Nothing more to my question, really. I just wonder if the words share an etymological root. Thanks.
Emperor is from Latin imperator, a commander, from imperare, to order. A related word is imperative as in the grammatical term imperative voice, i.e. the voice in which commands are given. "Empire", "emperor" came from this via Old French.
Empirical is from Greek experienced by way of Latin empiricus: no relation to imperator.