The Hammers face Arsenal on Wednesday night after their original Boxing Day clash was moved due to the threat of a tube strike.
The match looks to have come at a good time for West Ham given Arsenal are yet to win a league fixture in 2013, something Allardyce believes is down in no small part to the club not replacing Van Persie when he joined Manchester United.
"When you sell your best players, which is not normally expected of Arsenal, then it is always difficult to replace them," Allardyce said.
"There is no doubt that the players that have come in are very, very good players, but are they better than the ones they have sold?
"You are selling players for nearly £30million. It is a big ask, even for Arsene Wenger with his huge recruitment policy, to replace those players immediately.
"I think that is why they have suffered somewhat with the level of results they get. The football is still very, very good, but when you lose that many goals [it is hard].
"They have suffered particularly with the loss of Van Persie. They are missing more chances than they were before, hence they are not quite getting as many results.
"It's not surprising because I don't know where you go and buy a player of Van Persie's quality unless you're going to pay more than £24million.
"Everybody says it's great business in terms of his age, but it's not really great business for Arsene when he can't replace the goals that he's lost.
"That's all our frustrations in terms of those who are the best strikers generally in the end continue to get the best results."
Arsenal come into the match on the back of a disappointing 2-1 defeat at Chelsea on Sunday.
Early goals from Juan Mata and Frank Lampard proved the difference at Stamford Bridge, where the Gunners were overwhelmed in the first half.
"We have got to look to try to frustrate them in terms of not letting them get into that nice, free-flowing football that they like to play," Allardyce said.
"I saw how well Chelsea disrupted them in the first half on Sunday and how they went about their task to stop them, even though Chelsea were playing at home.
"They stopped Arsenal playing first then went about exposing some of the spaces there are left.
"Chelsea did so well closing them down and pinching the ball back off them. Obviously the quality of the finishing then was the reason they got into quite a comfortable lead.
"That would be the order of the day for us, making sure we don't make life easy for Arsenal when they are in possession because that is their main strength. That is what they do best."