[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-117584-en":3,"doc-seo-117584-105":30,"detail-sidebar-cat-0-en-105":83},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},117584,1374391974468,"Eden","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Statistical Inference for Noisy Incomplete Binary Matrix","The work addresses statistical inference for noisy incomplete binary matrices, often called 1-bit matrices, focusing beyond traditional point estimation and prediction used in categorical matrix completion. Under a nonlinear factor analysis model with a logit link, the study provides a point estimator and proves its asymptotic normality. A flexible missing-entry design is adopted, avoiding restrictive random sampling assumptions. Under suitable conditions, the estimator is statistically efficient, attaining the asymptotic Cramér-Rao lower bound. Applications include linking different educational test forms and comparing U.S. Senate roll-call voting records across years.","Statistical Inference for Noisy Incomplete Binary Matrix  \nYunxiao Chen [y.chen186@lse.ac.uk](y.chen186@lse.ac.uk)  \nDepartment of Statistics  \nLondon School of Economics and Political Science London WC2A 2AE, UK  \nChengcheng Li [lccvic@umich.edu](lccvic@umich.edu)  \nJing Ouyang [jingoy@umich.edu](jingoy@umich.edu)  \nGongjun Xu [gongjun@umich.edu](gongjun@umich.edu)  \nDepartment of Statistics University of Michigan Ann Arbor, MI 48109, USA  \nEditor: Ali Shojaie  \nAbstract  \nWe consider the statistical inference for noisy incomplete binary (or 1-bit) matrix. Despite the importance of uncertainty quanti􀀌cation to matrix completion, most of the categorical matrix completion literature focuses on point estimation and prediction. This paper moves one step further toward statistical inference for binary matrix completion. Under a popular nonlinear factor analysis model, we obtain a point estimator and derive its asymptotic normality. Moreover, our analysis adopts a 􀀍exible missing-entry design that does not require a random sampling scheme as required by most of the existing asymptotic results for matrix completion. Under reasonable conditions, the proposed estimator is statistically e􀀎cient and optimal in the sense that the Cramer-Rao lower bound is achieved asymptotically for the model parameters. Two applications are considered, including (1) linking two forms of an educational test and (2) linking the roll call voting records from multiple years in the United States Senate. The 􀀌rst application enables the comparison between examinees who took di􀀋erent test forms, and the second application allows us to compare the liberal-conservativeness of senators who did not serve in the Senate at the same time.  \nKeywords: 1-bit matrix; Matrix completion; Binary data; Asymptotic normality; Nonlinear latent variable model.  \n1. Introduction  \nNoisy low-rank matrix completion is concerned with the recovery of a low-rank matrix when only a fraction of noisy entries are observed. This topic has received much attention as a result of its vast applications in practical contexts such as collaborative 􀀌ltering (Goldberg et al., 1992), system identi􀀌cation (Liu and Vandenberghe, 2010) and sensor localization (Biswas et al., 2006) . While the majority of the literature considers the completion of real-valued observations (Cand􀀒es and Recht, 2009; Cand􀀒es and Tao, 2010; Keshavan et al. , 2010; Koltchinskii et al., 2011; Negahban and Wainwright, 2012; Chen et al., 2020), many practical problems involve categorical-valued matrices, such as the famous Net􀀍ix challenge. Several works have been done on matrix completion involving categorical variables, includ-  \n􀀍c2023 Yunxiao Chen, Chengcheng Li, Jing Ouyang and Gongjun Xu.  \nLicense: CC-BY 4.0, see [https://creativecommons.org/licenses/by/4.0/](https://creativecommons.org/licenses/by/4.0/. Attribution)[. Attribution](https://creativecommons.org/licenses/by/4.0/. Attribution) requirements are provided  \nat [http://jmlr.org/papers/v24/22-0214.html](http://jmlr.org/papers/v24/22-0214.html).  \nChen, Li, Ouyang and Xu  \ning Davenport et al. (2014) and Bhaskar and Javanmard (2015) for 1-bit matrix whose entries take binary values, and Klopp et al. (2015) and Bhaskar (2016) for categorical matrix, and Chen and Li (2022) for matrix of binary, count, and continuous variables. In these works, low-dimensional nonlinear probabilistic models are assumed.  \nDespite the importance of uncertainty quanti􀀌cation to matrix completion, most of the matrix completion literature focuses on point estimation and prediction, while statistical inference has received attention only recently. Speci􀀌cally, Chen et al. (2019) and Xia and Yuan (2021) considered statistical inference under the linear models and derived asymptotic normality results. The statistical inference for categorical matrices is more challenging due to the involvement of nonlinear models. To our best knowledge, no work has been done to provide statistical inferen","cbCaigonmRoq9gJC","https://ap.wps.com/l/cbCaigonmRoq9gJC","pdf",935339,1,66,"English","en",105,"# Introduction\n## Background and motivation\n## Nonlinear factor analysis model and challenges\n## Scope and contributions\n## Applications","[{\"question\":\"What efficiency guarantee does the estimator provide?\",\"answer\":\"Under reasonable conditions, the estimator is statistically efficient and achieves the asymptotic Cramér-Rao lower bound for the model parameters.\"}]","Statistical Inference for Noisy Incomplete Binary Matrix | PDF",1785677102,166,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":78,"head_meta":80,"extra_data":82,"updated_unix":28},"statistical-inference-for-noisy-incomplete-binary-matrix","",{"@graph":36,"@context":77},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/statistical-inference-for-noisy-incomplete-binary-matrix/117584/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-02",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71],{"name":72,"@type":73,"acceptedAnswer":74},"What efficiency guarantee does the estimator provide?","Question",{"text":75,"@type":76},"Under reasonable conditions, the estimator is statistically efficient and achieves the asymptotic Cramér-Rao lower bound for the model parameters.","Answer","https://schema.org",{"og:url":52,"og:type":79,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":81,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":84},[85,89,93,97,102,107,112,115,120,123,127],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":86,"show_sort_weight":87,"slug":88},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":90,"show_sort_weight":91,"slug":92},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Exam",70,"exam",{"id":98,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},5,"Comic",60,"comic",{"id":103,"doc_module":4,"doc_module_name":46,"category_name":104,"show_sort_weight":105,"slug":106},6,"Technology",50,"technology",{"id":108,"doc_module":4,"doc_module_name":46,"category_name":109,"show_sort_weight":110,"slug":111},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":113,"slug":114},30,"research-report",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":46,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":98,"slug":130},19,"General","general"]