{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,6]],"date-time":"2026-10-06T21:52:19Z","timestamp":1791323539172,"version":"4.3.1"},"reference-count":57,"publisher":"Institute of Mathematical Statistics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Ann. Statist."],"published-print":{"date-parts":[[2026,6,1]]},"abstract":"<jats:p>This paper develops a novel two-step estimating procedure for heavy-tailed AR models with nonzero median GARCH-type noises, allowing for time-varying volatility. We first establish the self-weighted quantile regression estimator (SQE) across all quantile levels \u03c4\u2208(0,1) for the AR parameters \u03b80. We show that the SQE, less a bias, converges weakly to a Gaussian process at a rate of n\u22121\/2. The bias is zero if and only if \u03c4 equals \u03c40, the probability that the noise is less than zero. Based on the SQE, we propose an approach to estimate \u03c40 in the second step and feed the estimated \u03c40 back into the SQE to estimate \u03b80. Both the estimated \u03c40 and \u03b80 are shown to be consistent and asymptotically normal. A random weighting bootstrap method is developed to approximate the complex distribution. The problem we study is nonstandard because \u03c40 may not be identifiable in conventional quantile regression, and the usual methods cannot verify the existence of the SQE bias. Unlike existing procedures for heavy-tailed time series, our method does not require prior information about the symmetry, tail index, or the parametric form of the noise, nor does it require classical identification conditions, such as zero-mean or zero-median.<\/jats:p>","DOI":"10.1214\/25-aos2610","type":"journal-article","created":{"date-parts":[[2026,6,25]],"date-time":"2026-06-25T13:51:05Z","timestamp":1782395465000},"page":"1425-1450","source":"Crossref","is-referenced-by-count":0,"title":["A two-step estimating approach for heavy-tailed AR models with nonzero median GARCH-type noises"],"prefix":"10.1214","volume":"54","author":[{"given":"Rui","family":"She","sequence":"first","affiliation":[{"name":"Center of Statistical Research, School of Statistics and Data Science, The Southwestern University of Finance and Economics"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Linlin","family":"Dai","sequence":"additional","affiliation":[{"name":"Center of Statistical Research, School of Statistics and Data Science, The Southwestern University of Finance and Economics"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4232-7744","authenticated-orcid":false,"given":"Shiqing","family":"Ling","sequence":"additional","affiliation":[{"name":"Department of Mathematics, The Hong Kong University of Science and Technology"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"14","doi-asserted-by":"publisher","unstructured":"<small>Francq, C.<\/small> and <small>Zako\u00efan, J.-M.<\/small> (2010). <i>GARCH Models<\/i>: <i>Structure, Statistical Inference and Financial Applications<\/i>. 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