{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T19:53:31Z","timestamp":1771012411658,"version":"3.50.1"},"reference-count":30,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2009,2,11]],"date-time":"2009-02-11T00:00:00Z","timestamp":1234310400000},"content-version":"unspecified","delay-in-days":6647,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Econom. Theory"],"published-print":{"date-parts":[[1990,12]]},"abstract":"<jats:p>During the Oxford Conference of the Econometric Society in 1936, Ragnar Frisch proposed a problem of characterization of distributions based on the property of linear regression of one linear function of random variables on the other. This problem has been solved, partially by Allen [1], and then completely by Rao [24,25], Fix [7], and Laha [13] relaxing the conditions imposed on the component random variables. The purpose of this paper is to solve the above mentioned problem for the multivariate case, characterizing multivariate distributions based on the multivariate linear regression of one linear function of not necessarily i.i.d. random vectors with matrix coefficients on the other. We make some mild assumptions concerning the component random vectors and the related constant matrices. It is shown that the property of multivariate linear regression yields a system of partial differential equations (p.d.e.'s) satisfied by the characteristic functions of the component random vectors. A general solution of this system of p.d.e.'s is given by certain functional forms. Special cases of the general solution give characterizations of the \u201cmultivariate generalized stable laws\u201d and the multivariate semistable laws, and a method is presented to characterize the multivariate stable laws.<\/jats:p>","DOI":"10.1017\/s0266466600005454","type":"journal-article","created":{"date-parts":[[2009,3,10]],"date-time":"2009-03-10T15:44:48Z","timestamp":1236699888000},"page":"445-458","source":"Crossref","is-referenced-by-count":6,"title":["Functional Forms of Characteristic Functions and Characterizations of Multivariate Distributions"],"prefix":"10.1017","volume":"6","author":[{"given":"Yasuko","family":"Chikuse","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2009,2,11]]},"reference":[{"key":"S0266466600005454_ref006","first-page":"259","article-title":"A characterization of the normal law on Hilbert space","volume":"31","author":"Eaton","year":"1969","journal-title":"Sankhya A"},{"key":"S0266466600005454_ref001","first-page":"60","article-title":"A theorem concerning the linearity of regression","volume":"2","author":"Allen","year":"1938","journal-title":"Statistical Research Memorandum"},{"key":"S0266466600005454_ref018","volume-title":"Applications of characteristic functions","author":"Lukacs","year":"1964"},{"key":"S0266466600005454_ref028","first-page":"1","volume-title":"Statistical distributions in scientific work","volume":"3","author":"Rao","year":"1975"},{"key":"S0266466600005454_ref023","first-page":"125","article-title":"Some results on characterizations of the normal and generalized stable laws","volume":"30","author":"Ramachandran","year":"1968","journal-title":"Sankhya A"},{"key":"S0266466600005454_ref003","volume-title":"Methods of mathematical physics. 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Seminar on Probability, Vol. 52 (In Japanese), 1981."},{"key":"S0266466600005454_ref024","doi-asserted-by":"publisher","DOI":"10.2307\/1905484"},{"key":"S0266466600005454_ref026","volume-title":"Linear statistical inference and its applications","author":"Rao","year":"1968"},{"key":"S0266466600005454_ref027","first-page":"163","volume-title":"Proceedings of the international congress of mathematics","author":"Rao","year":"1974"},{"key":"S0266466600005454_ref030","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1969-0238365-3"},{"key":"S0266466600005454_ref014","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177706968"},{"key":"S0266466600005454_ref004","doi-asserted-by":"publisher","DOI":"10.2307\/1401511"}],"container-title":["Econometric Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0266466600005454","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,18]],"date-time":"2019-05-18T19:39:42Z","timestamp":1558208382000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0266466600005454\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1990,12]]},"references-count":30,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1990,12]]}},"alternative-id":["S0266466600005454"],"URL":"https:\/\/doi.org\/10.1017\/s0266466600005454","relation":{},"ISSN":["0266-4666","1469-4360"],"issn-type":[{"value":"0266-4666","type":"print"},{"value":"1469-4360","type":"electronic"}],"subject":[],"published":{"date-parts":[[1990,12]]}}}