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Minimax nonparametric hypothesis testing for ellipsoids and Besov bodies:

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Bibliographic Details
Main Authors:	Ingster, Jurij I. (Author), Suslina, Irina A. (Author)
Format:	Book
Language:	German
Published:	Berlin Weierstraß-Inst. für Angewandte Analysis und Stochastik 1997
Series:	Weierstrass-Institut für Angewandte Analysis und Stochastik <Berlin>: Report 12
Links:	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007752631&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
Physical Description:	88 S. graph. Darst.

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adam_text	WEIERSTRAFI-INSTITUT FUR ANGEWANDTE ANALYSIS UND STOCHASTIK IM FORSCHUNGSVERBUND BERLIN E.V. MINIMAX NONPARAMETRIC HYPOTHESIS TESTING FOR ELLIPSOIDS AND BESOV BODIES YURI I. INGSTER 1 , IRINA A. SUSLINA 2 SUBMITTED: 7TH AUGUST 1997 1 ST. PETERSBURG TRANSPORT UNIVERSITY 2 ST. PETERSBURG INSTITUTE DEP. OF APPLIED MATHEMATICS MOSKOWSKII AV. 9 191003 ST. PETERSBURG RUSSIA OF EXACT MECHANICS AND OPTICS (TECHNICAL UNIVERSITY) SABLINSKAYA STR. 14 197101 ST. PETERSBURG RUSSIA SUB G&TTLNGEN 204340 020., REPORT NO. 12 BERLIN 1997 1991 MATHEMATICS SUBJECT CLASSIFICATION. 62G10, 62G20. KEY WORDS AND PHRASES, NONPARAMETRIC HYPOTHESES TESTING, MINIMAX HYPOTHESE TESTING, ASYMP- TOTICS OF ERROR PROBABILITIES. RESEARCH WAS PARTIALLY SUPPORTED BY RUSSIAN FUND OF FUNDAMENTAL INVESTIGATIONS GRANT 96-01- 00-684, BY THE GRANT INTAS 93-1585 EXT AND BY WEIERSTRASS INSTITUTE FOR APPLIED ANALYSIS AND STOCHASTICS. CONTENTS 1 INTRODUCTION 4 1.1 SETTING 4 1.2 ALTERNATIVES 5 1.2.1 ELLIPSOIDAL CASE 5 1.2.2 BESOV BODIES CASE 6 1.2.3 DISCUSSION 7 1.3 MAIN RESULTS 7 2 NON-GAUSSIAN ASYMPTOTICS . 9 2.1 CLASSICAL TYPE (C) 9 2.2 TRIVIAL TYPE (T) 10 2.3 DEGENERATE TYPE 11 3 GAUSSIAN ASYMPTOTICS 12 3.1 TYPES GI AND GI 12 3.2 BOUNDARY LOG-TYPES OF GAUSSIAN ASYMPTOTICS 15 3.3 GRAPHICAL REPRESENTATION 17 4 TEST PROCEDURES 20 4.1 DEGENERATE CASE * . 20 4.2 GAUSSIAN CASE 21 5 GAUSSIAN ASYMPTOTICS: REDUCTION TO EXTREME PROBLEM 22 5.1 HILBERT STRUCTURE 24 5.2 LOWER BOUNDS 24 5.3 UPPER BOUNDS 27 5.4 EXTREME PROBLEM 33 6 THE EXTREME PROBLEM FOR ELLIPSOIDS 36 6.1 THE CASE P = Q OO 36 6.1.1 PROOF OF THE THEOREMS 4 AND 5 FOR P = Q 2 37 6.1.2 PROOF OF THE THEOREMS 4 AND 5 FOR P = Q 2 38 6.2 THE CASE P ^ Q: SEPARATION EQUATIONS SYSTEM FOR THE EXTREME PROBLEM 39 6.2.1 ONE-DIMENSIONAL PROBLEMS 40 6.2.2 SOLUTION OF ONE-DIMENSIONAL PROBLEMS 40 6.2.3 EQUATIONS SYSTEM FOR EXTREME PROBLEM, Q OO 41 6.3 THE SOLUTION OF THE SYSTEM (6.83) WITH C% = 0 41 6.3.1 SOLUTIONS OF EXTREME PROBLEM FOR N G H^ 46 6.3.2 PROOF THE THEOREMS 4 AND 6 FOR K G H^ 48 6.4 THE SOLUTION OF THE SYSTEM (6.83) WITH D 0 49 6.5 SOLUTION OF EXTREME PROBLEM FOR K G HJ 52 6.5.1 THE CASE / 0 52 6.5.2 THE CASE 7 = 0 53 6.5.3 THE CASE / 0 54 6.5.4 PROOF THE THEOREMS 4 AND 6 FOR K G H^ 56 6.6 THE EXTREME PROBLEM FOR Q = OO 58 6.6.1 PROOF THE THEOREMS 4 AND 6 FOR P 2 59 6.6.2 PROOF THE THEOREMS 4 AND 6 FOR P 2 60 6.7 SOME ADDITIONAL PROPERTIES OF THE SOLUTION OF (6.83) 61 6.7.1 CONTINUOUS PROPERTIES OF SOLUTIONS OF THE EXTREME PROBLEM . 61 6.7.2 CORRELATION PROPERTIES 62 7 THE EXTREME PROBLEM FOR BESOV BODIES 64 7.1 INVESTIGATION OF THE SYSTEM (7.154) ( 66 7.1.1 THE CASE CJ; = B 3 *. = 0 66 7.1.2 THE CASE CJ = 0, BJ 0 66 7.1.3 THE CASE C 3 0, BJ 0 68 7.1.4 THE CASE CJ 0,BJ = 0 69 7.1.5 THE SOLUTIONS OF (7.154) 69 7.2 SOLUTIONS OF EXTREME PROBLEMS AND UPPER BOUNDS 71 7.2.1 THE CASES P 2, P QOIQ P 2 AND CII 0 71 7.2.2 THE CASES P 2, Q P OR 2 P Q AND H 0 72 7.2.3 THE CASE 2 P Q, A X 0 73 7.2.4 THE CASE 2 P Q, B X 0 74 7.2.5 UPPER BOUNDS 74 7.2.6 RATE PROPERTIES OF SOLUTIONS OF THE EXTREME PROBLEM FOR BESOV BODIES 75 7.3 LOWER BOUNDS 76 8 DEGENERATE TYPE: PROOF OF THE THEOREMS 3 AND 7 77 8.1 UPPER BOUNDS: ELLIPSOIDS 77 8.2 UPPER BOUNDS: BESOV BODIES 78 8.3 LOWER BOUNDS: ELLIPSOIDS 80 8.4 LOWER BOUNDS: BESOV BODIES 81 9 TRIVIAL TYPE: PROOF OF THE THEOREM 2 81 9.1 ELLIPSOIDAL CASE 81 9.2 BESOV BODIES CASE 84
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author	Ingster, Jurij I. Suslina, Irina A.
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publisher	Weierstraß-Inst. für Angewandte Analysis und Stochastik
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series	Weierstrass-Institut für Angewandte Analysis und Stochastik <Berlin>: Report
series2	Weierstrass-Institut für Angewandte Analysis und Stochastik <Berlin>: Report
spellingShingle	Ingster, Jurij I. Suslina, Irina A. Minimax nonparametric hypothesis testing for ellipsoids and Besov bodies Weierstrass-Institut für Angewandte Analysis und Stochastik <Berlin>: Report
title	Minimax nonparametric hypothesis testing for ellipsoids and Besov bodies
title_auth	Minimax nonparametric hypothesis testing for ellipsoids and Besov bodies
title_exact_search	Minimax nonparametric hypothesis testing for ellipsoids and Besov bodies
title_full	Minimax nonparametric hypothesis testing for ellipsoids and Besov bodies Yuri I. Ingster ; Irina A. Suslina
title_fullStr	Minimax nonparametric hypothesis testing for ellipsoids and Besov bodies Yuri I. Ingster ; Irina A. Suslina
title_full_unstemmed	Minimax nonparametric hypothesis testing for ellipsoids and Besov bodies Yuri I. Ingster ; Irina A. Suslina
title_short	Minimax nonparametric hypothesis testing for ellipsoids and Besov bodies
title_sort	minimax nonparametric hypothesis testing for ellipsoids and besov bodies
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007752631&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
volume_link	(DE-604)BV009889758
work_keys_str_mv	AT ingsterjuriji minimaxnonparametrichypothesistestingforellipsoidsandbesovbodies AT suslinairinaa minimaxnonparametrichypothesistestingforellipsoidsandbesovbodies

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