The man's body was found on Wednesday.
He was suitably experienced and carried a personal locator beacon and a two-way radio.
The family were grateful for the support of Ultimate Hikes and emergency services, but would make no further comment at this stage, the statement said.
Police said the man's body was recovered about 6pm on Thursday near Giant Gate Falls.
A rāhui is in place in the Giant Gate Falls and Lake Ada area until April 17.
A police spokeswoman said there would be a postmortem to formally identify the body.
It is understood the man has family in Dunedin.