Theory: Restricted means, extreme quantiles, and interpretation

For heavy-tailed outcomes, means may be unstable or even infinite under a GPD tail. CausalMixGPD therefore pairs:

• restricted-mean contrasts for stable center/“average below a cutoff” comparisons, and
• quantile contrasts for tail-aware distributional comparisons (including extreme upper quantiles).

Restricted means for stability in heavy tails

Under a spliced model, the GPD shape parameter \(\xi_a\) (by arm) controls tail heaviness. A key consequence is that the conditional mean of the spliced distribution can be infinite when \(\xi_a\ge 1\).

To keep mean-type contrasts finite, the package uses restricted mean summaries: instead of \(\mathbb{E}(Y(a)\mid x)\) it computes a cutoff-restricted version that remains finite even when unrestricted means fail.

In practice:

• type="mean" is flagged as infinite when posterior mass supports $_a`;
• type="rmean" remains finite by construction.

Standardized (marginal) estimands: restricted-mean ATE + QTE

For binary treatment \(A\in\{0,1\}\), the package’s standardized estimands are contrasts computed from marginal (population-standardized) treatment-specific targets.

You can view the restricted-mean ATE as a stable average effect below a cutoff \(c\):

\[ \mathrm{ATE}_{rmean}(c) = \mu_{1,r}(c) - \mu_{0,r}(c), \]

and QTE as a quantile-indexed contrast:

\[ \mathrm{QTE}(\tau)=Q_1^{m}(\tau)-Q_0^{m}(\tau), \]

for \(\tau\in(0,1)\).

The main modeling point is that, because quantiles are computed under the spliced bulk+tail representation, QTE naturally reflects improved identifiability/stability in the extremes compared to separate bulk-only or tail-only approaches.

Conditional estimands: representative profiles and CQTE

To describe heterogeneity, CausalMixGPD computes conditional contrasts at covariate profiles.

A common presentation uses two “representative” profiles:

1. a typical profile (e.g., at sample medians), and
2. a high-profile (e.g., at an upper percentile of a key covariate such as “previous earnings”).

Conditional restricted-mean effects use CATE at those profiles, while conditional quantile effects use CQTE:

\[ \mathrm{CQTE}(\tau\mid x)=Q_1(\tau\mid x)-Q_0(\tau\mid x). \]

Tail behavior as an interpretable model output

A major advantage of splicing is that the tail index \(\xi_a\) is estimated directly for each arm.

• Larger \(\xi_a\) indicates a heavier upper tail and a greater propensity for extreme outcomes.
• Because the same posterior object supports both prediction and causal contrasts, tail behavior is interpretable in the same model used to compute QTE / CQTE.

Interpretation (how to read the results)

Typical interpretation in the package’s heavy-tail causal workflow is:

• restricted-mean ATE indicates whether the treatment produces a positive average effect below an empirical cutoff;
• QTE strengthens in the upper quantiles, indicating larger treatment impacts in the distribution’s tail;
• conditional restricted-mean CATE can be positive at both representative profiles, and
• CQTE can show stronger upper-tail gains at the “high” profile (when heterogeneity interacts with tail heaviness).

Finally, if one arm has the larger posterior tail index \(\xi_a\), this explains why unrestricted mean contrasts can be much less stable than restricted-mean contrasts in heavy-tailed applications.

References (key)

• Balkema & de Haan (1974), Residual Life Time at Great Age — doi:10.1214/aop/1176996548
• Pickands (1975), Statistical Inference Using Extreme Order Statistics — doi:10.1214/aos/1176343003
• Behrens et al. (2004), Bayesian Analysis of Extreme Events with Threshold Estimation — doi:10.1191/1471082X04st075oa
• Rosenbaum & Rubin (1983), Propensity Score in Observational Studies — doi:10.1093/biomet/70.1.41
• Gelman et al. (2013), Bayesian Data Analysis — doi:10.1201/b16018

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