Research notes#
Working notes that are not part of the library contract. They use the
public API only; nothing here changes normix.
The first note asks whether a portfolio can reveal the latent subordinator \(Y\) in a normal mean–variance mixture, and then measures that claim on daily S&P 500 returns.
Subordinator tracking — one-page summary#
In the mixture \(X \stackrel{d}{=} \mu + \gamma Y + \sqrt{Y}\,Z\), the linear functional of \(X\) with the least Gaussian noise per unit of \(Y\)-loading is unique: \(w^\star \propto \Sigma^{-1}\gamma\). The achievable signal-to-noise is the scalar \(\tilde q = \gamma^\top\Sigma^{-1}\gamma\) that already appears in the GH density and the GIG posterior. That direction is also the best linear predictor of \(Y\), the model’s max-skewness portfolio (for VG, NIG, and NInvG; for GH under a cumulant check), and the Markowitz tangency fund when the whole risk premium is compensation for \(Y\).
A second, non-tradable channel learns \(Y\) from the orthogonal Mahalanobis radius \(q_\perp(x)\). It does not need \(\gamma\), and it gets stronger with dimension.
On the S&P 500 current-constituent panel (2015–2026, \(d\le 468\)) the algebra is not contradicted — the MSE laws hold in simulation, and every fitted VG/NIG/GH model has the tracker as its max-skewness direction — but the linear channel is absent. Fitted fluctuation SNR sits at or below a day-wise sign-flip null at every \(d\). The posterior mean \(E[Y\mid X]\), which uses \(q_\perp\), tracks realized volatility. Online EM does not turn the tracker into a clock.
Read next. Subordinator-tracking portfolios for the derivations; Subordinator tracking on the S&P 500 for the design, the meaning of each metric and cohort, and the results that held and the ones that did not.
Status. Theory 2026-08-10; S&P 500 Phases 0–3, 2026-08-13. No package change.
Gauges — study plan#
The mixture has a scaling orbit
\((\gamma,\Sigma,Y)\mapsto(\gamma/s,\Sigma/s,sY)\); every fitted
\((\gamma,\Sigma,E[Y],\mathrm{Var}(Y))\) is quoted in a gauge.
Gauges for normal mean–variance mixtures classifies gauges by two requirements —
comparable parameters across families fitted to the same data,
and comparable subordinator moments across dimensions (including
against an ensemble of univariate fits) — shows that \(E[Y]=1\) is
the only candidate satisfying both (with 'a_eq_b' failing the
first and \(\lvert\Sigma\rvert=1\) the second), proves batch EM is
gauge-equivariant so the choice is post-estimation reporting except
through floors, stopping rules, and the online-EM η recursion, and
pre-registers the numerical phases.
Status. Plan, 2026-08-19. Not yet executed.