You find both accusative pronouns (me/him/her/them) and nominative pronouns (I/he/she/they) in this syntactic position in standard English. The forms with the nominal genitive pronouns (mine/yours/hers etc.) are a red herring because they stand for something possessed rather than the person themself.
The traditional rule for comparison with a person is that you must use nominative. However, according to my research, accusative is more common.
I searched the Corpus of Contemporary American English for this syntactic structure, followed by a comma or a period to ensure we are not looking for cases like faster than he is, with a verb following the pronoun, in which case nominative is obligatory.
Ich habe mich, glaube ich, auch an die Akkusativ-Version gewöhnt, obwohl es ursprünglich wohl wirklich "I am faster than he" korrekt war. oo Und es scheint ja eh beides mitlerweile Standard-Englisch zu sein. ~
Oh, Grafik dazu, praktisch: