a couple applying for my rental only wants the wife to undergo background check

[Jimmy Hunt]

I am trying to rent my house for the first time and I hired a realtor to advertise and show my house and screen renters. A couple (with three children) would like to move in, but the wife stated that she would be the primary applicant and would not provide any information other than income for her husband (no background/criminal check or credit score). They said they would keep looking elsewhere if I required the background check. I am in a hurry to rent my house. How would you respond?