{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,30]],"date-time":"2023-10-30T21:14:46Z","timestamp":1698700486006},"reference-count":17,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2010,10,18]],"date-time":"2010-10-18T00:00:00Z","timestamp":1287360000000},"content-version":"unspecified","delay-in-days":6713,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Econom. Theory"],"published-print":{"date-parts":[[1992,6]]},"abstract":"Positive definiteness of income effect matrices provides a sufficient condition for the law of demand<\/jats:italic> to hold. Given cross section household expenditure data, empirical evidence for the law of demand can be obtained by estimating such matrices. H\u00e4rdle, Hildenbrand, and Jerison used the bootstrap method to simulate the distribution of the smallest eigenvalue of random matrices and to test their positive definiteness. Here, theoretical aspects of this bootstrap test of positive definiteness are considered. The asymptotic distribution of the smallest eigenvalue , of the matrix estimate is obtained. This theory applies generally to symmetric, asymptotically normal random matrices. A bootstrap approximation to the distribution of is shown to converge in probability to the asymptotic distribution of . The bootstrap test is illustrated using British family expenditure survey data.<\/jats:p>","DOI":"10.1017\/s0266466600012809","type":"journal-article","created":{"date-parts":[[2010,10,18]],"date-time":"2010-10-18T05:12:04Z","timestamp":1287378724000},"page":"276-292","source":"Crossref","is-referenced-by-count":9,"title":["A Bootstrap Test for Positive Definiteness of Income Effect Matrices"],"prefix":"10.1017","volume":"8","author":[{"given":"Wolfgang","family":"H\u00e4rdle","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jeffrey D.","family":"Hart","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2010,10,18]]},"reference":[{"key":"S0266466600012809_ref007","doi-asserted-by":"publisher","DOI":"10.1080\/01621459.1990.10474974"},{"key":"S0266466600012809_ref016","volume-title":"The Algebraic Eigenvalue Problem","author":"Wilkinson","year":"1965"},{"key":"S0266466600012809_ref006","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176350933"},{"key":"S0266466600012809_ref008","doi-asserted-by":"crossref","unstructured":"8. H\u00e4rdle W. Applied Nonparametric Regression. Econometric Society Monograph Series 19, Cambridge University Press, 1990.","DOI":"10.1017\/CCOL0521382483"},{"key":"S0266466600012809_ref013","volume-title":"Introduction to the Theory of Nonparametric Statistics","author":"Randies","year":"1979"},{"key":"S0266466600012809_ref003","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176350170"},{"key":"S0266466600012809_ref001","doi-asserted-by":"publisher","DOI":"10.1093\/biomet\/74.3.457"},{"key":"S0266466600012809_ref004","unstructured":"4.Family Expenditure Survey, Annual Base Tapes (1968-1983) Department of Employment. Statistics Division, Her Majesty's Stationery Office, London 1968\u20131983. The data utilized in this hook were made available by the ESRC Data Archive at the Universtiy of Essex."},{"key":"S0266466600012809_ref002","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176346579"},{"key":"S0266466600012809_ref005","volume-title":"Matrices With Applications in Statistics","author":"Graybill","year":"1983"},{"key":"S0266466600012809_ref009","first-page":"986","article-title":"Investigating smooth multiple regression by the method of average derivatives","volume":"84","author":"H\u00e4rdle","year":"1989","journal-title":"Journal of the American Statistical Association"},{"key":"S0266466600012809_ref010","doi-asserted-by":"publisher","DOI":"10.2307\/2938277"},{"key":"S0266466600012809_ref011","volume-title":"Asymptotically optimal kernels for average derivative estimation","author":"Mammitzsch","year":"1989"},{"key":"S0266466600012809_ref012","doi-asserted-by":"publisher","DOI":"10.1002\/9780470316559"},{"key":"S0266466600012809_ref014","doi-asserted-by":"publisher","DOI":"10.1002\/9780470316481"},{"key":"S0266466600012809_ref015","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176349548"},{"key":"S0266466600012809_ref017","unstructured":"17. XploRe 2.0 XploRe A Computing Environment for exploratory Regression and Data Analysis. Available from CORE (1990)."}],"container-title":["Econometric Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0266466600012809","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,16]],"date-time":"2019-05-16T17:39:42Z","timestamp":1558028382000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0266466600012809\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1992,6]]},"references-count":17,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1992,6]]}},"alternative-id":["S0266466600012809"],"URL":"https:\/\/doi.org\/10.1017\/s0266466600012809","relation":{},"ISSN":["0266-4666","1469-4360"],"issn-type":[{"value":"0266-4666","type":"print"},{"value":"1469-4360","type":"electronic"}],"subject":[],"published":{"date-parts":[[1992,6]]}}}