I had a similar case with yours. My car got hit by a motorcycle from back when I stopped to yield pedestrians. I only have liability, didn't call the police and did not have a witness. I had to deal with motorcycle driver’s insurance company by myself. I called and reported the accident and was asked to wait. They need to verify with their insuree. I was told that by law, the insurance company has 40 days investigation period. During this period, they can let you estimate your car and get it fixed (or pay you the money) if they make sure it is their insuree's mistake, or they will give you a report by the day 40th. Then you can file a law suit within 3 years. What I did was keeping calling the insurance once every week; estimating the damage and letting them know how bad the damage was; keeping taking notes for every conversation between me and insurance includes date, time and every little details, even it was the voice mail incase I need to go to the court. I also came here and got help and suggestions from Lao Mao and other good guys (Thanks again). Luckily, I got my case solved in 30 days. Hope my experience can help you a little. Good luck.