Investigators estimated Saturday that the bodies of a Greenwood family of five had been in their Lake Minnetonka mansion for several days before being found Thursday around noon by police answering a call for a welfare check.

The Hennepin County medical examiner's office released confirmation of the family members' names and causes of death. Brian Scott Short, 45, died of a self-inflicted shotgun blast. His wife, Karen Anne Short, 48, and their children Cole, 17; Madison, 15, and Brooklyn, 14, each died of a shotgun wound to the head.

South Lake Minnetonka police, describing the case as a murder-suicide investigation, estimated that the killings took place either late Monday or early Tuesday, but would not speculate on a motive. Sources with direct knowledge of the investigation said Friday that Brian Short, an entrepreneur who founded a networking and resources website for professional nurses, went from room to room, shooting Cole, Madison, Brooklyn and his wife, before turning the shotgun on himself. The children were all students at Minnetonka High School.

South Lake Minnetonka interim Police Chief Mike Siitari confirmed Friday at a news conference that a shotgun was found in the house, though he would not say where.

Earlier Saturday the medical examiner listed the time of death as 12:07 p.m. Thursday, when the bodies were found.

The bodies were found by officers carrying out a welfare check at the request of Short's co-workers after he didn't show at work for a couple of days.

His children missed school Tuesday and Wednesday. Siitari again on Saturday asked for patience as police investigate what is being described as a complex and "gruesome" case. "There are many moving parts in an investigation of this magnitude," he said.

Several relatives of the Shorts declined to comment Friday.

The website Brian Short founded, AllNurses.com, posted a message about the deaths that read in part: "No matter what the details are, the results are still the same … a very tragic loss for the extended families, friends, co-workers and this nursing community." By Saturday, several hundred people had left condolences on the website.

Kaitlyn Ancello, of Minotola, N.J., a friend of Cole's on Facebook, said she spoke with the teen over the phone and over Skype frequently after they became friends playing video games across the Internet about a year ago. Cole loved his family, his friends and his dog, she said, and they exchanged photos of their pets. He seemed to come from a nice, well-to-do family, she said.

They often vented to each other about their daily lives, Ancello said, and frequently played video games when Cole wasn't working at a local pet store. His sister Brooklyn sometimes played, too, Ancello said, though she never really spoke to her.

Ancello was shocked to learn that her friend across the country had been killed. The last time she spoke to Cole on the phone — Saturday or Sunday, she thought — he wasn't very talkative: "He seemed like not in the mood, I think he was upset they were moving soon."

Cole had told her that the family had to move because his dad's business was not going well and was losing money and they were struggling financially, Ancello said.

Ancello said Cole didn't know where they were going to move, but thought it would be to a different house with a smaller mortgage in the same area.

Cole seemed funny, caring and nonjudgmental, she said, and she had hoped to meet in person some day.

"I grew up actually being bullied and he sat there, being my friend, he didn't judge me or nothing," she said, adding that she couldn't understand how his father could shoot his whole family. "You have kids who haven't even started their lives yet who have an opportunity to do anything and be anything they want."

Staff writers David Chanen and Kelly Smith contributed to this report.