(If you use any portion of these notes, please give credit to me, Lois Sorensen, and to other sources as noted herein.)
In 1870 Arthur is age 17, living with his parents and working as a store clerk in Hudson, Columbia Co., NY.
In 1880 he may be the Arthur Remington age 27 living at Omaha, NE, working as a store clerk and living in a rooming house. However, the census gives his birthplace as Iowa. There are other Remington relatives there (Wager H. Remington and his wife), so I suspect he may be this Arthur.
In 1910 there is an Arthur Remington living at Yorkshire, Cattaraugus Co., NY.