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Subsequently, we propose a novel functional projection approach based on GLSS for hypothesis testing on eigenspaces of \u201cpopulation-spiked\u201d covariance matrices, showcasing a universality phenomenon in the magnitude of the spikes. The theoretical accuracy of our results established for GLSS and the advantages of the newly suggested testing procedure are demonstrated through various numerical studies.<\/jats:p>","DOI":"10.1214\/25-aos2601","type":"journal-article","created":{"date-parts":[[2026,4,2]],"date-time":"2026-04-02T13:05:41Z","timestamp":1775135141000},"page":"1028-1052","source":"Crossref","is-referenced-by-count":0,"title":["Generalized linear spectral statistics of high-dimensional sample covariance matrices and its applications"],"prefix":"10.1214","volume":"54","author":[{"given":"Yanlin","family":"Hu","sequence":"first","affiliation":[{"name":"Department of Statistics & Finance, School of Management, University of Science and Technology of China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Qing","family":"Yang","sequence":"additional","affiliation":[{"name":"Department of Statistics & Finance, School of Management, University of Science and Technology of China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiao","family":"Han","sequence":"additional","affiliation":[{"name":"Department of Statistics & Finance, School of Management, University of Science and Technology of China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"1","doi-asserted-by":"publisher","unstructured":"BAI, Z.<\/small>, JIANG, D.<\/small>, YAO, J.-F.<\/small> and ZHENG, S.<\/small> (2009). 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