{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,4]],"date-time":"2026-09-04T22:42:27Z","timestamp":1788561747655,"version":"build-2803163510"},"reference-count":30,"publisher":"Oxford University Press (OUP)","issue":"3","license":[{"start":{"date-parts":[[2020,5,6]],"date-time":"2020-05-06T00:00:00Z","timestamp":1588723200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,7,1]]},"abstract":"<jats:title>Summary<\/jats:title><jats:p>Two-stage sampling designs are commonly used for household and health surveys. To produce reliable estimators with associated confidence intervals, some basic statistical properties like consistency and asymptotic normality of the Horvitz\u2013Thompson estimator are desirable, along with the consistency of associated variance estimators. These properties have been mainly studied for single-stage sampling designs. In this work, we prove the consistency of the Horvitz\u2013Thompson estimator and of associated variance estimators for a general class of two-stage sampling designs, under mild assumptions. We also study two-stage sampling with a large entropy sampling design at the first stage and prove that the Horvitz\u2013Thompson estimator is asymptotically normally distributed through a coupling argument. When the first-stage sampling fraction is negligible, simplified variance estimators which do not require estimating the variance within the primary sampling units are proposed and shown to be consistent. An application to a panel for urban policy, which is the initial motivation for this work, is also presented.<\/jats:p>","DOI":"10.1111\/rssb.12368","type":"journal-article","created":{"date-parts":[[2020,5,6]],"date-time":"2020-05-06T07:39:06Z","timestamp":1588750746000},"page":"797-815","source":"Crossref","is-referenced-by-count":13,"title":["Inference for Two-Stage Sampling Designs"],"prefix":"10.1093","volume":"82","author":[{"given":"Guillaume","family":"Chauvet","sequence":"first","affiliation":[{"name":"\u00c9cole Nationale de la Statistique et de l\u2019Analyse de l\u2019Information , Rennes , France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Audrey-Anne","family":"Vall\u00e9e","sequence":"additional","affiliation":[{"name":"Universit\u00e9 Laval \u00a0 Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2020,5,6]]},"reference":[{"key":"2023022408363949800_","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1111\/sjos.12243","article-title":"Empirical processes in survey sampling with (conditional) Poisson designs","volume":"44","author":"Bertail","year":"2017","journal-title":"Scand. J. Statist."},{"key":"2023022408363949800_","doi-asserted-by":"crossref","first-page":"1967","DOI":"10.1214\/12-EJS736","article-title":"Approximation of rejective sampling inclusion probabilities and application to high order correlations","volume":"6","author":"Boistard","year":"2012","journal-title":"Electron. J. Statist."},{"key":"2023022408363949800_","doi-asserted-by":"crossref","first-page":"1728","DOI":"10.1214\/16-AOS1507","article-title":"Functional central limit theorems for single-stage sampling designs","volume":"45","author":"Boistard","year":"2017","journal-title":"Ann. Statist."},{"key":"2023022408363949800_","doi-asserted-by":"crossref","first-page":"830","DOI":"10.1111\/j.1467-9469.2011.00766.x","article-title":"Negative dependence in sampling","volume":"39","author":"Br\u00e4nd\u00e9n","year":"2012","journal-title":"Scand. J. Statist."},{"key":"2023022408363949800_","doi-asserted-by":"crossref","first-page":"1026","DOI":"10.1214\/aos\/1015956706","article-title":"Local polynomial regresssion estimators in survey sampling","volume":"28","author":"Breidt","year":"2000","journal-title":"Ann. Statist."},{"key":"2023022408363949800_","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1214\/009053607000000703","article-title":"Endogenous post-stratification in surveys: classifying with a sample-fitted model","volume":"36","author":"Breidt","year":"2008","journal-title":"Ann. Statist."},{"key":"2023022408363949800_","doi-asserted-by":"crossref","first-page":"822","DOI":"10.1080\/01621459.2015.1058264","article-title":"Nonparametric variance estimation under fine stratification: an alternative to collapsed strata","volume":"111","author":"Breidt","year":"2016","journal-title":"J. Am. Statist. Ass."},{"key":"2023022408363949800_","first-page":"651","article-title":"On variance estimation for the French master sample","volume":"27","author":"Chauvet","year":"2011","journal-title":"J. Off. Statist."},{"key":"2023022408363949800_","doi-asserted-by":"crossref","first-page":"1320","DOI":"10.3150\/11-BEJ380","article-title":"On a characterization of ordered pivotal sampling","volume":"18","author":"Chauvet","year":"2012","journal-title":"Bernoulli"},{"key":"2023022408363949800_","doi-asserted-by":"crossref","first-page":"2484","DOI":"10.1214\/15-AOS1348","article-title":"Coupling methods for multistage sampling","volume":"43","author":"Chauvet","year":"2015","journal-title":"Ann. Statist."},{"key":"2023022408363949800_","volume-title":"Sampling Techniques","author":"Cochran","year":"1977"},{"key":"2023022408363949800_","doi-asserted-by":"crossref","first-page":"234","DOI":"10.1007\/s13571-014-0083-x","article-title":"On the estimation of the distribution function of a finite population under high entropy sampling designs, with applications","volume":"76","author":"Conti","year":"2014","journal-title":"Sankhya"},{"key":"2023022408363949800_","doi-asserted-by":"crossref","DOI":"10.1017\/9781316779422","volume-title":"Random Graphs and Complex Networks","author":"van Der Hofstad","year":"2016"},{"key":"2023022408363949800_","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1093\/biomet\/85.1.89","article-title":"Unequal probability sampling without replacement through a splitting method","volume":"85","author":"Deville","year":"1998","journal-title":"Biometrika"},{"key":"2023022408363949800_","volume-title":"Contribution to Statistics (J. Hajek Memorial Volume)","author":"Dupakova","year":"1975"},{"key":"2023022408363949800_","volume-title":"Sampling Statistics","author":"Fuller","year":"2011"},{"key":"2023022408363949800_","article-title":"Pour comprendre Le Recensement de la population","volume-title":"S\u00e9r. Insee M\u00e9th","author":"Godinot","year":"2005"},{"key":"2023022408363949800_","doi-asserted-by":"crossref","first-page":"1491","DOI":"10.1214\/aoms\/1177700375","article-title":"Asymptotic theory of rejective sampling with varying probabilities from a finite population","volume":"35","author":"H\u00e1jek","year":"1964","journal-title":"Ann. Statist."},{"key":"2023022408363949800_","volume-title":"Sampling from a Finite Population","author":"H\u00e1jek","year":"1981"},{"key":"2023022408363949800_","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1214\/aoms\/1177731356","article-title":"On the theory of sampling from finite populations","volume":"14","author":"Hansen","year":"1943","journal-title":"Ann. Math. Statist."},{"key":"2023022408363949800_","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1080\/01621459.1982.10477770","article-title":"Survey design under the regression superpopulation model","volume":"77","author":"Isaki","year":"1982","journal-title":"J. Am. Statist. Ass."},{"key":"2023022408363949800_","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1002\/cjs.11339","article-title":"Statistical inference using generalized linear mixed models under informative cluster sampling","volume":"45","author":"Kim","year":"2017","journal-title":"Can. J. Statist."},{"key":"2023022408363949800_","doi-asserted-by":"crossref","first-page":"1010","DOI":"10.1214\/aos\/1176345580","article-title":"Inference from stratified samples: properties of the linearization, jackknife and balanced repeated replication methods","volume":"9","author":"Krewski","year":"1981","journal-title":"Ann. Statist."},{"key":"2023022408363949800_","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1007\/BF00340058","article-title":"Asymptotic normality for two-stage sampling from a finite population","volume":"81","author":"Ohlsson","year":"1989","journal-title":"Probab. Theory Reltd Flds"},{"key":"2023022408363949800_","first-page":"489","volume-title":"Handbook of Statistics","author":"Pr\u00e1\u0161kov\u00e1","year":"2009"},{"key":"2023022408363949800_","doi-asserted-by":"crossref","first-page":"234","DOI":"10.1111\/j.1467-842X.1982.tb00829.x","article-title":"On the convergence of the Horvitz-Thompson estimator","volume":"24","author":"Robinson","year":"1982","journal-title":"Aust. J. Statist."},{"key":"2023022408363949800_","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1093\/biomet\/54.3-4.499","article-title":"On sampling without replacement with unequal probabilities of selection","volume":"54","author":"Sampford","year":"1967","journal-title":"Biometrika"},{"key":"2023022408363949800_","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-4378-6","volume-title":"Model Assisted Survey Sampling","author":"S\u00e4rndal","year":"1992"},{"key":"2023022408363949800_","volume-title":"Sampling Algorithms","author":"Till\u00e9","year":"2006"},{"key":"2023022408363949800_","doi-asserted-by":"crossref","first-page":"556","DOI":"10.1111\/j.2517-6161.1968.tb00755.x","article-title":"An exact \u03c0ps sampling scheme\u2013generalization of a method of Hanurav","volume":"30","author":"Vijayan","year":"1968","journal-title":"J. R. Statist. Soc."}],"container-title":["Journal of the Royal Statistical Society Series B: Statistical Methodology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1111%2Frssb.12368","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/82\/3\/797\/49323635\/jrsssb_82_3_797.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/82\/3\/797\/49323635\/jrsssb_82_3_797.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,5]],"date-time":"2024-08-05T10:03:19Z","timestamp":1722852199000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/article\/82\/3\/797\/7056034"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,5,6]]},"references-count":30,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2020,5,6]]},"published-print":{"date-parts":[[2020,7,1]]}},"URL":"https:\/\/doi.org\/10.1111\/rssb.12368","relation":{},"ISSN":["1369-7412","1467-9868"],"issn-type":[{"value":"1369-7412","type":"print"},{"value":"1467-9868","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2020,7]]},"published":{"date-parts":[[2020,5,6]]}}}