There's still a great deal that science doesn't quite understand about human sleep patterns, and the studies on auditory stimulation and sleep have been small. One 2017 experiment at Oxford University on eight sleepers found that subjects fell asleep around 40 percent faster while listening to white noise. Overall sleep time was mostly unchanged, though. A 2016 study showed that 16 young adults had slightly improved recollection of vocabulary words if they slept under pink noise. And another 2017 study at Northwestern University (of 13 older adults) linked pink noise with deeper sleep and improved ability to recall words.
A larger study conducted by the Journal of Caring Sciences in Iran looked at 60 elderly coronary patients, with half of them sleeping under white noise, and half with regular hospital ambient sounds. In the control group, scientists found that quality of sleep degraded as the patients spent multiple nights in the hospital. For those getting the white noise treatment, however, quality of sleep remained roughly the same throughout their stay. There haven't been any research studies on the effects of brown noise on sleep.
The effects of white, pink and brown noises will most likely remain subjective until experiments can be conducted with larger sample sizes and a more diverse array of participants.
"What I tell my patients is, 'I really don't know which is going to be better. Why don't you just try them out to see which is relaxing for you?'" said Dr. Phyllis Zee, director the Center for Circadian and Sleep Medicine at Northwestern University Feinberg School of Medicine, in a CNN article. Zee was one of the researchers in the 2017 study of pink noise and older adults. You could even try blending all three as in the video below.