"We as a family are heartbroken," it said.
"Sadly, deteriorating weather conditions overwhelmed him while he was on the lake.
"He was a most dearly loved son of Barb and Rob, brother of Mark, Ben and William. He will also be hugely missed by his grandparents, aunties, uncles, cousins, and many, many friends."
The family said they appreciated the "care and love" they had received from Private Duncan's NZDF mates, as well as the army.
"We will be eternally grateful for their support," the statement said.
"We are also grateful to the military, police, and the navy in their search for Hamish at Lake Coleridge, one of the most beautiful places in the world.
"Hamish was loving his time in the military and basic training was a challenge that he was proud to have achieved. He had commenced his training as a medic within the NZ Army, and the camaraderie he was enjoying with his friends on the course was all a young man could wish for.
"We have been fortunate to meet many of his classmates who are a credit to their families and the services.
"We are grateful for the continuing support and prayers we have received and will carry Hamish with us in our hearts forever."
A Defence Force spokesman also spoke of the sadness in not being able to find Private Duncan.
"His peers at Defence Health School will remember him as an outgoing, talented, smart and kind young man," a statement said.
"The Defence Force will continue to support Hamish's colleagues and his family throughout this difficult time."
Canterbury police are currently reviewing and evaluating the search for Private Duncan in consultation with the coroner and members of the police national dive squad.
