Hi Graeme: I agree with Shelly. Additional information about the family, such as the names of Arthur's parents', where he entered the U.S., who he came with, what his occupation was, etc. would help.
I did find Arthur and siblings with his mother, Mary Ann Roberts, as head of the family in the 1871 U.K. census in Gillingham.
In Pennsylvania, marriages have always been registered at the county level. And until 1906 births were also registered at the county level. Unfortunately, Pennsylvania has 67 counties. Unless you can narrow down the search, you'll have a daunting task on your hands. Some marriages and births in Pennsylvania in the late 1800s were not registered at all.
Incidentally, I looked at the two Pennsylvania marriage databases on FamilySearch for the marriage of Arthur Roberts and Mary Davis, but found nothing.
There was an arrival in New York on July 16, 1890 of an Arthur Roberts, 21 years old and born England. He was a laborer and came on the SS Teutonic sailing from Liverpool and Queenstown. Found that on Castlegarden.org (
www.Castlegarden.org), which is a database of passengers arriving in New York between 1820 and 1913.
He might also have come by way of Philadelphia or even Boston.
John