The prototypical psychopath has an extremely grandiose sense of self-worth. He believes himself to not only be better than others, but more important than others as well. At its very extreme, some psychopaths believe they can cheat death. More typically, this facet of the condition is reflected in extreme self-esteem and self-importance.
It rarely matters what proof there is to justify such a perspective by the psychopath. I recall a time at my previous job when I was convinced that I was the most adept person in the building – an honest-to-god savant. I had no statistics to back this up, I simply felt it. I would constantly remind others of their “incompetence” while touting my own superiority. Humility was not a strong suit. Looking back with clarity, I can see that such a perspective was flawed. However, I still exhibit my own grandiosity in my own way.