This is based on:
* Oral traditions: Māori oral traditions describe a long journey from Hawaiki (a mythical ancestral homeland, possibly in Polynesia), with some stories mentioning specific ancestors and their voyages.
* Archaeological evidence: Excavations and analysis of archaeological sites show evidence of Māori settlements dating back to this period. This includes artifacts, food remains, and settlement structures.
* Radiocarbon dating: Carbon dating of bones and other organic materials found at archaeological sites provides further confirmation of the age of Māori settlements.
It's important to note that these dates are estimates and there's ongoing research to refine them. The exact date of arrival may vary depending on the specific location in New Zealand.