The very last scene and why the doctor was wrong
First off...
I had a problem with the doctor's physical reaction to all of the events as well as his explanation of what was happening. I liked his character, but some things just seemed off. Like if a sky scraper were to just collapse in front of you and a friend and your friend turned to you and said "well golly, didn't see that coming". The reaction doesn't fit what just happened.
Regardless.....
The very last scene proves the doctor was wrong for this reason:
Remember earlier on in the movie when the doctor had the dad send his young boy off with their grandfather? Originally the doctor intended on having the daughter go too. He wanted both of them out of the house to see if the happenings still occurred (loud bangs, things flying around). His reason being that if it was a poltergeist then it would be related to one of the two kids (and in his explanation.... therefore not some supernatural haunting).
If both kids were gone then under his beliefs nothing should happen.
However, the daughter stayed behind and didn't go with the grandfather.
What happens? Alot of weird crap goes down. So the doctor comes to the conclusion that he was right after all.... all of the freaky happenings was ultimately related to the daughter (schizophrenia paired with some strong psychic abilities.... therefore, not a haunting from beyond). When the doctor was having the heart to heart with the father he said (paraphrased) "this isn't some supernatural occurrence.... there is no ghost from beyond (the mother) haunting this family. It's because of your daughter. She's schizophrenic and has abilities that we can't explain. It doesn't mean they're supernatural, it's just not something we understand yet."
Then towards the very end the daughter is on a stretcher and is rolled out of the house. The staff leaves too but leaves one camera. Then that camera, at the very end, catches the proof that the doctor was wrong and it was indeed a haunting from beyond. Because in his earlier explanation (why the kids were supposed to go with the grandfather) if the psychic/poltergeist/schizo person was removed from the house then nothing should happen in that house.
........therefore, it was indeed the deceased mother haunting the family.