{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,25]],"date-time":"2026-04-25T16:59:19Z","timestamp":1777136359133,"version":"3.51.4"},"reference-count":41,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,9,9]],"date-time":"2022-09-09T00:00:00Z","timestamp":1662681600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["Grants 11571219"],"award-info":[{"award-number":["Grants 11571219"]}]},{"name":"National Natural Science Foundation of China","award":["2017FR044"],"award-info":[{"award-number":["2017FR044"]}]},{"name":"Shanghai Research Center for Data Science and Decision Technology","award":["Grants 11571219"],"award-info":[{"award-number":["Grants 11571219"]}]},{"name":"Shanghai Research Center for Data Science and Decision Technology","award":["2017FR044"],"award-info":[{"award-number":["2017FR044"]}]},{"name":"Startup Foundation for Talents in Zhejiang Agriculture and Forestry University","award":["Grants 11571219"],"award-info":[{"award-number":["Grants 11571219"]}]},{"name":"Startup Foundation for Talents in Zhejiang Agriculture and Forestry University","award":["2017FR044"],"award-info":[{"award-number":["2017FR044"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we address a class of heterogeneous spatial autoregressive models with all n(n\u22121) spatial coefficients taking m distinct true values, where m is independent of the sample size n, and we establish asymptotic properties of the maximum likelihood estimator and the quasi-maximum likelihood estimator for all parameters in the class of models, extending Lee\u2019s work (2004). The rates of convergence of those estimators depend on the features of values taken by elements of the spatial weights matrix in this model. Under the situations where, based on the values of the weights, each individual will not only influence a few neighbors but also be influenced by only a few neighbors, the estimator can enjoy an n-rate of convergence and be asymptotically normal. However, when each individual can influence many neighbors or can be influenced by many neighbors and their number does not exceed o(n), singularity of the information matrix may occur, and various components of the estimators may have different (usually lower than n) rates of convergence. An inconsistent estimator is provided if some important assumptions are violated. Finally, simulation studies demonstrate that the finite sample performances of maximum likelihood estimators are good.<\/jats:p>","DOI":"10.3390\/sym14091894","type":"journal-article","created":{"date-parts":[[2022,9,13]],"date-time":"2022-09-13T22:37:28Z","timestamp":1663108648000},"page":"1894","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Asymptotic Properties of Quasi-Maximum Likelihood Estimators for Heterogeneous Spatial Autoregressive Models"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2626-3354","authenticated-orcid":false,"given":"Feng","family":"Qiu","sequence":"first","affiliation":[{"name":"School of Mathematics and Computer Science, Zhejiang Agriculture and Forestry University, Hangzhou 311300, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hao","family":"Ding","sequence":"additional","affiliation":[{"name":"Samoyed Cloud Technology Group Holdings Limited, Shanghai 200124, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jianhua","family":"Hu","sequence":"additional","affiliation":[{"name":"School of Statistics and Management, Shanghai University of Finance and Economics, Shanghai 200433, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1162\/003465300558641","article-title":"Applying the generalized-moments estimation approach to spatial problems involving microlevel data","volume":"82","author":"Bell","year":"2000","journal-title":"Rev. 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