Do you have an issue where the phone believes it's always plugged in, ever get a popup saying battery voltage too high?
I have this issue but only because I submerged my phone in apple juice. It's broken the USB connector slightly. Although everything works, the phone believes it's constantly plugged in. When turned off, it will either display the charging screen or just boot up randomly. I would say your choices are to clean the usb with rubbing alcohol or send off to insurance.
One final thought, there is a piece of software called NoMoarPowah that can manipulate your Off/Charging screen. I wonder if someone has used this method to create a piece of malware, ensuring your phone is always on and sending data/texts?