18-EFT.WP.Methods.CrossStats v1.0 | Chapter 4 — Estimation and Intervals (Frequentist/Bayesian Harmonization)

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Chapter 4 — Estimation and Intervals (Frequentist/Bayesian Harmonization)

One-Line Objective

I. Scope and Objects

1. Scope
   • Applies to point estimation, interval estimation, and posterior summaries for weighted samples, complex designs, and online streams.
   • Covers uncertainty propagation for means/proportions/rates, regression (GLM), ratios, and functional metrics g( theta ).
2. Objects
   • Inputs: data D = { (y_i, x_i, w_i, t_i) }, sampling info pi(i) or replicate weights, window Delta_t, arrival-time fields T_arr.
   • Outputs: hat{theta}, SE(hat{theta}), CI_{1-alpha} or posterior intervals, U = k * u_c, and manifest.stats.estim.*.
   • Constraints: consistent units and dimensions; sum(w_i)/N_hat ≈ 1; compute on tau_mono, publish on ts.

II. Terms and Variables

1. Basics
   • theta (parameter vector), hat{theta} (estimator), SE (standard error), V (covariance), CI_{1-alpha} (interval).
   • Weighted mean: hat{mu}_w = ( ∑ w_i y_i ) / ( ∑ w_i ).
   • Ratio: R = ( ∑ w_i a_i ) / ( ∑ w_i b_i ).
2. GLM and robust variance
   • Score equations: U( theta ) = ∑ x_i * ( y_i - mu_i( theta ) ) / v_i( theta ) = 0.
   • Sandwich variance: V_hat = ( A^{-1} ) * B * ( A^{-1} )^T, with A = - ∂U/∂theta, B = ∑ u_i u_i^T.
3. Bayesian elements
   p(theta), L(theta; D), posterior p(theta | D) ∝ L * p(theta), posterior predictive p(y_new | D) = ( ∫ p(y_new | theta) p(theta | D) d theta ).
4. Metrology and units
   unit(hat{theta}) = unit(theta), dim(hat{theta}) = dim(theta); run check_dim( y - f(x) ) prior to release.
5. Time and arrival time
   Statistical window: window( t; Delta_t, tau_mono ); record both T_arr conventions and delta_form in parallel.

III. Axioms P304-*

• P304-1 (Explicit weights): any weighted analysis must document the generation and versioning of w_i, with sum(w_i)/N_hat ≈ 1.
• P304-2 (Dimensional conservation): unit(expr) and dim(expr) must be consistent and traceable; implicit unit conversion is prohibited.
• P304-3 (Robust variance by default): under heteroskedasticity/correlation, prefer robust or replicate-weight variance.
• P304-4 (Verifiable coverage): intervals must state the target coverage 1 - alpha and provide empirical coverage approximation or PPC results.
• P304-5 (Unified timebase): compute on tau_mono, publish on ts with offset/skew/J.
• P304-6 (Parallel T_arr conventions): when estimates depend on T_arr, record both conventions and delta_form, with asserted thresholds.
• P304-7 (Numerical stability): optimization/sampling must report convergence criteria, mcse, or iterative residuals; failing thresholds cannot be released as a “strong” caliber.

IV. Minimal Equations S304-*

1. S304-1 (Weighted mean and variance)
   • hat{mu}_w = ( ∑ w_i y_i ) / ( ∑ w_i ).
   • Linearization variance (SRS approximation; use replication for complex designs):
     Var( hat{mu}_w ) ≈ ( ∑ w_i^2 ( y_i - hat{mu}_w )^2 ) / ( ( ∑ w_i )^2 ).
2. S304-2 (Proportion/rate intervals)
   • Wilson proportion interval:
     p_w = ( y + z^2 / 2 ) / ( n + z^2 ),
     half = z * sqrt( ( p_hat ( 1 - p_hat ) + z^2 / ( 4 n ) ) / ( n + z^2 ) ),
     CI = [ p_w - half , p_w + half ].
   • Poisson rate (exposure E): lambda_hat = ( k / E ), normal-approx interval lambda_hat ± z * sqrt( k ) / E (use exact or Byar for small samples).
3. S304-3 (Delta method)
   • Scalar: Var( g( hat{theta} ) ) ≈ ( g'( theta ) )^2 Var( hat{theta} );
   • Vector: Var( g( hat{theta} ) ) ≈ G V G^T, with G = ∂g/∂theta |_{hat{theta}}.
4. S304-4 (Ratio estimator via Delta)
   With R = A / B,
   Var( R ) ≈ ( 1 / B^2 ) Var( A ) + ( A^2 / B^4 ) Var( B ) - ( 2 A / B^3 ) Cov( A, B ).
5. S304-5 (GLM normal-approx intervals)
   CI_{1-alpha}( theta_j ) = hat{theta}_j ± z_{1-alpha/2} * SE( hat{theta}_j ); use t_{df} for small samples.
6. S304-6 (Bootstrap intervals)
   Percentile: CI = [ q_{alpha/2}( theta^* ), q_{1-alpha/2}( theta^* ) ]; BCa as the default robust option.
7. S304-7 (Bayesian intervals and coverage factor)
   • Central or HPD: CI = [ q_{alpha/2}( p(theta|D) ), q_{1-alpha/2} ];
   • Metrology mapping: align U = k * u_c with frequentist intervals; under normal approximation, k ≈ z_{1-alpha/2}.
8. S304-8 (Posterior predictive checks)
   ppc = P( T( y_rep ) ≥ T( y_obs ) | D ); publish the chosen statistic T(•) and the ppc value.
9. S304-9 (Two T_arr conventions discrepancy)
   delta_form = | ( 1 / c_ref ) * ( ∫ n_eff d ell ) - ( ∫ ( n_eff / c_ref ) d ell ) |, with delta_form ≤ tol_Tarr asserted.
10. S304-10 (Placeholder for multiple comparisons)
    Family-wise error control is executed in Chapter 6; here, interval widths are computed under the given alpha_budget, with alpha_used ≤ alpha_budget.

V. Statistical Process M30-4 (Ready → Estimate → Intervals → Diagnostics → Release)

• Readiness
  Load w_i/replicate weights, design info, and Delta_t; run check_dim and evaluate effective sample size n_eff.
• Estimation
  Compute hat{theta} (weighted/GLM/ratio); choose variance method (robust/replicate/bootstrap).
• Intervals
  Construct CI_{1-alpha}: prefer robust or replicate variance; use appropriate conventions for proportions/rates; in Bayesian mode, report posterior intervals and the U = k * u_c mapping.
• Diagnostics
  Coverage or PPC, residuals, and bias-variance analysis; record mcse or optimization convergence; if T_arr is involved, compute both conventions and delta_form.
• Release
  Write manifest.stats.estim.*: conventions, alpha, SE, CI, ppc/coverage, offset/skew/J, TraceID, and signature.

VI. Contracts and Assertions (C30-4xx)

• C30-401 (Weight normalization): | ( ∑ w_i / N_hat ) - 1 | ≤ tol_w_norm.
• C30-402 (Variance consistency): relative gap | SE_rep - SE_robust | / SE_robust ≤ tol_var_gap.
• C30-403 (Coverage/ppc): offline playback coverage error | cov_hat - ( 1 - alpha ) | ≤ tol_cov or ppc ∈ [tol_ppc_low, tol_ppc_high].
• C30-404 (Numerical stability): mcse( theta_j ) ≤ tol_mcse or optimization residual ≤ tol_opt.
• C30-405 (Arrival-time discrepancy): delta_form ≤ tol_Tarr.
• C30-406 (Dimensional conservation): assert check_dim( y - f(x) ) == true; missing units constitute a violation.
• C30-407 (Multiple-comparison budget): alpha_used ≤ alpha_budget (linked to Chapter 6).

VII. Implementation Bindings I30-*

• I30-41 fit_glm(ds, formula, family, variance="robust") -> {hat_theta, V_hat, SE}
  Supports family ∈ {gaussian, binomial, poisson, gamma}, variance ∈ {model, robust, cluster, replicate}.
• I30-42 fit_bayes(ds, model_spec, priors, draws, chains) -> posterior
  Returns posterior samples, mcse, rhat, HPD, and PPC; includes sampling seed and version.
• I30-43 bootstrap_metric(fn, ds, B, scheme) -> {est, SE, CI}
  scheme ∈ {pairs, residual, wild}, with support for weights and stratification.
• I30-44 delta_propagate(g, hat_theta, V_hat) -> {est_g, SE_g, CI_g}
  Auto-derives Jacobian G and computes G V G^T.
• I30-45 compose_interval(est, SE, alpha, mode) -> CI
  mode ∈ {normal, t, wilson, byar, bca}.
• I30-46 emit_estimation_manifest(results, policy) -> manifest.stats.estim
  Writes contract evaluations, conventions, thresholds, and signature; links to manifest.stats.sampling.

VIII. Cross-References

• Sampling and weight provenance: Chapter 3 of this volume.
• Multiple comparisons and sequential control: Chapter 6 of this volume.
• Drift monitoring and baseline refresh: Chapter 7 of this volume.
• Timeline and the two T_arr conventions: Methods.Cleaning v1.0, Ch. 5–6.
• Metrology and unit consistency: Methods.Cleaning v1.0, Ch. 4.

IX. Quality and Risk Control

1. SLI/SLO
   Coverage error | cov_hat - ( 1 - alpha ) |, interval width width_p50/p90, var_gap, mcse_p95, latency_ms_p99.
2. Risk control and rollback
   • Triggers: failures of C30-402/403/405 or rhat > cap_rhat.
   • Actions: switch between robust and replicate variance; downgrade the release to “experimental”; revert to the previous signed manifest and alert.

Summary