Add proportion-standardized Pillai separation estimates

[berrygrant]

Add proportion-standardized Pillai separation estimates

Summary

This PR adds opt-in proportion-standardized Pillai estimates to pillai_overlap() through the new argument:

proportion_standardized = TRUE

The existing default behavior is unchanged. In particular, $pillai remains present and retains its existing value.

When enabled, the function additionally returns:

• realized class counts (n1, n2) and harmonic mean (H);
• the plug-in squared Mahalanobis separation estimate (d2_plugin);
• the unbiased estimate given by Lachenbruch and Mickey (1968) (d2_unbiased);
• Becker’s (1986) proportion-standardized Pillai score (pillai_eq);
• explicit fallback and minority-fragility diagnostics.

Fallback behavior

When d2_unbiased < 0, the result is explicitly flagged:

• pillai_eq is NA;
• pillai_eq_fallback is TRUE;
• the fallback separation estimate is returned separately as d2_fallback;
• its known bias term is reported as bias_2p_over_H.

The fallback is never silently reported as a proportion-standardized Pillai score.

Scope and definedness

The new estimator chain:

• applies only to exactly two classes;
• assumes multivariate normality and a common within-class covariance;
• requires at least two observations per class and invertible within-class scatter;
• returns undefined new fields with a warning when nu_e - p - 1 <= 0;
• flags minority classes with fewer than p + 1 observations as fragile;
• does not add covariate or multiclass support.

Documentation

The function documentation, README, and NEWS have been updated with:

• definitions and interpretation of the new estimates;
• model assumptions and definedness conditions;
• fallback semantics;
• the distinction between raw Pillai and the standardized score;
• the relevant Becker (1986) and Lachenbruch and Mickey (1968) attributions.

Testing

Added tests covering:

• all supplied numerical test vectors;
• fallback and near-boundary behavior;
• undefined and fragile cases;
• exactly-two-class enforcement;
• backward compatibility of the default return value;
• agreement with base R’s manova() Pillai statistic;
• end-to-end estimator-chain calculations.

Validation completed under R 4.3.3:

• focused tests: passed;
• complete testthat suite: passed;
• R CMD build --no-manual: passed;
• R CMD check --no-manual: status OK;
• R CMD check --as-cran --no-manual: 0 errors and 0 warnings (one environment-related clock note).