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pith:CD5ZZZQI

pith:2026:CD5ZZZQIYZBXJAJJLXYEJRCFAS

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On Stability and Decomposition of Sample Quantiles under Heavy-Tailed Distributions

Choudur Lakshminarayan

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.

arxiv:2605.18370 v1 · 2026-05-18 · stat.ML · cs.LG · math.ST · stat.TH

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Claims

C1strongest 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.

C2weakest 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.

C3one 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.

References

17 extracted · 17 resolved · 0 Pith anchors

[1] Bahadur, R. R. (1966). A note on quantiles in large samples.Annals of Mathematical Statistics, 37, 577–580 1966

[2] Kiefer, J. (1967). On Bahadur’s representation of sample quantiles.Annals of Mathematical Statis- tics,38, 1323–1342 1967

[3] (1984).Convergence of Stochastic Processes 1984

[4] 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. 1999

[5] Markowitz, H. (1952). Portfolio selection.Journal of Finance,7, 77–91 1952

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First computed	2026-05-20T00:05:57.651996Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

10fb9ce608c6437481295df044c44504a48a19f3066473888cc08ee583f2c7e1

Aliases

arxiv: 2605.18370 · arxiv_version: 2605.18370v1 · doi: 10.48550/arxiv.2605.18370 · pith_short_12: CD5ZZZQIYZBX · pith_short_16: CD5ZZZQIYZBXJAJJ · pith_short_8: CD5ZZZQI

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curl -sH 'Accept: application/ld+json' https://pith.science/pith/CD5ZZZQIYZBXJAJJLXYEJRCFAS \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 10fb9ce608c6437481295df044c44504a48a19f3066473888cc08ee583f2c7e1

Canonical record JSON

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    "abstract_canon_sha256": "3c522fbc23153872bba7f9ebc02bea3504fac73ffbedd7f7209b0e0e14aac0bb",
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      "stat.TH"
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    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "stat.ML",
    "submitted_at": "2026-05-18T13:19:29Z",
    "title_canon_sha256": "8a9618dd30e7441bac6d2b1693c4af7be94ea7fee742e8d63ec418dc812ed5da"
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}