Anscombe was born on 8 February 1874 in Lindfield, Sussex, England. His parents Edmund and Eliza Anscombe (née Mason) emigrated on 27 June 1874 to New Zealand on the "Christian McAusland" in the assisted immigration scheme. They arrived in Dunedin with seven-month-old Edmund and his two-year-old sister Eliza. His father is described as a carpenter aged 25 years old from Sussex. His mother was 28. They arrived in Otago, New Zealand on 30 September 1874. His sister Edith Violet was born in Dunedin on 1 April 1885.