technical University, Athens, Greece",[],{"title":402,"gsAuthor":404},{"VI":403},"Maria Adam",{"VOID":405},"YjBr3sIAAAAJ",{"id":407,"sortIndex":84,"researcher":22,"roles":408,"affiliations":409,"properties":416,"displayName":418,"givenName":22,"familyName":22},"73a4e242-3fd6-4851-b28e-78b9430f50b2",[120],[410],{"id":394,"sortIndex":23,"affiliation":411,"properties":22},{"id":394,"createTime":22,"updateTime":22,"relativeEntities":412,"slug":22,"properties":413,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":415,"statistic":22},[],{"title":414},{"VI":399},[],{"title":417},{"VI":418},"John Maroulas",{"url":387,"publisher":420,"properties":462},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":421,"slug":10,"properties":422,"entityType":20,"verifyStatus":21,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":23,"subjectFields":426,"manageAffiliations":431,"indexDatabases":442,"url":22,"thumbnailPath":22,"statistic":457,"gsStatistic":22,"type":88,"analyzePriority":22},[],{"issn":423,"title":424,"eissn":425},{"VOID":15},{"EN":17},{"VOID":13},[427],{"id":26,"createTime":22,"updateTime":22,"relativeEntities":428,"label":429,"description":430,"parentId":22,"standard":22,"scholarHubFieldId":22},[],{"EN":29},{},[432,437],{"id":33,"createTime":22,"updateTime":22,"relativeEntities":433,"slug":22,"properties":434,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":436,"statistic":22},[],{"title":435},{"EN":37},[39],{"id":41,"createTime":22,"updateTime":22,"relativeEntities":438,"slug":22,"properties":439,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":441,"statistic":22},[],{"title":440},{"EN":45},[39],[443,450],{"id":49,"indexDatabase":444,"url":62,"indexYears":22,"academicFieldIds":449,"indexDatabaseRanking":22},{"id":51,"createTime":22,"updateTime":22,"relativeEntities":445,"label":446,"description":447,"key":58,"publicationTags":448,"standard":22},[],{"EN":54,"VI":54},{"EN":56,"VI":57},[60,61],[64],{"id":66,"indexDatabase":451,"url":77,"indexYears":78,"academicFieldIds":456,"indexDatabaseRanking":81},{"id":68,"createTime":22,"updateTime":22,"relativeEntities":452,"label":453,"description":454,"key":74,"publicationTags":455,"standard":22},[],{"EN":71,"VI":71},{"EN":71,"VI":73},[76],[80],{"impactFactor":23,"impactFactorByYear":458,"i10Index":23,"i10IndexLast5Year":23,"totalPublication":84,"totalPublicationByYear":459,"totalCitation":23,"totalCitationByYear":460,"totalCitationPerPublication":23,"totalCitationPerPublicationByYear":461,"hindexLast5Year":23,"hindex":23},{},{"1979":84},{},{},{"pages":463,"volume":465},{"VOID":464},"655-666",{"VOID":466},"56",3,{"total":467,"publishYear":469,"statisticByYear":470},2004,{"2006":84,"2013":84,"2019":84},"2004-12-01","ERROR_IN_ANALYZE_CITATION","2026-08-18T22:41:32.437+00:00",[81,60],[476,482,488,491,494,497,500,503,506],{"id":477,"text":478,"url":479,"identifiers":480},"486010d5-ee67-4752-9158-10035bc6ac17","Baksalary, J. K., Puntanen, S. and Yanai, H. (1992). Canonical correlations associated with symmetric reflexive generalized inverses of the dispersion matrix,Linear Algebra and Its Applications,176, 61–74.","https:\u002F\u002Fwww.sciencedirect.com\u002Fscience\u002Farticle\u002Fpii\u002F0024379592902102",{"doi":481},"10.1016\u002F0024-3795(92)90210-2",{"id":483,"text":484,"url":485,"identifiers":486},"4c68646b-0035-4279-8000-0006b275d4fa","Bérubé, J., Hartwig, R. and Styan, G. P. H. (1993). On canonical correlations and the degrees of non-orthogonality in the three-way layout,Statistical Science and Data Analysis: Proceedings of the Third Pacific Area Statistical Conference (eds. K. Matusita, M. L. Puri and T. Hayakawa), 245–252, VSP.","https:\u002F\u002Flink.springer.com\u002F10.1007\u002Fs10440-022-00541-7",{"doi":487},"10.1007\u002Fs10440-022-00541-7",{"id":483,"text":489,"url":485,"identifiers":490},"Hotelling, H. (1936). Relations between two sets of variates,Biometrika 28, 321–377.",{"doi":487},{"id":483,"text":492,"url":485,"identifiers":493},"Khatri, C. G. (1976). A note on multiple and canonical correlations for a singular covariance matrix,Psychometrika,41, 465–470.",{"doi":487},{"id":22,"text":495,"url":22,"identifiers":496},"Lancaster, P. and Tismenetsky, M. (1984).The Theory of Matrices, Academic Press, Orlando.",{},{"id":22,"text":498,"url":22,"identifiers":499},"Rao, C. G. and Yanai, H. (1979). General definition of a projector, its decomposition and application to statistical problems,Journal of Statistical Planning Inference,3, 1–17.",{},{"id":22,"text":501,"url":22,"identifiers":502},"Styan, G. P. H. (1983). Schur complements and linear statistical models,Proceedings of the First Tampere Seminar Linear Models, 37–75, Department of Mathematical Sciences, University of Tampere.",{},{"id":483,"text":504,"url":485,"identifiers":505},"Styan, G. P. H. (1986). Canonical correlations in the three-way layout,Pacific Statistical Congress (eds. I. S. Francis, B. F. J. Manly and F. C. Lam), 433–438, North-Holland, Amsterdam.",{"doi":487},{"id":507,"text":508,"url":509,"identifiers":510},"307d605e-1105-4cfc-99a0-f199172cf067","Yanai, H. and Takane, Y. (1992). Canonical correlation analysis with linear constraints,Linear Algebra and Its Applications,176, 75–89.","https:\u002F\u002Fwww.sciencedirect.com\u002Fscience\u002Farticle\u002Fpii\u002F002437959290211R",{"doi":511},"10.1016\u002F0024-3795(92)90211-r",{"id":513,"createTime":514,"updateTime":515,"relativeEntities":516,"slug":517,"properties":518,"entityType":109,"verifyStatus":110,"verifyTime":527,"verifyNote":112,"languages":22,"translateLanguages":22,"viewCount":23,"primaryUrl":528,"fullTextUrl":22,"authors":529,"publicationType":133,"publisherRelationship":562,"citationCount":610,"citationInfo":611,"publishDate":615,"publishYear":612,"citationAnalyzeStatus":472,"lastCitationAnalyze":616,"indexDatabases":617,"openAccess":22,"references":618,"isForceReanalyzing":185},"01063ce8-7afa-4e82-800e-ec2f659d14ce","2024-01-09T16:53:53.776+00:00","2026-08-15T00:02:37.126+00:00",[],"Bootstrapping-a-Bayes-estimator-of-a-survival-function-with-censored-data",{"abstract":519,"title":521,"gsPaper":523,"doi":525},{"EN":520},"The large-sample frequentist property of a frequentist bootstrap for a posterior mean with respect to a Dirichlet prior of the survival function for a randomly censored data is given. The weak convergence of a bootstrap version of the Susarla-Van Ryzin estimator is established on the whole real line. An illustration of the technique and some Monte Carlo studies are also given.",{"EN":522},"Bootstrapping a Bayes estimator of a survival function with censored data",{"VOID":524},"[\"9997460660197083826\"]",{"VOID":526},"10.1007\u002FBF00773512","2024-04-29T04:45:35.616+00:00","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002FBF00773512",[530,547],{"id":531,"sortIndex":23,"researcher":22,"roles":532,"affiliations":533,"properties":542,"displayName":544,"givenName":22,"familyName":22},"0d7448c6-4c82-4427-82b3-eec8d9654f7e",[120],[534],{"id":535,"sortIndex":23,"affiliation":536,"properties":22},"a6b838b3-0cd3-4cad-a253-d497520e9b47",{"id":535,"createTime":22,"updateTime":22,"relativeEntities":537,"slug":22,"properties":538,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":541,"statistic":22},[],{"title":539},{"VI":540},"Statistics Center, Cornell University, Ithaca, U.S.A.",[],{"title":543,"gsAuthor":545},{"VI":544},"Martin T. 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G. (1986). Bootstrapping the Kaplan-Meier estimator,J. Amer. Statist. Assoc.,81, 1032–1038.",{"doi":487},{"id":623,"text":624,"url":625,"identifiers":626},"281aae91-5a9e-4657-9052-9df1fd3e98ea","Billingsley, P. (1968).Convergence of Probability Measures, Wiley, New York.","https:\u002F\u002Fwww.goodreads.com\u002Fbook\u002Fshow\u002F411432.Convergence_of_Probability_Measures",{"isbn":627,"isbn13":628},"0471197459","9780471197454",{"id":483,"text":630,"url":485,"identifiers":631},"Breslow, N. and Crowley, J. (1974). A large sample study of the life table and product-limit estimates under random censorship,Ann. Statist.,2, 437–453.",{"doi":487},{"id":483,"text":633,"url":485,"identifiers":634},"Efron, B. (1981). Censored data and the bootstrap,J. Amer. Statist. Assoc.,76, 312–319.",{"doi":487},{"id":483,"text":636,"url":485,"identifiers":637},"Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems,Ann. Statist.,1, 209–230.",{"doi":487},{"id":639,"text":640,"url":641,"identifiers":642},"d8eb7653-e902-4fa7-8d31-6317759efe7d","Ferguson, T. S., Phadia, E. G. and Tiwari, R. C. (1992). Bayesian nonparametric inference,Current Issues in Statistical Inference: Essays in Honor of D. Basu (eds. M. Ghosh and P. K. Pathak), IMS Lecture Note & Monograph Series, Vol. 17, 127–150.","https:\u002F\u002Fwww.sciencedirect.com\u002Fscience\u002Farticle\u002Fpii\u002FS0169716105250125",{"doi":643},"10.1016\u002Fs0169-7161(05)25012-5",{"id":483,"text":645,"url":485,"identifiers":646},"Gill, R. D. (1980). Censoring and Stochastic Integrals, Mathematical Centre Tract 124, Mathematical Centrum, Amsterdam.",{"doi":487},{"id":483,"text":648,"url":485,"identifiers":649},"Gill, R. D. (1983). Large sample behavior of the product limit estimator on the whole line,Ann. Statist.,11, 49–58.",{"doi":487},{"id":483,"text":651,"url":485,"identifiers":652},"Jacod, J. (1975). Multivariate point processes: predicable projection, Radon-Nykodym derivatives, representation of martingales,Z. Wahrsch. verw. Gebiete,31, 235–253.",{"doi":487},{"id":22,"text":654,"url":22,"identifiers":655},"Lipster, R. S. and Shiryayev, A. N. (1978).Statistics of Random Processes II: Applications, Springer, New York.",{},{"id":483,"text":657,"url":485,"identifiers":658},"Lo, A. Y. (1987). A large sample study of the Bayesian bootstrap,Ann. Statist.,15, 360–375.",{"doi":487},{"id":483,"text":660,"url":485,"identifiers":661},"Lo, A. Y. (1988). A Bayesian bootstrap for a finite population,Ann. Statist.,16, 1684–1685.",{"doi":487},{"id":483,"text":663,"url":485,"identifiers":664},"Lo, A. Y. (1993). A Bayesian bootstrap for censored data,Ann. Statist.,21, 100–123.",{"doi":487},{"id":666,"text":667,"url":668,"identifiers":669},"bb91a3a4-d766-4eff-abc7-058dd2de14c3","Rebolledo, R. (1980). Central limit theorems for local martingales,Z. Wahrsch. verw. Gebiete,51, 269–289.","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002FBF00587353",{"doi":670},"10.1007\u002FBF00587353",{"id":483,"text":672,"url":485,"identifiers":673},"Reid, N. (1981). Estimating the median survival time,Biometrika,68, 601–608.",{"doi":487},{"id":483,"text":675,"url":485,"identifiers":676},"Rubin, D. B. (1981). The Bayesian bootstrap,Ann. Statist.,9, 130–134.",{"doi":487},{"id":483,"text":678,"url":485,"identifiers":679},"Susarla, V. and Van Ryzin, J. (1976). Nonparametric Bayesian estimation of survival curves from incomplete data,J. Amer. Statist. Assoc.,71, 897–902.",{"doi":487},{"id":483,"text":681,"url":485,"identifiers":682},"Susarla, V. and Van Ryzin, J. (1978). Large sample theory for a Bayesian nonparametric survival curve estimator based on censored samples,Ann. Statist.,6, 755–768.",{"doi":487},{"id":483,"text":684,"url":485,"identifiers":685},"Weng, C. S. (1989). On a second order asymptotic property of the Bayesian bootstrap mean,Ann. Statist.,17, 705–710.",{"doi":487},{"id":483,"text":687,"url":485,"identifiers":688},"Ying, Z. (1989). A note on the asymptotic properties of the product limit estimator on the whole line,Statist. Probab. Lett.,7, 311–314.",{"doi":487},{"id":690,"createTime":691,"updateTime":692,"relativeEntities":693,"slug":694,"properties":695,"entityType":109,"verifyStatus":110,"verifyTime":706,"verifyNote":112,"languages":22,"translateLanguages":22,"viewCount":23,"primaryUrl":707,"fullTextUrl":22,"authors":708,"publicationType":133,"publisherRelationship":726,"citationCount":23,"citationInfo":774,"publishDate":777,"publishYear":775,"citationAnalyzeStatus":21,"lastCitationAnalyze":778,"indexDatabases":779,"openAccess":22,"references":22,"isForceReanalyzing":185},"a42b21f6-d343-4038-8106-d46bff9f9e76","2023-12-26T16:11:37.534+00:00","2026-07-28T06:15:04.444+00:00",[],"Measuring-the-baseline-sales-and-the-promotion-effect-for-incense-products-a-Bayesian-state-space-modeling-approach",{"abstract":696,"title":698,"gsPaper":700,"references":702,"doi":704},{"EN":697},"One of the most important research fields in marketing science is the analysis of time series data. This article develops a new method for modeling multivariate time series. The proposed method enables us to measure simultaneously the effectiveness of marketing activities, the baseline sales, and the effects of controllable\u002Funcontrollable business factors. The critical issue in the model construction process is the method for evaluating the usefulness of the predictive models. This problem is investigated from a statistical point of view, and use of the Bayesian predictive information criterion is considered. The proposed method is applied to sales data regarding incense products. The method successfully extracted useful information that may enable managers to plan their marketing strategies more effectively.",{"EN":699},"Measuring the baseline sales and the promotion effect for incense products: a Bayesian state-space modeling approach",{"VOID":701},"[\"659313778991138354\"]",{"VOID":703},"Abraham M.M., Lodish L.M. (1993) An implemented system for improving promotion productivity using store scanner data. Marketing Science 12: 248–269\nAndo, T. (2006a). Bayesian State space modeling approach for measuring the effectiveness of marketing activities and baseline sales from POS data. In Proceeding of IEEE International Conference on Data Mining (pp. 21–32).\nAndo T. (2006b) Bayesian inference for nonlinear and non-Gaussian stochastic volatility model with leverage effect. Journal of the Japan Statistical Society 36: 173–197\nAndo, T. (2006c). Bayesian credit rating analysis based on ordered probit regression model with functional predictor. In Proceeding of The Third IASTED International Conference on Financial Engineering and Applications (pp. 69–76).\nAndo T. (2007) Bayesian predictive information criterion for the evaluation of hierarchical Bayesian and empirical Bayes models. Biometrika 94: 443–458\nBarnard J., McCulloch R., Meng X. (2000) Modeling covariance matrices in terms of standard deviations and correlations, with application to shrinkage. Statistica Sinica 10: 1281–1311\nBass F.M. (1969) A simultaneous equation regression study of advertising and sales of cigarettes. Journal of Marketing Research 6: 291–300\nBeckwith N.E. (1972) Multivariate analysis of sales responses of competapplication brands to advertising. Journal of Marketing Research 9: 168–176\nBlattberg R.C., Eppen G., Liebermann J. (1981) A theoretical and empirical evaluation of price deals in consumer non-durables. Journal of Marketing 45: 116–129\nCarlin B., Louis T. (1996) Bayes and empirical Bayes methods for data analysis. Chapman and Hall, New York\nChib S., Nardarib F., Shephard N. (2002) Markov chain Monte Carlo methods for stochastic volatility models. Journal of Econometrics 108: 281–316\nDekimpe M.G., Hanssens M. (2000) Time-series models in marketing: Past, present and future. International Journal of Research in Marketing 17: 183–193\nGeweke J. (1992) Evaluating the accuracy of sampling-based approaches to calculating posterior moments. In: Bernado J.M., Berger J.O., Dawid A.P., Smith A.F.M.(eds) Bayesian statistics, Vol. 4 . Oxford University Press, Oxford, pp 169–193\nGilks, W.R., Richardson, S., Spiegelhalter, D.J. (eds) (1996) Markov Chain Monte Carlo in practice. Chapman and Hall, New York\nGupta S. (1988) Impact of sales promotions on when, what, and how much to buy. Journal of Marketing Research 25: 342–355\nHanssens D.M. (1980) Market response, competitive behavior, and time series analysis. Journal of Marketing Research 17: 470–485\nKim S., Shephard N., Chib S. (1998) Stochastic volatility: likelihood inference comparison with ARCH models. Review of Economic Studies 65: 361–393\nKitagawa G. (1996) Monte Carlo filter and smoother for non-Gaussian nonlinear state space models. Journal of Computational and Graphical Statistics 5: 1–25\nKitagawa G. (1987) Non-Gaussian state-space modeling of nonstationary time series. Journal of the American Statistical Association 82: 1032–1063\nKitagawa G., Gersch W. (1996) Smoothness priors analysis of time series. Springer, New York\nKitagawa G., Higuchi T., Kondo F.N. (2003) Smoothness prior approach to explore mean structure in large-scale time series. Theoretical Computer Science 292: 431–446\nKondo F.N., Kitagawa G. (2000) Time series analysis of daily scanner sales—Extraction of trend, day-of-week effect, and price promotion effect. Marketing Intelligence & Planning 18: 53–66\nLee J., Boatwright P., Kamakura W.A. (2003) A Bayesian model for prelaunch sales: forecasting of recorded music. Management Science 49: 179–196\nLeone R.P. (1983) Modeling sales-advertising relationships: an integrated time series—Econometric approach. Journal of Marketing Research 20: 291–295\nNaik P.A., Murali K.M., Alan S. (1998) Planning pulsing media schedules in the presence of dynamic advertising quality. Marketing Science 17: 214–235\nNaik P.A. (1999) Estimating the half-life of advertisements. Marketing Letters 10: 351–362\nNeelamegham R., Pradeep C. (1999) Bayesian model to forecast new product performance in domestic and international markets. Marketing Science 18: 115–136\nNeslin S., Henderson C., Quelch J. (1985) Coupon promotions and acceleration of product purchase. Marketing Science 4: 147–165\nNeslin S. (2002) Sales promotion. Marketing Science Institute, Cambridge\nPatrick R. (1982) An extension of Shapiro and Wilk’s W test for normality to large samples. Applied Statistics 31: 115–124\nPauwels K., Hanssens D.M., Siddarth S. (2004) The long-term effects of price promotions on category incidence, brand choice and purchase quantity. Journal of Marketing Research 29: 421–439\nRobert C.P., Titterington D.M. (2002) Discussion on “Bayesian measures of model complexity and fit” (by Spiegelhalter, D. J. et al). Journal of the Royal Statistical Society, Series B 64: 621–622\nSato T., Higuchi T., Kitagawa G. (2004) Statistical inference using stochastic switching models for the discrimination of unobserved display promotion from POS data. Marketing Letters 15: 37–60\nSpiegelhalter D.J., Best N.G., Carlin B.P., Vander Linde A. (2002) Bayesian measures of model complexity and fit (with Discussion). Journal of the Royal Statistical Society, Series B 64: 583–639\nTanizaki H., Mariano R.S. (1998) Nonlinear and non-Gaussian state-space modeling with Monte Carlo simulations. Journal of Econometrics 83: 263–290\nTellis G.J., Fred F.Z., Zufryden F. (1995) Tackling the retailer decision maze: which brands to discount, how much, when, and why?. Marketing Science 12: 271–299\nTierney L. (1994) Markov chains for exploring posterior distributions (with discussion). Annals of Statistics 22: 1701–1762\nVan Heerde H., Leeflang P., Wittink D. (2004a) Decomposing the sales promotion bump with store data. Marketing Science 23: 317–334\nVan Heerde H., Mela C., Manchandra P. (2004b) The dynamic effect of innovation on market structure. Journal of Marketing Research 41: 166–183\nWildt A.R. (1974) Multifirm analysis of competitive decision variables. Journal of Marketing Research 11: 50–62\nXie J., Song M.S., Wang Q. (1997) Kalman filter estimation of new product diffusion models. Journal of Marketing Research 34: 378–393\nYamaguchi R., Tsuchiya E., Higuchi T. (2004) State space modeling approach to decompose daily sales of a restaurant into time-dependent multi-factors (in Japanese). Transactions of the Operations Research Society of Japan 49: 52–60",{"VOID":705},"10.1007\u002Fs10463-008-0194-0","2024-05-16T12:47:05.626+00:00","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002Fs10463-008-0194-0",[709],{"id":710,"sortIndex":23,"researcher":22,"roles":711,"affiliations":712,"properties":721,"displayName":723,"givenName":22,"familyName":22},"18873491-e246-4653-8196-299fa3b3710f",[120],[713],{"id":714,"sortIndex":23,"affiliation":715,"properties":22},"7d73985b-e302-406b-9491-1a1dfa40f99b",{"id":714,"createTime":22,"updateTime":22,"relativeEntities":716,"slug":22,"properties":717,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":720,"statistic":22},[],{"title":718},{"VI":719},"Graduate School of Business Administration, Keio University, Yokohama, Kanagawa, Japan",[],{"title":722,"gsAuthor":724},{"VI":723},"Tomohiro Ando",{"VOID":725},"[\"qj3BgngAAAAJ\"]",{"url":707,"publisher":727,"properties":769},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":728,"slug":10,"properties":729,"entityType":20,"verifyStatus":21,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":23,"subjectFields":733,"manageAffiliations":738,"indexDatabases":749,"url":22,"thumbnailPath":22,"statistic":764,"gsStatistic":22,"type":88,"analyzePriority":22},[],{"issn":730,"title":731,"eissn":732},{"VOID":15},{"EN":17},{"VOID":13},[734],{"id":26,"createTime":22,"updateTime":22,"relativeEntities":735,"label":736,"description":737,"parentId":22,"standard":22,"scholarHubFieldId":22},[],{"EN":29},{},[739,744],{"id":33,"createTime":22,"updateTime":22,"relativeEntities":740,"slug":22,"properties":741,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":743,"statistic":22},[],{"title":742},{"EN":37},[39],{"id":41,"createTime":22,"updateTime":22,"relativeEntities":745,"slug":22,"properties":746,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":748,"statistic":22},[],{"title":747},{"EN":45},[39],[750,757],{"id":49,"indexDatabase":751,"url":62,"indexYears":22,"academicFieldIds":756,"indexDatabaseRanking":22},{"id":51,"createTime":22,"updateTime":22,"relativeEntities":752,"label":753,"description":754,"key":58,"publicationTags":755,"standard":22},[],{"EN":54,"VI":54},{"EN":56,"VI":57},[60,61],[64],{"id":66,"indexDatabase":758,"url":77,"indexYears":78,"academicFieldIds":763,"indexDatabaseRanking":81},{"id":68,"createTime":22,"updateTime":22,"relativeEntities":759,"label":760,"description":761,"key":74,"publicationTags":762,"standard":22},[],{"EN":71,"VI":71},{"EN":71,"VI":73},[76],[80],{"impactFactor":23,"impactFactorByYear":765,"i10Index":23,"i10IndexLast5Year":23,"totalPublication":84,"totalPublicationByYear":766,"totalCitation":23,"totalCitationByYear":767,"totalCitationPerPublication":23,"totalCitationPerPublicationByYear":768,"hindexLast5Year":23,"hindex":23},{},{"1979":84},{},{},{"pages":770,"volume":772},{"VOID":771},"763-780",{"VOID":773},"60",{"total":23,"publishYear":775,"statisticByYear":776},2008,{},"2008-09-02","2026-07-28T06:15:04.442+00:00",[81,60],{"id":781,"createTime":782,"updateTime":783,"relativeEntities":784,"slug":785,"properties":786,"entityType":109,"verifyStatus":110,"verifyTime":797,"verifyNote":112,"languages":22,"translateLanguages":22,"viewCount":23,"primaryUrl":798,"fullTextUrl":22,"authors":799,"publicationType":133,"publisherRelationship":832,"citationCount":23,"citationInfo":880,"publishDate":883,"publishYear":881,"citationAnalyzeStatus":884,"lastCitationAnalyze":885,"indexDatabases":886,"openAccess":22,"references":22,"isForceReanalyzing":185},"33b68903-5825-4994-80e4-81ecf01a4dcc","2024-02-14T15:55:31.523+00:00","2026-07-25T17:08:07.253+00:00",[],"On-principal-components-regression-with-Hilbertian-predictors",{"abstract":787,"title":789,"gsPaper":791,"references":793,"doi":795},{"EN":788},"We demonstrate that, in a regression setting with a Hilbertian predictor, a response variable is more likely to be more highly correlated with the leading principal components of the predictor than with trailing ones. This is despite the extraction procedure being unsupervised. Our results are established under the conditional independence model, which includes linear regression and single-index models as special cases, with some assumptions on the regression vector. These results are a generalisation of earlier work which showed that this phenomenon holds for predictors which are real random vectors. A simulation study is used to quantify the phenomenon.",{"EN":790},"On principal components regression with Hilbertian predictors",{"VOID":792},"[\"15254343574919094306\"]",{"VOID":794},"Arnold, B. C., Brockett, P. L. (1992). On distributions whose component ratios are cauchy. American Statistician, 46(1), 25–26.\nArtemiou, A., Li, B. (2009). On principal components regression: A statistical explanation of a natural phenomenon. Statistica Sinica, 19, 1557–1565.\nArtemiou, A., Li, B. (2013). Predictive power of principal components for single-index model and sufficient dimension reduction. Journal of Multivariate Analysis, 119, 176–184.\nCook, R. (2007). Fisher lecture: Dimension reduction in regression. Statistical Science, 22(1), 1–26.\nCox, D. R. (1968). Notes on some aspects of regression analysis. Journal of the Royal Statistical Society Series A (General), 131(3), 265–279.\nDauxois, J., Ferré, L., Yao, A.-F. (2001). Un modèle semi-paramétrique pour variables aléatoires hilbertiennes. Comptes Rendus de l’Académie des Sciences, 333(1), 947–952.\nFerré, L., Yao, A. F. (2003). Functional sliced inverse regression analysis. Statistics, 37(6), 475–488.\nHall, P., Yang, Y. J. (2010). Ordering and selecting components in multivariate or functional data linear prediction. Journal of the Royal Statistical Society Series B: Statistical Methodology, 72(1), 93–110.\nHsing, T., Eubank, R. (2015). Theoretical foundations of functional data analysis, with an introduction to linear operators. 1st ed. West Sussex: Wiley.\nKingman, J. F. C. (1972). On random sequences with spherical symmetry. Biometrika, 59(2), 492.\nLi, B. (2007). Comment: Fisher lecture—Dimension reduction in regression. Statistical Science, 22(1), 32–35.\nLi, B. (2018). Sufficient dimension reduction: Methods and applications with R. 1st ed. Boca Raton: CRC Press.\nLi, B., Song, J. (2017). Nonlinear sufficient dimension reduction for functional data. The Annals of Statistics, 45(3), 1059–1095.\nLi, Y. (2007). A note on hilbertian elliptically contoured distributions. Unpublished manuscript, Department of Statistics, University of Georgia.\nNi, L. (2011). Principal component regression revisited. Statistica Sinica, 21, 741–747.\nPinelis, I., Molzon, R. (2016). Optimal-order bounds on the rate of convergence to normality in the multivariate delta method. Electronic Journal of Statistics, 10(1), 1001–1063.\nRamsay, J., Silverman, B. W. (1997). Functional data analysis. 1st ed. New York: Springer.",{"VOID":796},"10.1007\u002Fs10463-018-0702-9","2024-05-16T15:07:23.726+00:00","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002Fs10463-018-0702-9",[800,817],{"id":801,"sortIndex":23,"researcher":22,"roles":802,"affiliations":803,"properties":812,"displayName":814,"givenName":22,"familyName":22},"fffbd364-8313-4e70-b831-d9b474e1cf9b",[120],[804],{"id":805,"sortIndex":23,"affiliation":806,"properties":22},"c281676f-0489-4d77-a463-76e81af63744",{"id":805,"createTime":22,"updateTime":22,"relativeEntities":807,"slug":22,"properties":808,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":811,"statistic":22},[],{"title":809},{"VI":810},"School of Mathematics, Cardiff University, Cardiff, UK",[],{"title":813,"gsAuthor":815},{"VI":814},"Ben Jones",{"VOID":816},"[\"dU7nc6gAAAAJ\"]",{"id":818,"sortIndex":84,"researcher":22,"roles":819,"affiliations":820,"properties":827,"displayName":829,"givenName":22,"familyName":22},"c55b80ae-1024-46c8-a5ec-2c1e71d348ac",[120],[821],{"id":805,"sortIndex":23,"affiliation":822,"properties":22},{"id":805,"createTime":22,"updateTime":22,"relativeEntities":823,"slug":22,"properties":824,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":826,"statistic":22},[],{"title":825},{"VI":810},[],{"title":828,"gsAuthor":830},{"VI":829},"Andreas Artemiou",{"VOID":831},"[\"MxJYE7gAAAAJ\"]",{"url":798,"publisher":833,"properties":875},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":834,"slug":10,"properties":835,"entityType":20,"verifyStatus":21,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":23,"subjectFields":839,"manageAffiliations":844,"indexDatabases":855,"url":22,"thumbnailPath":22,"statistic":870,"gsStatistic":22,"type":88,"analyzePriority":22},[],{"issn":836,"title":837,"eissn":838},{"VOID":15},{"EN":17},{"VOID":13},[840],{"id":26,"createTime":22,"updateTime":22,"relativeEntities":841,"label":842,"description":843,"parentId":22,"standard":22,"scholarHubFieldId":22},[],{"EN":29},{},[845,850],{"id":33,"createTime":22,"updateTime":22,"relativeEntities":846,"slug":22,"properties":847,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":849,"statistic":22},[],{"title":848},{"EN":37},[39],{"id":41,"createTime":22,"updateTime":22,"relativeEntities":851,"slug":22,"properties":852,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":854,"statistic":22},[],{"title":853},{"EN":45},[39],[856,863],{"id":49,"indexDatabase":857,"url":62,"indexYears":22,"academicFieldIds":862,"indexDatabaseRanking":22},{"id":51,"createTime":22,"updateTime":22,"relativeEntities":858,"label":859,"description":860,"key":58,"publicationTags":861,"standard":22},[],{"EN":54,"VI":54},{"EN":56,"VI":57},[60,61],[64],{"id":66,"indexDatabase":864,"url":77,"indexYears":78,"academicFieldIds":869,"indexDatabaseRanking":81},{"id":68,"createTime":22,"updateTime":22,"relativeEntities":865,"label":866,"description":867,"key":74,"publicationTags":868,"standard":22},[],{"EN":71,"VI":71},{"EN":71,"VI":73},[76],[80],{"impactFactor":23,"impactFactorByYear":871,"i10Index":23,"i10IndexLast5Year":23,"totalPublication":84,"totalPublicationByYear":872,"totalCitation":23,"totalCitationByYear":873,"totalCitationPerPublication":23,"totalCitationPerPublicationByYear":874,"hindexLast5Year":23,"hindex":23},{},{"1979":84},{},{},{"pages":876,"volume":878},{"VOID":877},"627-644",{"VOID":879},"72",{"total":23,"publishYear":881,"statisticByYear":882},2018,{},"2018-12-06","DONE_ANALYZE_CITATION","2026-07-25T17:08:07.252+00:00",[81,60],{"id":888,"createTime":889,"updateTime":890,"relativeEntities":891,"slug":892,"properties":893,"entityType":109,"verifyStatus":110,"verifyTime":904,"verifyNote":112,"languages":22,"translateLanguages":22,"viewCount":23,"primaryUrl":905,"fullTextUrl":22,"authors":906,"publicationType":133,"publisherRelationship":924,"citationCount":614,"citationInfo":970,"publishDate":973,"publishYear":971,"citationAnalyzeStatus":884,"lastCitationAnalyze":890,"indexDatabases":974,"openAccess":22,"references":22,"isForceReanalyzing":185},"c281c549-bee3-4ac1-810c-38af945cc36c","2024-01-30T19:00:21.776+00:00","2026-07-24T21:15:41.922+00:00",[],"Gradual-change-point-analysis-based-on-Spearman-matrices-for-multivariate-time-series",{"abstract":894,"title":896,"gsPaper":898,"references":900,"doi":902},{"EN":895},"It may happen that the behavior of a multivariate time series is such that the underlying joint distribution is gradually moving from one distribution to another between unknown times of change. Under this context of a possible gradual-change, tests of change-point detection in the dependence structure of multivariate series are developed around the associated sequence of Spearman matrices. It is formally established that the proposed test statistics for that purpose are asymptotically marginal-free under a general strong-mixing assumption, and written as functions of integrated Brownian bridges. Consistent estimators of the pair of times of change, as well as of the before-the-change and after-the-change Spearman matrices, are also proposed. A simulation study examines the sampling properties of the introduced tools, and the methodologies are illustrated on a synthetic dataset.",{"EN":897},"Gradual change-point analysis based on Spearman matrices for multivariate time series",{"VOID":899},"[\"8578503608611651507\"]",{"VOID":901},"Andrews, D. W. K. (1991). Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica, 59(3), 817–858.\nAue, A., Steinebach, J. (2002). A note on estimating the change-point of a gradually changing stochastic process. Statistics & Probability Letters, 56(2), 177–191.\nBissell, A. F. (1984). The performance of control charts and cusums under linear trend. Journal of the Royal Statistical Society: Series C (Applied Statistics), 33(2), 145–151.\nBrodsky, B. E., Darkhovsky, B. S. (1993). Nonparametric methods in change-point problems. Mathematics and its Applications, Vol. 243. Kluwer Academic Publishers Group, Dordrecht.\nBücher, A., Kojadinovic, I., Rohmer, T., Segers, J. (2014). Detecting changes in cross-sectional dependence in multivariate time series. Journal of Multivariate Analysis, 132, 111–128.\nBücher, A., Ruppert, M. (2013). Consistent testing for a constant copula under strong mixing based on the tapered block multiplier technique. Journal of Multivariate Analysis, 116, 208–229.\nCarlstein, E. (1988). Nonparametric change-point estimation. The Annals of Statistics, 16(1), 188–197.\nDehling, H., Vogel, D., Wendler, M., Wied, D. (2017). Testing for changes in Kendall’s tau. Econometric Theory, 33(6), 1352–1386.\nDehling, H., Vuk, K., Wendler, M. (2022). Change-point detection based on weighted two-sample U-statistics. Electronic Journal of Statistics, 16(1), 862–891.\nFermanian, J. -D., Radulović, D., Wegkamp, M. H. (2004). Weak convergence of empirical copula processes. Bernoulli, 10, 847–860.\nGan, F. (1992). Cusum control charts under linear drift. Journal of the Royal Statistical Society: Series D (The Statistician), 41(1), 71–84.\nGombay, E., Horváth, L. (1995). An application of \\(U\\)-statistics to change-point analysis. Acta Universitatis Szegediensis. Acta Scientiarum Mathematicarum, 60(1–2), 345–357.\nGombay, E., Horváth, L. (1999). Change-points and bootstrap. Environmetrics, 10(6), 725–736.\nHušková, M. (1999). Gradual changes versus abrupt changes. Journal of Statistical Planning and Inference, 76(1–2), 109–125.\nHušková, M, Meintanis, S. G. (2006a). Change point analysis based on empirical characteristic functions. Metrika, 63(2), 145–168.\nHušková, M., Meintanis, S. G. (2006b). Change-point analysis based on empirical characteristic functions of ranks. Sequential Analysis. Design Methods & Applications, 25(4), 421–436.\nInoue, A. (2001). Testing for distributional change in time series. Econometric Theory, 17(1), 156–187.\nKander, Z., Zacks, S. (1966). Test procedures for possible changes in parameters of statistical distributions occurring at unknown time points. Annals of Mathematical Statistics, 37, 1196–1210.\nKojadinovic, I., Quessy, J. -F., Rohmer, T. (2016). Testing the constancy of Spearman’s rho in multivariate time series. Annals of the Institute of Statistical Mathematics, 68(5), 929–954.\nLombard, F. (1987). Rank tests for changepoint problems. Biometrika, 74(3), 615–624.\nNasri, B. R., Rémillard, B. N., Bahraoui, T. (2022). Change-point problems for multivariate time series using pseudo-observations. Journal of Multivariate Analysis, 187, 104857.\nNelsen, R. B. (2006). An introduction to copulas. Springer Series in Statistics, second edition, Springer, New York.\nPage, E. S. (1955). A test for a change in a parameter occurring at an unknown point. Biometrika, 42, 523–527.\nParzen, E. (1962). On estimation of a probability density function and mode. Annals of Mathematical Statistics, 33, 1065–1076.\nPettitt, A. (1979). A non-parametric approach to the change point problem. Applied Statistics, 28(2), 126–135.\nQuessy, J. -F. (2019). Consistent nonparametric tests for detecting gradual changes in the marginals and the copula of multivariate time series. Statistical Papers, 60(3), 367–396.\nQuessy, J. -F, Saïd, M., Favre, A. -C. (2013). Multivariate Kendall’s tau for change-point detection in copulas. Canadian Journal of Statistics, 41(1), 65–82.\nRio, E. (2000). Théorie asymptotique des processus aléatoires faiblement dépendants. Mathématiques et Applications. Berlin: Springer-Verlag.\nSklar, A. (1959). Fonctions de répartition à \\(n\\) dimensions et leurs marges. Publications de l’Institut de statistique de l’Université de Paris, 8, 229–231.\nvan der Vaart, A. W., Wellner, J. A. (1996). Weak convergence and empirical processes: With applications to statistics. Springer Series in Statistics. New York: Springer-Verlag.\nVogt, M., Dette, H. (2015). Detecting gradual changes in locally stationary processes. The Annals of Statistics, 43(2), 713–740.\nWied, D., Dehling, H., van Kampen, M., Vogel, D. (2014). A fluctuation test for constant Spearman’s rho with nuisance-free limit distribution. 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C., Celeux, G. and Govaert, G. (2003). Strategies for getting the highest likelihoods in mixture models, Special Issue on Recent Developments in Mixture Models (Guest Editors: D. Böhning and W. Seidel),Computational Statistics & Data Analysis,41, 561–575.",{"doi":1092},"10.1016\u002FS0167-9473(02)00163-9",{"id":22,"text":1094,"url":22,"identifiers":1095},"Böhning, D. (2000).Computer-Assisted Analysis of Mixtures and Applications, Chapman & Hall\u002FCRC, Boca Raton.",{},{"id":22,"text":1097,"url":22,"identifiers":1098},"Böhning, D. and Schlattmann, P. (1992). Computer-assisted analysis of mixtures (C.A.MAN): Statistical algorithms,Biometrics,48, 283–303.",{"doi":1099},"10.2307\u002F2532756",{"id":22,"text":1101,"url":22,"identifiers":1102},"Böhning, D., Dietz, E., Schaub, R., Schlattmann, P. and Lindsay, B. (1994). The distribution of the likelihood ratio for mixtures of densities from the one-parameter exponential family,Annals of the Institute of Statistical Mathematics,46, 373–388.",{"doi":1103},"10.1007\u002FBF01720593",{"id":22,"text":1105,"url":22,"identifiers":1106},"Dacunha-Castelle, D. and Gassiat, E. (1999). Testing the order of a model using locally conic parametrization: Population mixtures and stationary ARMA processes,The Annals of Statistics,27, 1178–1209.",{"doi":1107},"10.1214\u002Faos\u002F1017938921",{"id":22,"text":1109,"url":22,"identifiers":1110},"Ghosh, J. K. and Sen, P. K. (1985). On the asymptotic performance of the log likelihood ratio statistic for the mixture model and related results,Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer (eds. L. M. Le Cam and R. A. Olshen),2, 789–806, Wadsworth, Belmont, California.",{},{"id":22,"text":1112,"url":22,"identifiers":1113},"Hartigan, J. A. (1985). A failure of likelihood asymptotics for normal mixtures,Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer (eds. L. M. Le Cam and R. A. Olshen),2, 807–810, Wadsworth, Belmont, California.",{},{"id":22,"text":1115,"url":22,"identifiers":1116},"Karlis, D. (2001). A cautionary note about the EM algorithm for finite exponential mixtures, Tech. Report, No. 150, Department of Statistics, Athens University of Economics.",{},{"id":22,"text":1118,"url":22,"identifiers":1119},"Karlis, D. and Xekalaki, E. (2003). Choosing initial values for the EM algorithm for finite mixtures, Special Issue on Recent Developments in Mixture Models (Guest Editors: D. Böhning and W. Seidel),Computational Statistics & Data Analysis,41, 577–590.",{"doi":1120},"10.1016\u002FS0167-9473(02)00177-9",{"id":22,"text":1122,"url":22,"identifiers":1123},"Lesperance, M. and Kalbfleisch, J. (1992). An algorithm for computing the nonparametric MLE of a mixing distribution,Journal of the American Statistical Association,87, 120–126.",{"doi":1124},"10.1080\u002F01621459.1992.10475182",{"id":22,"text":1126,"url":22,"identifiers":1127},"Lindsay, B. G. (1995).Mixture Models: Theory, Geometry and Applications, Institute of Mathematical Statistics, Hayward, California.",{"doi":1128},"10.1214\u002Fcbms\u002F1462106013",{"id":22,"text":1130,"url":22,"identifiers":1131},"Liu, X. and Shao, Y. (2003). Asymptotics for likelihood ratio tests under loss of identifiability,The Annals of Statistics,31, 807–832.",{"doi":1132},"10.1214\u002Faos\u002F1056562463",{"id":22,"text":1134,"url":22,"identifiers":1135},"McLachlan, G. and Peel, D. (2000).Finite Mixture Models, Wiley, New York.",{"doi":1136},"10.1002\u002F0471721182",{"id":22,"text":1138,"url":22,"identifiers":1139},"Seidel, W. and Ševčíková, H. (2002a). Efficient calculation of the NPMLE of a mixing distribution for mixtures of exponentials,Discussion Papers in Statistics and Quantitative Economics,96, Universität der Bundeswehr Hamburg.",{},{"id":22,"text":1141,"url":22,"identifiers":1142},"Seidel, W. and Ševčíková, H. (2002b). Tools for analyzing and maximizing likelihood functions in mixture models,Discussion Papers in Statistics and Quantitative Economics,104, Universität der Bundeswehr Hamburg.",{},{"id":22,"text":1144,"url":22,"identifiers":1145},"Seidel, W. and Ševčíková, H. (2003). A detailed investigation of likelihood maxima in two-component exponential mixture models and their implication on LR tests,Discussion Papers in Statistics and Quantitative Economics,106, Universität der Bundeswehr Hamburg.",{},{"id":22,"text":1147,"url":22,"identifiers":1148},"Seidel, W., Mosler, K. and Alker, M. (2000a) A cautionary note on likelihood ratio tests in mixture models,Annals of the Institute of Statistical Mathematics,52, 481–487.",{"doi":1149},"10.1023\u002FA:1004117419204",{"id":22,"text":1151,"url":22,"identifiers":1152},"Seidel, W., Mosler, K. and Alker, M. (2000b). Likelihood ratio tests based on subglobal optimization: A power comparison in exponential mixture models,Statistical Papers,41, 85–98.",{"doi":1153},"10.1007\u002FBF02925678",{"id":22,"text":1155,"url":22,"identifiers":1156},"Seidel, W., Ševčíková, H. and Alker, M. (2000c). On the power of different versions of the likelihood ratio test for homogeneity in an exponential mixture model,Discussion Papers in Statistics and Quantitative Economics,92, Universität der Bundeswehr Hamburg.",{},{"id":1158,"createTime":1159,"updateTime":1160,"relativeEntities":1161,"slug":1162,"properties":1163,"entityType":109,"verifyStatus":110,"verifyTime":1174,"verifyNote":112,"languages":22,"translateLanguages":22,"viewCount":23,"primaryUrl":1175,"fullTextUrl":22,"authors":1176,"publicationType":133,"publisherRelationship":1224,"citationCount":23,"citationInfo":1272,"publishDate":1275,"publishYear":1273,"citationAnalyzeStatus":21,"lastCitationAnalyze":1276,"indexDatabases":1277,"openAccess":22,"references":22,"isForceReanalyzing":185},"980e026a-4f7e-4e0f-811d-ba397cd70de7","2024-02-07T03:41:48.490+00:00","2026-07-16T18:49:22.802+00:00",[],"Estimation-of-a-multivariate-Box-Cox-transformation-to-elliptical-symmetry-via-the-empirical-characteristic-function",{"abstract":1164,"title":1166,"gsPaper":1168,"references":1170,"doi":1172},{"EN":1165},"Let X=(X\n1, X\n2,..., X\n\n                  d\n                )\n                  t\n                 be a random vector of positive entries, such that for some λ=(λ1,λ2,...,λ\n                  d\n                )\n                  t\n                , the vector X\n(λ) defined by % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GaamiwamaaDaaaleaamiaadMgaaSqaaWGaaiikaiabeU7aSnaaBaaa% baGaamyAaiaacMcaaeqaaaaakiabg2da9iaacIcadaWcgaqaaiaadI% fadaqhaaWcbaadcaWGPbaaleaamiabeU7aSnaaBaaabaGaamyAaaqa% baaaaOGaeyOeI0IaaGymaiaacMcaaeaacqaH7oaBdaWgaaWcbaadca% WGPbGaaiilaaWcbeaakiaadMgacqGH9aqpcaaIXaGaeSOjGSKaaiil% aiaadsgaaaaaaa!53BB!\\[X_i^{(\\lambda _{i)} }  = ({{X_i^{\\lambda _i }  - 1)} \\mathord{\\left\u002F {\\vphantom {{X_i^{\\lambda _i }  - 1)} {\\lambda _{i,} i = 1 \\ldots ,d}}} \\right. \\kern-\\nulldelimiterspace} {\\lambda _{i,} i = 1 \\ldots ,d}}\\]is elliptically symmetric. We describe a procedure based on the multivariate empirical characteristic function for estimating the λi's. Asymptotic results regarding consistency of the estimators are given and we evaluate their performance in simulated data. In a one-dimensional setting, comparisons are made with other available transformations to symmetry.",{"EN":1167},"Estimation of a multivariate Box-Cox transformation to elliptical symmetry via the empirical characteristic function",{"VOID":1169},"[\"2507218839976600259\"]",{"VOID":1171},"Andrews, D. F. and Herzberg, A. M. (1985). Data: A Collection of Problems from Many Fields for the Student and Research Worker, Springer, New York.\nAndrews, D. F., Gnanadesikan, R. and Warner, J. L. (1971). Transformations of multivariate data, Biometrics, 27, 825–840.\nArcones, M. A. and Giné, E. (1993). Limit theorems for U-processes, Ann. Probab., 21, 1494–1542.\nAssouad, P. (1983). Densité et dimension, Ann. Inst. Fourier Grenoble, 33, 233–282.\nBerry, D. A. (1987). Logarithmic transformation in anova, Biometrics, 3, 39–52.\nBoos, D. D. (1982). A test for asymmetry associated with a Hodges-Lehmann estimator, J. Amer. Statist. Assoc., 77, 647–651.\nCambanis, S., Huang, S. and Simons, G. (1981). On the theory of elliptically contoured distributions, J. Multivariate Anal., 11, 368–385.\nCsörgő, S. (1986). Testing for normality in arbitrary dimension, Ann. Statist., 14, 708–723.\nCsörgő, S. and Heathcote, C. R. (1987). Testing for symmetry, Biometrika, 74, 177–184.\nDevlin, S. J., Gnanadesikan, R. and Kettenring, J. R. (1976). Some multivariate applications of elliptical distributions, Essays in Probability and Statistics (eds. S. Ideka, T. Hayakawa, H. Hudimoto, M. Okamoto, M. Siotani and S. Yamaoto), 365–395, Shink Tsusho Co., Ltd. Tokyo.\nDudley, R. M. (1984). A course on empirical processes, Ecole d'Eté de Probabilités de Saint-Flour, XII-1982, Lecture Notes in Math., 1097, 1–142. Springer, New York.\nDudley, R. M. (1987). Universal Donsker classes and metric entropy, Ann. Probab., 15, 1306–1326.\nFang, K. T. and Anderson, T. W. (eds.) (1990). Statistical Inference in Elliptically Contoured and Related Distributions, Allerton Press, New York.\nFang, K. T., Kotz, S. and Ng, K. W. (1990). Symmetric multivariate and related distributions, Monographs Statist. Appl. Probab., 36, Chapman and Hall, London.\nFeuerverger, A. and Mureika, R. A. (1977). The empirical characteristic function and its applications, Ann. Statist., 5, 88–97.\nGhosh, S. and Ruymgaart, F. (1992). Applications of empirical characteristic functions in some multivariate problems, Canad. J. Statist., 20(4), 429–440.\nHinkley, D. V. (1977). On quick choice of power transformation, Appl. Statist., 26, 67–68.\nLoève, M. M. (1955). Probability Theory: Foundations, Random Sequences, Van Nostrand, New York.\nNakamura, M. and Ruppert, D. (1990). Semi-parametric estimation of symmetrizing transformations with application to the shifted power transformation (unpublished manuscript).\nNelson, C. H., Cox, D. D. and Ndjuenga, J. (1989). Mean variance portfolio choice: a test for elliptical symmetry, Tech. Report, No. 41, Department of Statistics, University of Illinois.\nPollard, D. (1984). Convergence of Stochastic Processes, Springer, New York.\nRandles, R. H. and Wolfe, D. A. (1979). Introduction to the Theory of Nonparametric Statistics, Wiley, New York.\nSerfling, R. J. (1980). Approximation Theorems of Mathematical Statistics, Wiley, New York.\nSherman, R. P. (1993). The limiting distribution of the maximum rank correlation estimator, Econometrica, 61, 123–137.\nShohat, J. A. and Tamarkin, J. D. (1943). The Problem of Moments, Mathematical Surveys, Number 1, American Mathematical Society, Rhode Island.\nTaylor, J. M. G. (1985). Power transformations to symmetry, Biometrika, 72, 145–152.\nVelilla, S. (1993). 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