May 14 (Reuters) – With private fitting rooms, members-only stores and clever apps, U.S. department stores are reinventing their old and tested loyalty programs to fend off the challenge from e-commerce giant Amazon.com Inc and other online rivals.

Over the past two years, major U.S. retailers, including Macy’s, Nordstrom, Kohls Corp and others, have spent billions of dollars to overhaul existing programs or launch new loyalty schemes. According to estimates compiled by market intelligence firm Beroe, the U.S. loyalty program market was worth between $47 billion and $55 billion in 2018 in terms of spending by companies and sector analysts expect it to keep growing by 2 percent to 4 percent a year between 2018-2020.

Data from research firm LoyaltyOne showed retailers now invest at