By Raul Cortes and Lizbeth Diaz

MEXICO CITY (Reuters) -Mexican authorities on Sunday confirmed the American and two Australian tourists who went missing in northern Mexico last week are dead after their bodies were identified by parents.

The bodies of Australian brothers Callum, 33, and Jake Robinson, 30, as well as American Carter Rhoad, 30, were found at the bottom of a well in Baja California state earlier this week after a days-long search.

The three foreigners went missing while on a vacation surfing near the popular tourist town of Ensenada, about 90 minutes south of the U.S.-Mexico border on the Pacific coast.

Baja California state Attorney General's Office said in a statement the confirmation was issued once the parents of the victims "were able to identify them, without the need of genetic tests".

State Attorney General Elena Andrade told the parents and diplomatic officials that there was "total institutional commitment to continue with the investigation of these unfortunate events until those responsible receive the full weight of the law", the statement added.

On Saturday, Andrade said that the bodies were found in an advanced state of decomposition at the bottom of a well more than 15 meters (50 ft) deep.

A source from the attorney general's office told Reuters that all three bodies had a shot in the head.

Australian Treasurer Jim Chalmers offered his condolences to the families of Callum and Jake and said the whole country mourned with them.

"We can only imagine what this ordeal has been like for them and for the loved ones of Callum and Jake," he said at a news conference.

"It has been an absolutely horrendous, absolutely horrific ordeal and our thoughts are with all of them today."

SHOT BY CAR THIEVES

Three people have so far been arrested, officials say. A burnt-out vehicle believed to have been used by the three surfers was also found in the area.

Mexican authorities' preliminary hypothesis is that the arrested individuals attempted to carjack the foreigners and when the surfers resisted they were shot and their bodies dumped in a well.

The three surfers were last seen on April 27 and reported missing a couple of days later, when authorities launched a multi-day search with the help from the U.S. Federal Bureau of Investigation (FBI).

Baja California is one of Mexico's most violent states, although the Ensenada area is considered safer. The U.S. State Department advises Americans to reconsider travel to the state due to crime and kidnapping.