Approximately calibrated small sample inference about means from bivariate normal data with missing values |
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| Affiliation: | 1. Instituto de Física, Universidade de São Paulo - C.P. 66318, CEP: 05315-970, São Paulo, Brazil;2. Laboratoire Univers & Particules de Montpellier (LUPM), Université de Montpellier (UMR-5299) Place Eugène Bataillon, F-34095 Montpellier Cedex 05, France;3. Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster, Wilhelm-Klemm-Straße 9, 48149 Münster, Germany;4. DARK, Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200 Copenhagen, Denmark;5. Department of Physics, Istanbul Technical University, Maslak 34469 Istanbul, Turkey;6. CERCA/ISO, Department of Physics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106, USA;7. Astrophysics Group & Imperial Centre for Inference and Cosmology, Department of Physics, Imperial College London, Blackett Laboratory, Prince Consort Road, London SW7 2AZ, United Kingdom;8. Laboratoire de Physique de l''École Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, F-75005 Paris, France;9. Observatoire de Paris, Université PSL, Sorbonne Université, LERMA, 75014 Paris, France;10. Department of Physics, University of Ioannina, GR-45110, Ioannina, Greece;11. Physics Department, Boston University, Boston, MA 02215, USA;12. Institute of Theoretical Physics, Heidelberg University, Philosophenweg 16, 69120 Heidelberg, Germany;13. Department of Physics and Astronomy, Lehman College, City University of New York, NY 10468, USA;14. Department of Physics, Graduate Center, City University of New York, NY 10016, USA;15. Department of Astrophysics, American Museum of Natural History, NY 10024, USA;p. Institute of Physics, Laboratory of Astrophysics, École Polytechnique Fédérale de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix, Switzerland;q. Oskar Klein Centre, Department of Physics, Stockholm University, SE-106 91 Stockholm, Sweden;r. E.A Milne Centre, University of Hull, Cottingham Road, Hull, HU6 7RX, United Kingdom;s. Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh, EH9 3HJ, United Kingdom;t. Dipartimento di Fisica e Astronomia, Università di Bologna, viale Berti Pichat 6/2, 40127 Bologna, Italy;u. INAF/OAS Bologna, via Piero Gobetti 101, I-40129 Bologna, Italy;v. INFN, Sezione di Bologna, viale Berti Pichat 6/2, 40127 Bologna, Italy;w. Department of Physics & Astronomy, University of the Western Cape, Cape Town 7535, South Africa;x. Department of Physics, University of Wisconsin, Madison, WI 53706, USA;y. Academy of Athens, Research Center for Astronomy and Applied Mathematics, Soranou Efesiou 4, 11527, Athens, Greece;z. National Observatory of Athens, Lofos Nymfon, 11852 Athens, Greece;11. Escola de Ciências e Tecnologia, Universidade Federal do Rio Grande do Norte, 59078-970, Natal, RN, Brazil;12. Dipartimento di Fisica, Sapienza Università di Roma, P.le A. Moro 2, I-00185, Roma, Italy;13. INFN Sezione di Roma, P.le A. Moro 2, I-00185, Roma, Italy;14. Jodrell Bank Centre for Astrophysics, University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom;15. Scuola Superiore Meridionale, Largo S. Marcellino 10, I-80138, Napoli, Italy;16. Istituto Nazionale di Fisica Nucleare (INFN) Sezione di Napoli, Complesso Universitario di Monte Sant''Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy;17. DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom;18. Kavli Institute of Cosmology (KICC), University of Cambridge, Madingley Road, Cambridge, CB3 0HA, United Kingdom;19. Queens'' College, Cambridge, CB3 9ET, United Kingdom;110. Theoretical Physics Department, Fermilab, P.O. Box 500, Batavia, IL 60510, USA;111. Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA;112. Kapteyn Astronomical Institute, University of Groningen, P.O. Box 9000, 9700 AA Groningen, the Netherlands;113. Van Swinderen Institute for Particle Physics and Gravity, University of Groningen, Nijenborgh 4, 9747 AG Groningen, the Netherlands;114. Kavli Institute for Particle Astrophysics and Cosmology and Department of Physics, Stanford University, Stanford, CA 94305, USA;115. NSF''s National Optical-Infrared Astronomy Research Laboratory (NOIRLab), Tucson, AZ 85719, USA;1p. Department of Physics, The University of Texas at Austin, Austin, TX 78712, USA;1q. Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150, 22290-180 Rio de Janeiro, RJ, Brazil;1r. Centro Federal de Educação Tecnológica Celso Suckow da Fonseca, Rodovia Márcio Covas, lote J2, quadra J - Itaguaí, Brazil;1s. Departamento de Física, Universidade Federal de Juiz de Fora, 36036-330, Juiz de Fora, MG, Brazil;1t. Dipartimento di Fisica e Astronomia “Augusto Righi”, Alma Mater Studiorum Università di Bologna, via Piero Gobetti 93/2, I-40129 Bologna, Italy;1u. INAF - Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, via Piero Gobetti 93/3, I-40129 Bologna, Italy;1v. Institut d''Astrophysique de Paris, CNRS & Sorbonne University, 98 bis Boulevard Arago, 75014, Paris, France;1w. Instituto de Fisica Teorica, Universidad Autonoma de Madrid, Madrid, 28049, Spain;1x. Univ Lyon, Ens de Lyon, Univ Lyon1, CNRS, Centre de Recherche Astrophysique de Lyon UMR5574, F–69007, Lyon, France;1y. Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USA;1z. Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL 60637, USA;21. School of Physics and Astronomy, Cardiff University, The Parade, Cardiff, CF24 3AA, United Kingdom;22. Department of Physics & Astronomy, Dartmouth College, 6127 Wilder Laboratory, Hanover, NH 03755 USA;23. PPGCosmo, Universidade Federal do Espírito Santo, 29075-910, Vitória, ES, Brazil;24. Dipartimento di Fisica “E. Pancini”, Università degli Studi di Napoli “Federico II”, Complesso Universitario di Monte S. Angelo, Ed. G, Via Cinthia, I-80126, Napoli, Italy;25. Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA;26. Department of Physics and Astronomy, University of California, Los Angeles CA 90095, USA;27. Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA;28. Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU, WPI), Kashiwanoha, Chiba, 277-8583, Japan;29. Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA;210. Department of Physics & Astronomy, McMaster University, 1280 Main Street West, Hamilton, ON L8S 4M1, Canada;211. Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM 87106, USA;212. Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotnikow 32/46, 02-668 Warsaw, Poland;213. National Astronomical Observatory of Japan, 2 Chome-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan;214. The Graduate University for Advanced Studies, SOKENDAI, Shonankokusaimura, Hayama, Miura District, Kanagawa 240-0193, Japan;215. Space Science Institute, Boulder, CO, USA;2p. Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma, Parco Area delle Scienze 7/A, 43124 Parma, Italy;2q. INFN Gruppo Collegato di Parma, 43124 Parma, Italy;2r. Department of Physics, Kansas State University, 116 Cardwell Hall, Manhattan, KS 66502 USA;2s. Departament de Matemática Aplicada, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain;2t. CNRS-UCB International Research Laboratory, Centre Pierre Binétruy, IRL2007, CPB-IN2P3, Berkeley, USA;2u. Laboratoire Astroparticule et Cosmologie (APC), CNRS/IN2P3, Université Paris Diderot, 75205 Paris Cedex 13, France;2v. IRFU, CEA, Université Paris Saclay, 91191 Gif-sur-Yvette, France;2w. Department of Astronomy, School of Physical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, PR China;2x. High Energy Theory Group, Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA;2y. Institute of Astronomy and Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA, United Kingdom;2z. Department of Physics, University of Arizona, Tucson, AZ 85721, USA;31. Department of Physics, University of Maryland, College Park, MD 20742, USA;32. School of Mathematics and Statistics, University of Sheffield, Hounsfield Road, Sheffield S3 7RH, United Kingdom;33. Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, PR China;34. Department of Astronomy, University of Geneva, Ch. d''Ecogia 16, CH-1290 Versoix, Switzerland;35. Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Circuito Exterior C.U., A.P. 70-543, México D.F. 04510, Mexico;36. Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA, 91109, USA;37. Osservatorio di Astrofisica e Scienza dello Spazio di Bologna & Istituto Nazionale di Astrofisica, via Gobetti 101, I-40129 Bologna, Italy;38. Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans, s/n, 08193 Barcelona, Spain;39. Institut d''Estudis Espacials de Catalunya (IEEC), Carrer Gran Capità 2-4, 08193 Barcelona, Spain;310. Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências da Universidade de Lisboa, Edificio C8, Campo Grande, P-1749016, Lisboa, Portugal;311. Galileo Galilei Institute for theoretical physics, National Centre of Advanced Studies of INFN Largo Enrico Fermi 2, I-50125, Firenze, Italy;312. Department of Physics, Michigan Technological University, Houghton, MI 49931, USA;313. Dipartamento di Fisica and INFN Sezione di Roma Tor Vergata, via della Ricerca Scientifica 1, 00133 Roma, Italy;314. Astrophysics Group, Cavendish Laboratory, J. J. Thomson Avenue, Cambridge, CB3 0HE, United Kingdom;315. Kavli Institute for Cosmology, Madingley Road, Cambridge, CB3 0HA, United Kingdom;3p. School of Physics and Astronomy, Cardiff University, Cardiff, CF24 3AA, United Kingdom;3q. The Institute of Mathematical Sciences, HBNI, CIT Campus, Chennai 600113, India;3r. Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical Institute (AIRUB), German Centre for Cosmological Lensing, 44780 Bochum, Germany;3s. Department of Physics, Columbia University, New York, NY 10027, USA;3t. Center for Computational Astrophysics, Flatiron Institute, New York, NY 10010, USA;3u. Instituto de Física Teórica UAM-CSIC, C/ Nicolás Cabrera 13-15, Universidad Autónoma de Madrid, Cantoblanco, Madrid 28049, Spain;3v. Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA;3w. Department of Physics and Enrico Fermi Institute, University of Chicago, Chicago IL 60637 USA;3x. Department of Physics and Helsinki Institute of Physics, PL 64, FI-00014 University of Helsinki, Finland;3y. Department of Physics, Shahid Beheshti University, 1983969411, Tehran, Iran;3z. Department of Physics, University of Michigan, 450 Church St, Ann Arbor, MI 48109-1040, USA;41. Leinweber Center for Theoretical Physics, University of Michigan, 450 Church St, Ann Arbor, MI 48109-1040, USA;42. Department of Physics, The University of Texas at Dallas, Richardson, TX 75080, USA;43. School of Natural Sciences, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA;44. Laboratoire de Univers et Particules de Montpellier, UMR5299-CNRS, Universite de Montpellier, 34095 Montpellier, France;45. ICCUB, University of Barcelona, Marti i Franques 1, Barcelona, 08028, Spain;46. ICREA, Pg. Lluis Companys 23, Barcelona, 08010, Spain;47. Waterloo Centre for Astrophysics, University of Waterloo, 200 University Ave W, Waterloo, ON N2L 3G1, Canada;48. Department of Physics and Astronomy, University of Waterloo, 200 University Ave W, Waterloo, ON N2L 3G1, Canada;49. Department of Physics, University of California Merced, 5200 North Lake Road, Merced, CA 95343, USA;410. Max Planck Institute for Astrophysics, Karl-Schwarzschild-Str. 1, D-85748 Garching, Germany;411. Kavli Institute for the Physics Mathematics of the Universe (Kavli IPMU, WPI), Todai Institutes for Advanced Study, The University of Tokyo, Kashiwa 277-8583, Japan;412. Kapteyn Astronomical Institute, University of Groningen, P.O. Box 800, 9700AV Groningen, the Netherlands;413. Department of Mathematics, Indira Gandhi University, Meerpur, Haryana-122502, India;414. Department of Physics, University of the Basque Country UPV/EHU, P.O. Box 644, 48080 Bilbao, Spain;415. Clare Hall, University of Cambridge, Herschel Rd, Cambridge CB3 9AL, United Kingdom;4p. Institute for Theoretical Particle Physics and Cosmology (TTK), RWTH Aachen University, D-52056 Aachen, Germany;4q. Institute of Space Sciences and Astronomy, University of Malta, Msida, MSD 2080, Malta;4r. Department of Physics, University of Malta, Msida, MSD 2080, Malta;4s. Astroparticle Physics Laboratory, NASA Goddard Space Flight Center, Greenbelt, MD, USA;4t. Department of Physics and Astronomy, Sejong University, Seoul 05006, Republic of Korea;4u. Service de Physique Théorique, Université Libre de Bruxelles, Boulevard du Triomphe, CP225, 1050 Brussels, Belgium;4v. Institute of Cosmology & Gravitation, University of Portsmouth, Portsmouth PO1 3FX, United Kingdom;4w. National Institute for Theoretical & Computational Sciences (NITheCS), Pretoria 2600, South Africa;4x. George P. and Cynthia W. Mitchell Institute for Fundamental Physics & Astronomy, Department of Physics & Astronomy, Texas A&M University, College Station, TX, USA;4y. Department of Physics and Astronomy, University of Hawaii, Honolulu, HI 96822, USA;4z. Núcleo de Astrofísica e Cosmologia (Cosmo-ufes) & Departamento de Física, Universidade Federal do Espírito Santo, 29075-910, Vitória, ES, Brazil;51. INAF – Osservatorio Astronomico di Trieste, via Tiepolo 11, 34131 Trieste, Italy;52. IFPU – Institute for Fundamental Physics of the Universe, via Beirut 2, 34151, Trieste, Italy;53. Centro de Astrofísica da Universidade do Porto, Rua das Estrelas, 4150-762 Porto, Portugal;54. Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, Rua das Estrelas, 4150-762 Porto, Portugal;55. Dipartimento di Fisica e Astronomia “Galileo Galilei”, Università degli Studi di Padova, via F. Marzolo 8, I-35131, Padova, Italy;56. INFN, Sezione di Padova, via F. Marzolo 8, I-35131, Padova, Italy;57. INAF-Osservatorio Astronomico di Padova, Vicolo dell''Osservatorio 5, I-35122 Padova, Italy;58. Gran Sasso Science Institute, Viale F. Crispi 7, I-67100 L''Aquila, Italy;59. Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur - 721302, India;510. Instituto de Física Corpuscular (CSIC-Universitat de València), Paterna, Spain;511. Department of Physics and Astronomy, University of North Carolina at Chapel Hill, NC, 27599, USA;512. Department of Physics and McDonnell Center for the Space Sciences, Washington University, St. Louis, MO 63130, USA;513. INFN, Sezione di Trieste, Via Bonomea 265, 34136 Trieste, Italy;514. Research Center for Nuclear Physics, Osaka University, Ibaraki, Osaka, 567-0047 Japan;515. C. N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, NY 11794, USA;5p. Departament de Física Quàntica i Astrofísica and Institute of Cosmos Sciences, Universitat de Barcelona, Av. Diagonal 647, E-08028 Barcelona, Catalonia, Spain;5q. Institute of Theoretical Astrophysics, University of Oslo, 0315 Oslo, Norway;5r. Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth, PO13FX, United Kingdom;5s. Perimeter Institute for Theoretical Physics, 31 Caroline Street N., Waterloo, Ontario, N2L 2Y5, Canada;5t. Niels Bohr Institute, Blegdamsvej 17, Copenhagen, Denmark;5u. Department of Physics, Northeastern University, Boston, MA 02115, USA;5v. Nordita, KTH Royal Institute of Technology and Stockholm University, Hannes Alfvéns väg 12, SE-106 91 Stockholm, Sweden;5w. Departament de Física Quàntica i Astrofisíca & Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona, Martí i Franquès 1, 08028 Barcelona, Spain;5x. Divisão de Astrofísica, Instituto Nacional de Pesquisas Espaciais, Avenida dos Astronautas 1758, São José dos Campos, 12227-010, SP, Brazil;5y. Center for Quantum Spacetime, Sogang University, Seoul 121-742, Republic of Korea;5z. Department of Physics, Sogang University, Seoul 121-742, Republic of Korea;61. Dipartimento di Fisica, Università di Torino, Via P. Giuria 1, I-10125, Torino, Italy;62. INFN, Sezione di Torino, Via P. Giuria 1, I-10125, Torino, Italy;63. Institute of Systems Science, Durban University of Technology, Durban 4000, South Africa;64. Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Valdivia, Chile;65. 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School of Arts and Sciences, Ahmedabad University, Ahmedabad, Gujarat, India;6y. Centre for Theoretical Physics, Jamia Millia Islamia, New Delhi 110025, India;6z. Physics and Astronomy Department, Stony Brook University, Stony Brook, NY 11794, USA;71. Korea Astronomy and Space Science Institute, Daejeon 34055, Republic of Korea;72. School of Physics, Institute for Research in Fundamental Sciences (IPM), P.O.Box 19395-5531, Tehran, Iran;73. Institut d''Astrophysique, 75014 Paris, France;75. Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia 1784, Tsarigradsko shosse 72, Bulgaria;76. Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI, 96822, USA;77. Department of Physics, Lafayette College, Easton, PA 18042, USA;78. Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Cuernavaca, Morelos, 62210, Mexico;79. Institut de Ciències del Cosmos, Universitat de Barcelona, Martí Franquès 1, Barcelona 08028, Spain;710. ICREA, Institució Catalana de Recerca i Estudis Avançats, Passeig Lluís Companys 23, Barcelona 08010, Spain;711. Tsung-Dao Lee Institute (TDLI) & School of Physics and Astronomy, Shanghai Jiao Tong University, Shengrong Road 520, 201210 Shanghai, PR China;712. National Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100012, PR China;713. CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, PR China;714. Department of Physics, Willamette University, Salem, OR 97301, USA;715. Syracuse University, Syracuse, NY, USA;7p. Center for Cosmology and Particle Physics, Department of Physics, New York University, New York, NY 10003, USA;7q. HEPCAT Group, Department of Mathematics and Applied Mathematics, University of Cape Town, 7700, South Africa;7r. Gravitation Astroparticle Physics Amsterdam (GRAPPA), Anton Pannekoek Institute for Astronomy and Institute for Physics, University of Amsterdam, Science Park 904, 1090 GL Amsterdam, the Netherlands;7s. United College of Engineering and Research, Greater Noida - 201310, India;7t. Department of Physics, Liaoning Normal University, Dalian, 116029, PR China;7u. National Astronomy Observatories, Chinese Academy of Sciences, Beijing, 100101, PR China;7v. University of Chinese Academy of Sciences, Beijing, 100049, PR China;7w. Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, D-14476 Potsdam-Golm, Germany |
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| Abstract: | Frequency properties of approximate Bayesian posterior probability intervals are considered for a small bivariate sample with data missing on one variable. For a class of priors, approximations to the posterior distribution based on matching moments of the t distribution are derived, and compared with the true distributions computed numerically. Coverage properties of highest posterior density intervals for three choices of prior are evaluated by simulation, and compared with other solutions. The simulations suggest that a second moment t approximation combined with the Jeffreys' prior for the bivariate distribution provides intervals that are quite well calibrated, in the sense of having approximate or slightly conservative coverage for a wide range of values of the underlying parameters. The use of calibration to select a suitable reference prior seems to have potential for a large number of problems. |
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