Marcel Goverde’s study “Helmets Make You Bicycle Faster” appears in AIR 15-5 (which you can read online). Goverde, who is based at the University of Basel, Switzerland, describes the big question and how he tackled it:
There seems to be a lack of profound, scientific studies investigating the risk compensation theory for helmet wearing cyclists. In the present study I investigate if helmet-wearing cyclists are driving more aggressive and thereby faster than non-helmeted cyclists. I measured for one year the time taken to travel from home to work and the return journey. Before leaving I choose either to wear the helmet or not by throwing a die….
UPDATE: According to a BBC news report, bicycling in the nude is permissible in New Zealand, provided that the cyclists wear helmets. The effect of helmets on nudist bicycling speed has not been directly measured.