{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T08:05:04Z","timestamp":1770969904186,"version":"3.50.1"},"reference-count":0,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2002,5,17]],"date-time":"2002-05-17T00:00:00Z","timestamp":1021593600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Econom. Theory"],"published-print":{"date-parts":[[2002,8]]},"abstract":"<jats:p>Often neither the exact density nor the exact cumulative \ndistribution function (c.d.f.) of a statistic of interest is \navailable in the statistics and econometrics literature (e.g., \nthe maximum likelihood estimator of the autocorrelation coefficient \nin a simple Gaussian AR(1) model with zero start-up value). \nIn other cases the exact c.d.f. of a statistic of interest is \nvery complicated despite the statistic being \u201csimple\u201d \n(e.g., the circular serial correlation coefficient, or a quadratic \nform of a vector uniformly distributed over the unit \n<jats:italic>n<\/jats:italic>-sphere). The first part of the paper tries to explain \nwhy this is the case by studying the analytic properties of \nthe c.d.f. of a statistic under very general assumptions. \nDifferential geometric considerations show that there can be \npoints where the c.d.f. of a given statistic is not analytic, \nand such points do not depend on the parameters of the model \nbut only on the properties of the statistic itself. The second \npart of the paper derives the exact c.d.f. of a ratio of quadratic \nforms in normal variables, and for the first time a closed form \nsolution is found. These results are then specialized to the \nmaximum likelihood estimator of the autoregressive parameter \nin a Gaussian AR(1) model with zero start-up value, which is \nshown to have precisely those properties highlighted in the \nfirst part of the paper.<\/jats:p>","DOI":"10.1017\/s0266466602184015","type":"journal-article","created":{"date-parts":[[2002,11,14]],"date-time":"2002-11-14T17:02:47Z","timestamp":1037293367000},"page":"823-852","source":"Crossref","is-referenced-by-count":18,"title":["THE EXACT CUMULATIVE DISTRIBUTION FUNCTION OF A RATIO OF \nQUADRATIC FORMS IN NORMAL VARIABLES, WITH APPLICATION TO THE \nAR(1) MODEL"],"prefix":"10.1017","volume":"18","author":[{"given":"G.","family":"Forchini","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2002,5,17]]},"container-title":["Econometric Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0266466602184015","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,1]],"date-time":"2019-04-01T15:28:28Z","timestamp":1554132508000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0266466602184015\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,5,17]]},"references-count":0,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2002,8]]}},"alternative-id":["S0266466602184015"],"URL":"https:\/\/doi.org\/10.1017\/s0266466602184015","relation":{},"ISSN":["0266-4666","1469-4360"],"issn-type":[{"value":"0266-4666","type":"print"},{"value":"1469-4360","type":"electronic"}],"subject":[],"published":{"date-parts":[[2002,5,17]]}}}