{"paper":{"title":"On Stability and Decomposition of Sample Quantiles under Heavy-Tailed Distributions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The error in a sample quantile for an estimated heavy-tailed projection decomposes into a direction-induced population shift, a fixed-direction empirical fluctuation, and a Bahadur remainder.","cross_cats":["cs.LG","math.ST","stat.TH"],"primary_cat":"stat.ML","authors_text":"Choudur Lakshminarayan","submitted_at":"2026-05-18T13:19:29Z","abstract_excerpt":"We study sample quantiles of distributions indexed by estimated parameters, with a on Value-at-Risk related to linear projections of financial returns that whose underlying probability law is heavy-tailed. In this setting, the projection direction and the empirical quantile threshold are estimated from the data, so the standard Bahadur representation under a fixed distribution does not separate the distinct sources of instability. A canonical starting point is Bahadur's representation, which expresses the sample quantile through the empirical distribution function plus a remainder term \\cite{b"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"The difference between the empirical quantile computed using the estimated projection direction and the population quantile at the reference direction can be decomposed as hat q_alpha(hat w) - q_alpha(w_0) = D1 + D2 + D3, where D1 measures population quantile movement from perturbing the projection direction, D2 measures empirical quantile fluctuation with direction held fixed, and D3 is the Bahadur-type remainder.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the Q-Q orthogonality formulation cleanly separates projection-direction effects from quantile-threshold effects without requiring the global uniform-convergence condition that empirical-process theory normally imposes on local quantile-stability problems.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Introduces Q-Q orthogonality to decompose the difference between empirical and population quantiles into direction-induced population movement, fixed-direction empirical fluctuation, and Bahadur remainder.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The error in a sample quantile for an estimated heavy-tailed projection decomposes into a direction-induced population shift, a fixed-direction empirical fluctuation, and a Bahadur remainder.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"ca1182e91b004d0dadcde4893b2c5858c2a7a22c6e7f3d019a561a6e6819193b"},"source":{"id":"2605.18370","kind":"arxiv","version":1},"verdict":{"id":"406f7cd5-63e6-4231-8117-1d6bae4f230a","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T23:58:16.710598Z","strongest_claim":"The difference between the empirical quantile computed using the estimated projection direction and the population quantile at the reference direction can be decomposed as hat q_alpha(hat w) - q_alpha(w_0) = D1 + D2 + D3, where D1 measures population quantile movement from perturbing the projection direction, D2 measures empirical quantile fluctuation with direction held fixed, and D3 is the Bahadur-type remainder.","one_line_summary":"Introduces Q-Q orthogonality to decompose the difference between empirical and population quantiles into direction-induced population movement, fixed-direction empirical fluctuation, and Bahadur remainder.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the Q-Q orthogonality formulation cleanly separates projection-direction effects from quantile-threshold effects without requiring the global uniform-convergence condition that empirical-process theory normally imposes on local quantile-stability problems.","pith_extraction_headline":"The error in a sample quantile for an estimated heavy-tailed projection decomposes into a direction-induced population shift, a fixed-direction empirical fluctuation, and a Bahadur remainder."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.18370/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"citation_quote_validity","ran_at":"2026-05-19T23:50:05.570997Z","status":"skipped","version":"0.1.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T23:33:30.009046Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"external_links","ran_at":"2026-05-19T23:31:49.158518Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T23:21:58.777523Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"758c43f588c412f58c59e30f5ad3a731e694afcbc30f4d423a1f122047557c1f"},"references":{"count":17,"sample":[{"doi":"","year":1966,"title":"Bahadur, R. R. (1966). A note on quantiles in large samples.Annals of Mathematical Statistics, 37, 577–580","work_id":"2ff2569c-e1f7-467e-a082-00052488b79f","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1967,"title":"Kiefer, J. (1967). On Bahadur’s representation of sample quantiles.Annals of Mathematical Statis- tics,38, 1323–1342","work_id":"d06ade93-1a84-4d0a-80b9-5848d77829de","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1984,"title":"(1984).Convergence of Stochastic Processes","work_id":"0dbdf554-3a52-4093-8ec7-18d3794fb2bc","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1999,"title":"Dudley, R. M. (1999).Uniform Central Limit Theorems. Cambridge University Press, Cambridge. van der Vaart, A. W. (1998).Asymptotic Statistics. Cambridge University Press, Cambridge. van der Vaart, A. 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