{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,7]],"date-time":"2026-02-07T01:01:35Z","timestamp":1770426095943,"version":"3.49.0"},"reference-count":0,"publisher":"Cambridge University Press (CUP)","issue":"5","license":[{"start":{"date-parts":[[2002,7,17]],"date-time":"2002-07-17T00:00:00Z","timestamp":1026864000000},"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":[[2002,10]]},"abstract":"<jats:p>It is well known that a one-step scoring estimator that starts \nfrom any <jats:italic>N<\/jats:italic><jats:sup>1\/2<\/jats:sup>-consistent estimator has the same \nfirst-order asymptotic efficiency as the maximum likelihood estimator. \nThis paper extends this result to <jats:italic>k<\/jats:italic>-step estimators and \ntest statistics for <jats:italic>k<\/jats:italic> \u2265 1, higher order asymptotic \nefficiency, and general extremum estimators and test statistics.<\/jats:p><jats:p>The paper shows that a <jats:italic>k<\/jats:italic>-step estimator has the same \nhigher order asymptotic efficiency, to any given order, as the \nextremum estimator toward which it is stepping, provided (i) \n<jats:italic>k<\/jats:italic> is sufficiently large, (ii) some smoothness and moment \nconditions hold, and (iii) a condition on the initial estimator \nholds.<\/jats:p><jats:p>For example, for the Newton\u2013Raphson <jats:italic>k<\/jats:italic>-step \nestimator based on an initial estimator in a wide class, we \nobtain asymptotic equivalence to integer order <jats:italic>s<\/jats:italic> provided \n2<jats:sup><jats:italic>k<\/jats:italic><\/jats:sup> \u2265 <jats:italic>s<\/jats:italic> + 1. Thus, for <jats:italic>k<\/jats:italic> = 1, 2, \nand 3, one obtains asymptotic equivalence to first, third, and seventh \norders, respectively. This means that the maximum differences between \nthe probabilities that the (<jats:italic>N<\/jats:italic><jats:sup>1\/2<\/jats:sup>-normalized) \n<jats:italic>k<\/jats:italic>-step and extremum estimators lie in any convex set \nare <jats:italic>o<\/jats:italic>(1), o(<jats:italic>N<\/jats:italic><jats:sup>\u22123\/2<\/jats:sup>), and \n<jats:italic>o<\/jats:italic>(<jats:italic>N<\/jats:italic><jats:sup>\u22123<\/jats:sup>), respectively.<\/jats:p>","DOI":"10.1017\/s0266466602185021","type":"journal-article","created":{"date-parts":[[2002,11,14]],"date-time":"2002-11-14T22:03:05Z","timestamp":1037311385000},"page":"1040-1085","source":"Crossref","is-referenced-by-count":12,"title":["EQUIVALENCE OF THE HIGHER ORDER ASYMPTOTIC EFFICIENCY OF \n<i>k<\/i>-STEP AND EXTREMUM STATISTICS"],"prefix":"10.1017","volume":"18","author":[{"given":"Donald W.K.","family":"Andrews","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2002,7,17]]},"container-title":["Econometric Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0266466602185021","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,7]],"date-time":"2019-05-07T00:32:48Z","timestamp":1557189168000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0266466602185021\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,7,17]]},"references-count":0,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2002,10]]}},"alternative-id":["S0266466602185021"],"URL":"https:\/\/doi.org\/10.1017\/s0266466602185021","relation":{},"ISSN":["0266-4666","1469-4360"],"issn-type":[{"value":"0266-4666","type":"print"},{"value":"1469-4360","type":"electronic"}],"subject":[],"published":{"date-parts":[[2002,7,17]]}}}