Add BCa-at-scale section to quantiles user guide

[rwilson4]

Summary

Follow-up to #29. Extends docs/user_guides/quantiles_at_scale.rst with a "BCa intervals at scale" section that closes the BCa loop the previous guide left open.

The previous guide showed how to bypass the bootstrap loop for sample quantiles via the Bin(n+1, q) index trick, but punted on BCa: the jackknife step is O(n) statistic evaluations and dominates at large n. This section derives a closed-form jackknife and a closed-form acceleration that eliminate that step entirely.

What's in the new section

• Closed-form jv. Removing observation X_i either leaves the leave-one-out quantile at X_{(k')} or shifts it to X_{(k'+1)} depending on whether X_i's rank is above or at-or-below k' = ceil((n-1) q). Two distinct values across the entire jackknife array; no statistic evaluations.

• Closed-form a_hat. Plugging the two-valued jv into the BCa acceleration sum, the data-dependence δ³ cancels via affine invariance, leaving

  a_hat = (2k' - n) / (6 * sqrt(n * k' * (n - k')))
  

  Depends only on n and q. Falls out: median ⇒ a_hat = 0 (BCa→BC), |a_hat| grows for tail quantiles, a_hat = O(1/sqrt(n)) so BCa→percentile asymptotically.

• Validation table. Closed-form a_hat agrees with bootstrap_stat.jackknife_values to 6 decimals at n = 200, 1000, 10000, 100000 (verified empirically before drafting).

• General-principle subsection ties the two-hook BCa pattern back to zero_inflated.rst: same recipe (split theta_star and jv, exploit structure on each axis), different exploitable structure (linearity vs. ranks).

• Caveats updated to drop the obsolete "BCa is out of scope" paragraph and replace it with a note about matching the quantile convention (method='inverted_cdf') across the bootstrap, observed q_hat, and jackknife.

Test plan

• cd docs && uv run make html builds cleanly (4 unrelated intersphinx warnings from sandbox).
• Numerical verification: closed-form jv matches jackknife_values() exactly at n ∈ {200, 1000, 10000, 100000}; closed-form a_hat matches both to 6 decimals.
• Eyeball the rendered HTML on the deployed site after merge — section structure uses the same ^^^^ underline depth as the rest of the guide, but worth a glance.

https://claude.ai/code/session_01CRkE1qgfLrZz9oxcfA2Msu

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