There is sometimes uncertainty over whether the –ing form of a verb can be preceded by a personal pronoun or a noun unmarked for possession, or whether it can be preceded only by a possessive determiner or a noun with a possessive inflection. The two possibilities can be seen in these examples:
1a. I don’t like your going out looking like that.
1b. I don’t like you going out looking like that.
The answer, as so often, is that it all depends. We first have to understand that the -ing form of the verb can function both as a verb and a noun. Arriving functions as a verb in Arriving home late, he offered his apologies. It functions as a noun in Arriving at your destination is often a disappointment. We next have to understand that the possessive determiners (that’s my, your, his, hers, its, our, their) must be followed by a noun and not a verb. So, too, must nouns which show possession by using the apostrophe. The opposite is also true: any word that is preceded by either of these must be a noun.
In sentence 1a, going is preceded by your, so the -ing form of the verb must here be functioning as a noun, and going out is the object of the verb don’t like. How are we to analyse sentence 1b? If it read simply I don’t like you we would have a perfectly formed sentence. The addition of going out looking like that makes no difference to the basic structure: it gives us more information about what aspect of the person addressed it is that the speaker doesn’t like. It’s rather as if the speaker were saying I don’t like the ‘going out looking like that’ you’.
The difference between the two constructions can perhaps be better illustrated with the sentences:
2a. My wife doesn’t like my cooking.
2b. My wife doesn’t like me cooking.
Most of us would understand 2a as meaning that the way the husband cooked in general was not to the wife’s taste, rather than that she didn’t like it whenever her husband usurped her chosen place in the kitchen. That is the sense that 2b conveys. Together, the sentences resolve uncertainty about the matter. They show that ‘when the possessive alternative is used, it focuses attention on the action described in the ‘-ing’ clause. In contrast [the other] form puts more emphasis on the person doing the action’ (‘Longman Student Grammar of Spoken and Written English’).
It is perhaps worth adding that in both 2a and 2b cooking forms the object of each sentence. It is in such a position that the role of the -ing form is most susceptible to a choice between noun and verb. Of the following pair, where cooking forms the subject, 3b would be found only in those non-standard dialects where me is used as a possessive determiner:
3a. My cooking doesn’t please everyone.
3b. Me cooking doesn’t please everyone.
As Pam Peters reports in the ‘Cambridge Guide to English Usage’, ‘Webster’s English Usage’ (1989) shows that the construction without the possessive has been in use for centuries. Opposition to it, like the irrational opposition to other quite normal usages, began in the 18th century. Database evidence, again reported by Peters, shows that both constructions are current in American, British, Australian and Canadian English. Analyses in ‘The Comprehensive Grammar of the English Language’ (1985) show both constructions to be grammatical. Insistence on the construction with the possessive alone seems to be yet another linguistic fetish which has no basis in the way English is used.
David Crystal has now posted on this topic here.