Title: Concave Processes for Multi-Task Career Trajectories: Evaluating Aging Across Multiple Measures of NBA Performance
Author: Abhijit Brahme, Alexander Franks
Institution: Department of Statistics and Applied Probability, University of California, Santa Barbara; arXiv preprint (stat.AP)
Publication Date: October 1, 2026 (arXiv v2; first posted September 27, 2026)
The Problem: One Number Can't Describe How a Player Ages
Front offices commit multi-year money based on how they expect production to hold up, yet most aging curve research tracks a single summary score. That hides the fact that shot blocking, shooting and playmaking rise and fall on different clocks.
Existing approaches also fix the peak age in advance, group players by rigid positions or preset archetypes, and are exposed to survivorship bias: weaker players leave the league early, which flattens the average curve. The authors point to Brooklyn's 2013 trade for 35 year old Paul Pierce and 37 year old Kevin Garnett as the cost of misjudging a trajectory.
Methodology: Four Pieces, One Model
Brahme and Franks fit one Bayesian model to 17 performance metrics for roughly 2,300 NBA players, covering ages 18 to 38 across the 1997 to 2026 seasons.
Concave process prior (the shot arc): Every curve must rise, peak once and fall, like a shot with a single apex, but the age and height of that peak are learned per player instead of set beforehand.
Player embedding (the comp finder): Each player gets a ten dimensional profile seeded by height, draft slot and position, so similar players share similar curves, like scouting comps built from production.
Joint multi-metric fit (the full scouting report): All metrics train together, so rebounding and minutes trends inform a scoring forecast.
Survival model (the roster-cut correction): Career length is modeled alongside production, so players who washed out early still shape the curve.
Why it Matters: Better Forecasts, Honest Limits
The single peak rule earns its keep: Across five holdout tests, it beat an unconstrained Gaussian process by 0.25 to 0.46 log predictive density per held out observation, with 95% intervals covering 90.7 to 92.6% of outcomes versus 84.5 to 88.0%.
Joint beats solo: Refitting on OBPM alone significantly degraded OBPM forecasts.
Skills age on different clocks: Blocks and offensive rebounds peak in the early twenties, shooting skill in the mid-twenties or later, and three point volume keeps climbing into the early thirties. Minutes per game fall faster than they rise.
Read it as a preprint: Intervals still run two to four points under nominal, inputs are season level statistics for players who reached the NBA, and young player forecasts are wide. Victor Wembanyama's projected peak OBPM is 4.2, with a 95% interval of 1.7 to 7.3.
ACTIONABLE TAKEAWAYS
Project contracts skill by skill: Before a multi-year offer, forecast minutes, availability and offensive impact separately. The paper finds those change most with age, while blocks, steals and defensive rebounds stay relatively intact.
Point veteran development at skills that age well: Shooting and playmaking peak later than athleticism driven production, so shift workload there as explosiveness fades.
Treat young player projections as ranges: Ask for threshold probabilities and historical comps, not one number. Wembanyama's chance of a peak OBPM above 4 is 54%.