Inc. is liable for billing parents for in-app purchases that their children made without permission, a federal judge has judge has decided, in a ruling that resolves accusations similar to ones that Apple Inc. and Google Inc. settled two years ago.

U.S. District Judge John Coughenour in Washington state on Tuesday granted a Federal Trade Commission motion for summary judgment on liability in a fight between Amazon and the agency over the unauthorized purchases.

"The millions of dollars billed to Amazon customers without a mechanism for consent, the thousands of customers complaining about unauthorized charges, and the time spent seeking refunds for those charges, all demonstrate substantial injury," he wrote.

"It is Amazon’s stated policy that in-app purchases are final and nonrefundable, likely discouraging much of its customer base from attempting to seek refunds."

The judge also noted that 1,573 customers who sought refunds did not receive them.

Coughenour denied the FTC's request for an injunction that would have subjected Amazon to 20 years of oversight. Amazon has made changes in how in-app purchases are made since the FTC began probing the issue, the court said.

The judge did not immediately decide on a remedy. The FTC, which filed the lawsuit in 2014, said it would press for full refunds for affected Amazon customers.

The agency has accused the online retailer of failing to make proper disclosures to parents regarding purchases made by their children while using apps such as "Pet Shop Story."

The judge cited in his ruling what he said was a confidential document about Amazon’s marketing plan in which it acknowledged that "'IAP' isn’t a concept widely known by customers."

Judge Coughenour also quoted Aaron Rubenson, who says on his Linkedin account that he is head of the Amazon Appstore, that customer complaints about the purchases were "near house on fire." The judge also quoted Rubenson as saying, "We’re clearly causing problems for a large percentage of our customers."

FTC data analyst Julie Miller estimated that Amazon had made $86 million from the in-app purchases and refunded $10 million. She estimated that fully 42 percent of the total purchases were unauthorized, the judge said. The judge said he believed that number could be "inflated" and asked for a further briefing.

Amazon called Miller’s estimate "fundamentally flawed," the judge said. Amazon did not respond to requests for comment for this story.

The FTC settled similar cases with Apple and Google in 2014. Apple agreed to refund to customers at least $32.5 million in unauthorized charges, while Google agreed to pay at least $19 million.