In a nominative proxy signature scheme, an original singer delegates his signing power to a proxy signer, who generates a nominative signature on behalf of the original signer. In a nominative proxy signature scheme, only the nominee can verify the signature and if necessary, only the nominee can prove its validity to the third party. In this paper, we first classify the nominative proxy signature into two types, original-nominative proxy signature and proxy-nominative proxy signature. Then we analyze the nominative proxy scheme proposed by Park and Lee. We show that the scheme suffers from universal verification. We also point out that the scheme presented by Seo and Lee is insecure and the scheme cannot provide nonrepudiation. Finally we present our nominative proxy signature schemes which overcome the weakness mentioned above.