A skewed t model had lower daily Nasdaq density loss than a symmetric t model at equal sign probabilities in 2018–2025

We tested whether a reported daily Nasdaq Composite density forecast gain remained when both models had the same sign probabilities. We used frozen forecasts from an earlier dated-input study. Each source model had a fitted mean. We split log loss into sign loss and return-size loss within each sign. We also made proper densities that gave both models the same past positive-return probability. In 2,011 sessions in 2018–2025, the skewed Student-t model had a mean gain of 0.007826 nats per session over the symmetric Student-t model at equal sign probabilities. The paired 95% interval was 0.002802 to 0.012902 with 21-session blocks. The original full gain was 0.008046 nats. The sign component was 0.000220 nats, with an interval that included zero. Neither original model clearly beat past signs on binary log loss. The size gain remained in fixed fresh-input and without-2020 checks. Independent calculations, raw-source timing checks, and an exact replay passed. Code, forecasts, source records, aggregate results, figures, and negative findings are preserved. Restricted source-derived outcomes are retained separately. This is an audit on a known historical sample, not a new market test or evidence of trading profit.

density forecasts · stock returns · Nasdaq Composite · sign probabilities · conditional return size · log score · point-in-time data · chronological evaluation · reproducibility