Apple only has one method to backup and restore data on your device and that that is from iTunes. You should be doing this on a PC or laptop already to back up your information in iTunes. If you are using the iCloud, there is no guarantee that your data will be backed up. This would be the only method to transfer statistics.
If so, you should be able to restore from a backup of your old device to your new device. Your data will then be restored back to the device with your statistics intact. This is the only method possible. You can only go from an iPad to iPad or iPhone to iPhone, but you cannot go from iPad to iPhone.
Then you would just log into Game Center on your new phone with the same username and password as on your old device.