My theory on this is very simple. At first there will still be accidents caused by driverless cars. But the data gathered by the driverless cars will easily be enough to reproduce those accidents. With each accident, a team of brilliant engineers will pour over the data and figure out how to deal with that accident. It wont only be solved for that accident but the changes will be tested against a zillion hours of difficult data to make sure that the new code doesn't cause new accidents. All cars will then get an upgrade and will now be safer.
After a while the driverless cars will have hit a point where solving for the tiny remaining number of edge cases will actually reduce overall safety. But these edge cases will be so few and far between that they will make national news when they happen. Thus, the stats behind driverless cars will be so extraordinarily safe that for a single human driver to "prove" that they are safer would take lifetimes of flawless driving to prove. This number will only spread as time grows.
I would not be surprised if the experimental driverless cars are in the top 1% of the top 1% of drivers in the world.
The key being that a single driver gets to accumulate only their own stats, while 1,000,000 driverless cars will do more than a taxi driver's lifetime driving nearly every day or so.
Then to make it worse, we meatbags are variable. Even the best driver in the world with a flawless record, might be forced to drive in non-optimal conditions. They haven't slept in a few days, they are sick, and a sudden emergency forces them to drive someone to the hospital. None of that applies to a driverless car.
For instance. I have a 20+ year flawless driving record. Part of that is that I know not to drive when I have not slept well the night before, or any time between 1am and 7am.
One other bit is that driverless cars will soon have some interesting abilities. Things such as gathering data from other driverless cars. Thus if something has dropped onto the highway, one driverless car can alert the rest about it. Or if one car hits a surprising icy slick, it will not only notify other cars, it will start to build a pattern of when that road is icy. These would be the few initial areas where people would be better; where you know that a certain intersection seems to have ice on it in these conditions. Or observing the behavior of other cars to possibly be alerted to problem such as trash on the road.