The deadly shooting of a man by an off-duty Los Angeles police officer last week at a Costco store in Corona, Calif., has led to calls for the authorities to release more information amid conflicting accounts of what happened.

The man who was killed, Kenneth French, 32, had been shopping with his parents, Paola and Russell French, before the shooting on Friday. They were critically wounded in the shooting, which the Corona police have said came in response to an unprovoked assault of the officer.

Mr. French’s family has called for the arrest of the officer, identified as Salvador Sanchez by The Los Angeles Times. The victim’s cousin, Rick Shureih, who has spoken on behalf of the family, has characterized Mr. French as having a mental disability and being nonverbal, but he did not elaborate on his condition. Mr. French had lived in Riverside, Calif., with his parents.

“When the Corona Police Department finishes its investigation, they will submit the case to our office for review for any and all potential criminal charges,” John Hall, a spokesman for the Riverside County district attorney’s office, said in an email to The New York Times on Monday. “So, at this time, the case is still with Corona police detectives and they are gathering facts and conducting numerous interviews of witnesses. We would ask the public for patience as detectives complete the investigation.”