The family of one of the hikers whose bodies were found in Joshua Tree National Park earlier this week say investigators have told them that they believe the couple died in a "sympathetic murder-suicide" after becoming lost in the sprawling desert with little food or water.

Relatives of 20-year-old Rachel Nguyen said investigators had based their belief on the circumstances and positioning of the bodies.

The bodies of Nguyen and Joseph Orbeso, 22, were found in a steep canyon Sunday. They had been missing for nearly three months after going for a hike in late July and failing to return to their bed-and-breakfast.

Autopsies found both had gunshot wounds and evidence at the scene led detectives to believe Orbeso shot Nguyen and then himself, the San Bernardino County Sheriff's Department said in a statement Friday. The stunning announcement came days after Orbeso's father, who was with searchers who made the discovery, said the bodies were locked in an embrace.

"We hold no grudges against Joseph or the Orbeso family," Nguyen's family said in a statement. "We thank God that we'll be able to give Rachel a proper burial and lay her to rest."

The Orange County Register reports there was evidence the pair had been battling the elements. The bodies were found under a tree, with clothing covering their legs to protect them from the blazing sun. They appeared to have been rationing food and had no water.

Orbeso and Nguyen had once dated and he took her to Joshua Tree to celebrate her birthday, his best friend Austin Young told the newspaper.

A gun found near the bodies was registered to Orbeso. Young said Orbeso worked as a security guard, and may have carried the gun to protect Nguyen.

"I think they got lost and were suffering in 100-degree heat," Young said.

"They had a choice of a slow and painful death or a quick death," he said. "And they made the choice of a quick death."

Authorities continued to investigate Orbeso's actions, though his motive may never be known.

"Since we don't have any evidence (note, message) to tell us why the shooting occurred, we are left to speculate," sheriff's spokeswoman Cindy Bachman said. "Knowing that they rationed food, had no water and were seeking shade certainly sheds a different light on the apparent dire circumstances."

The Joshua Tree Search and Rescue team said on its Facebook page that it's easy to get lost in the vast park covering more than 1,200 square miles of the Mojave and Colorado deserts, 130 miles east of Los Angeles.

"You can take a short hike and lose sight of the road, and, if you didn't pay attention to what the landmarks look like, in the opposite direction, you could start wandering," the team said.

The Associated Press contributed to this report.