Bilic took over from Allardyce at Upton Park, and he has brought in a number of new players to Upton Park.
The Croatian boss has been backed with a lot of funds, and Allardyce believes that his counterpart’s signings have helped to progress the Hammers' squad.
“[When I was at West Ham] we’d secured our Premier League status then so we could start building for the year after,” Allardyce said.
“That was the only time in my four years that we were in any real trouble.
“From thereon, we built an even better side and they’ve built an even better side now with the signings they’ve made.”
Bilic has made a series of notable additions at West Ham, with Dimitri Payet and Manuel Lanzini being two of his standout captures.
The attacking midfield pair have seriously strengthened the Hammers' forward options, and they have made the London side much more of a threat.
As a result, the Hammers have climbed up into sixth place in the Premier League table and they are in with a chance of claiming a European spot.
Allardyce, meanwhile, is now with Sunderland and he is trying to help the Wearsiders avoid relegation, after a disappointing first half of the campaign.