Allardyce told the club's official website: "I think that it says a lot for the experienced British manager. It tells you that while a lot of foreign coaches have been selected by other clubs, there's enough quality and experience on your doorstep to do the job in this very difficult league.
"I think that he has proven beyond any question of a doubt his qualities as a manager. Contrary to what anyone says, his team does not play like his Stoke team did.
"There are some similarities, but he has a bunch of players who are different to the ones he had at Stoke and he gets the best out of them. That's all you can do as a manager, work with the tools you've got and get the best out of them. Certainly he's doing that at the moment."
West Ham take on Crystal Palace in the Premier League on Saturday at Upton Park.