If a girl under 16 gives birth and will not identify the father, a new Mississippi law — most likely the first of its kind in the country — says the authorities must collect umbilical cord blood and run DNA tests to prove paternity as a step toward prosecuting statutory rape cases. Supporters say the law is intended to chip away at Mississippi’s teenage pregnancy rate, which has long been one of the highest in the nation. But critics say that though the procedure is painless, it invades the medical privacy of the mother, father and baby. The law took effect on July 1 but has not been used yet. Cord blood samples would have to be taken immediately after birth, and the state medical examiner is setting administrative rules for how the blood will be collected and stored.