{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,19]],"date-time":"2026-09-19T18:40:18Z","timestamp":1789843218667,"version":"4.0.1"},"reference-count":64,"publisher":"Oxford University Press (OUP)","issue":"5","license":[{"start":{"date-parts":[[2018,6,25]],"date-time":"2018-06-25T00:00:00Z","timestamp":1529884800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"name":"Liang\u2019s research","award":["DMS-1612924"],"award-info":[{"award-number":["DMS-1612924"]}]},{"name":"Liang\u2019s research","award":["DMS\/NIGMS R01-GM117597"],"award-info":[{"award-number":["DMS\/NIGMS R01-GM117597"]}]},{"DOI":"10.13039\/100000057","name":"NIGMS","doi-asserted-by":"publisher","award":["R01-GM126089"],"award-info":[{"award-number":["R01-GM126089"]}],"id":[{"id":"10.13039\/100000057","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2018,11,1]]},"abstract":"<jats:title>Summary<\/jats:title><jats:p>Missing data are frequently encountered in high dimensional problems, but they are usually difficult to deal with by using standard algorithms, such as the expectation\u2013maximization algorithm and its variants. To tackle this difficulty, some problem-specific algorithms have been developed in the literature, but there still lacks a general algorithm. This work is to fill the gap: we propose a general algorithm for high dimensional missing data problems. The algorithm works by iterating between an imputation step and a regularized optimization step. At the imputation step, the missing data are imputed conditionally on the observed data and the current estimates of parameters and, at the regularized optimization step, a consistent estimate is found via the regularization approach for the minimizer of a Kullback\u2013Leibler divergence defined on the pseudocomplete data. For high dimensional problems, the consistent estimate can be found under sparsity constraints. The consistency of the averaged estimate for the true parameter can be established under quite general conditions. The algorithm is illustrated by using high dimensional Gaussian graphical models, high dimensional variable selection and a random-coefficient model.<\/jats:p>","DOI":"10.1111\/rssb.12279","type":"journal-article","created":{"date-parts":[[2018,10,26]],"date-time":"2018-10-26T06:31:20Z","timestamp":1540535480000},"page":"899-926","source":"Crossref","is-referenced-by-count":23,"title":["An Imputation\u2013Regularized Optimization Algorithm for High Dimensional Missing Data Problems and Beyond"],"prefix":"10.1093","volume":"80","author":[{"given":"Faming","family":"Liang","sequence":"first","affiliation":[{"name":"Purdue University , West Lafayette , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bochao","family":"Jia","sequence":"additional","affiliation":[{"name":"University of Florida , Gainesville , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jingnan","family":"Xue","sequence":"additional","affiliation":[{"name":"Houzz , Palo Alto , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Qizhai","family":"Li","sequence":"additional","affiliation":[{"name":"Chinese Academy of Sciences , Beijing , People\u2019s Republic of China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ye","family":"Luo","sequence":"additional","affiliation":[{"name":"University of Florida , Gainesville , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2018,6,25]]},"reference":[{"key":"2023022002394610600_","doi-asserted-by":"crossref","first-page":"192","DOI":"10.1111\/j.2517-6161.1974.tb00999.x","article-title":"Spatial interaction and the statistical analysis of lattice systems (with discussion)","volume":"36","author":"Besag","year":"1974","journal-title":"J. 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