According to Allen in his Vox Latina, initial h- "was omitted...in words of rustic origin", like anser. No doubt anser originally had an initial h- (cf. Sanskrit haṃsas, "swan"; Greek χήν, "goose"), but interestingly, all the other examples Allen provides for words that lost the initial h- have a form attested with it except anser. Words like arena, umor, aedus, and ircus have amply attested forms with their initial aspirant.
I note that Cato the Elder, Plautus, Lucilius, and Hemina all attest to anser without an initial h-, yet nowhere in PHI can you find a form with it. Is anser unique in this regard? Or are there other words that similarly lost that initial h- very early on? If not, what conclusions can we draw from its unique status?