Guy Hedger, 61, died in hospital after he was shot by intruders who entered a house in St Ives, near Ringwood, Dorset, just after 03:00 BST on Sunday.
Detectives have launched a murder investigation.
They said a second person in the property at the time was "deeply affected by the incident" and is being supported by specially-trained officers.
Mr Hedger's next-of-kin are aware, a Dorset Police spokesman added.
Det Ch Insp Sarah Derbyshire said police do not believe Mr Hedger was known to the intruders, although all lines of inquiry are being pursued.
She said: "At this stage we are still trying to establish exactly what happened at the address and how and why the victim was shot dead.
"I am appealing for anyone who may have seen or heard anything suspicious in the area, or was in the area at that time, to contact Dorset Police as soon as possible.
"There will be a heavy police presence in the vicinity and officers from the local Neighbourhood Policing Team will be carrying out high-visibility patrols to offer reassurance to the community. They can be contacted with any concerns."
A man shot dead at a house in East Dorset has been named.