You mention a "Christian Hiliard" in your posting here, however the 16th century Elizabethan era limner, or miniature portraitist to the Royals at that time was a Nicholas Hilliard (1547 - 1619). If your miniature of Queen Elizabeth I is indeed authentic to its time and by the hand of Nicholas Hilliard then it would be quite desirable on the market. The estimate is based on this possibly being an original 16th century miniature by Nicholas Hilliard, though it has not been authenticated by me.