Wald Confidence Intervals from Sandwich Variance

The Wald confidence interval for parameter $\theta_p$ is $$ \mathrm{CI}_{1-\alpha}(\theta_p) = \hat\theta_p \pm z_{1-\alpha/2}\, \sqrt{V_{\mathrm{sand},pp}}, $$ where $z_{1-\alpha/2}$ is the standard normal quantile. Because the sandwich variance is design-consistent under mild regularity conditions (Williams and Savitsky, 2021, Theorem 3.2), Wald intervals achieve correct frequentist coverage for the pseudo-true parameter.

Usage

compute_wald_ci(theta_hat, V_sand, level = 0.95, param_labels = NULL)

Arguments

theta_hat: Numeric vector of length D containing point estimates (typically posterior means).
V_sand: Numeric symmetric positive-(semi)definite matrix of dimension D by D, the sandwich variance. Only the diagonal entries are used for marginal Wald intervals.
level: Numeric scalar in $(0,1)$. Confidence level. Default is 0.95.
param_labels: Optional character vector of length D giving parameter names. If NULL (default), names are taken from names(theta_hat) or generated as param_1, param_2, etc.

Value

A data frame with one row per parameter and columns:

parameter: Character: parameter label.
post_mean: Numeric: point estimate.
se: Numeric: sandwich standard error.
z_stat: Numeric: Wald z-statistic.
p_value: Numeric: two-sided p-value.
ci_lo: Numeric: lower confidence limit.
ci_hi: Numeric: upper confidence limit.
ci_width: Numeric: interval width.
significant: Logical: TRUE if p-value is below the nominal significance level.

Details

Computes Wald confidence intervals using the sandwich variance $V_{\mathrm{sand}}$. This is a standalone function that does not require a fitted model object or MCMC draws.

Recommendation

Wald intervals are recommended as the primary inference device because:

They depend only on the point estimate and $V_{\mathrm{sand}}$, not on MCMC sampling variability.
They are algebraically equivalent to the marginal quantiles of the Cholesky-corrected draws in large samples.
They avoid the non-trivial Monte Carlo error in tail quantile estimation from finite MCMC samples.

Significance testing

The Wald z-statistic is $z_p = \hat\theta_p / \sqrt{V_{\mathrm{sand},pp}}$, with two-sided p-value $2\,\Phi(-|z_p|)$. A parameter is flagged significant when the p-value falls below $1 - \mathrm{level}$.

References

Williams, M. R. and Savitsky, T. D. (2021). Uncertainty estimation for pseudo-Bayesian inference under complex sampling. International Statistical Review, 89(1), 72–107. doi:10.1111/insr.12376

Examples

# Minimal example with known values
theta_hat <- c(alpha_1 = -0.324, beta_1 = 0.090, log_kappa = 1.92)
V_sand <- diag(c(0.0027, 0.00035, 0.0041))
compute_wald_ci(theta_hat, V_sand, level = 0.95)
#>   parameter post_mean         se    z_stat       p_value       ci_lo      ci_hi
#> 1   alpha_1    -0.324 0.05196152 -6.235383  4.506744e-10 -0.42584272 -0.2221573
#> 2    beta_1     0.090 0.01870829  4.810702  1.504008e-06  0.05333243  0.1266676
#> 3 log_kappa     1.920 0.06403124 29.985362 1.522994e-197  1.79450107  2.0454989
#>     ci_width significant
#> 1 0.20368543        TRUE
#> 2 0.07333514        TRUE
#> 3 0.25099786        TRUE