The best way of doing this depends on your DBMS, you can either use NOT EXISTS or LEFT JOIN/IS NULL
NOT EXISTS
SELECT  *
FROM    PersonAlias_Table pat
WHERE   NOT EXISTS
        (   SELECT  1
            FROM    Person_Table pt
            WHERE   pt.Person_ID = pat.Person_ID
            AND     pt.Uniqe_ID = pat.Uniqe_ID
        )
LEFT JOIN/IS NULL
SELECT  *
FROM    PersonAlias_Table pat
        LEFT JOIN Person_Table pt
            ON pt.Person_ID = pat.Person_ID
            AND pt.Uniqe_ID = pat.Uniqe_ID
WHERE    pt.Person_ID IS NULL
According to this answer in MySQL NOT EXISTS is a little bit less efficient, In SQL Server LEFT JOIN / IS NULL is less efficient, in PostgreSQL and Oracle both methods are the same, so it is really personal preference.