Comparing and Bucketing Dates

I am looking for help, I have two dates that I need to put in a 3rd collumn depending on 3 scenarios.

F2 = Due Date (This will always have a date)
H2 = Published Date (This will be null until published then have the publish date)
I2 = On Time (This is where I am calculateing this needs to equal Not Due, Ontime, Past Due)

I have not been able to work through the appropirate if statements to make this work, I try to compare against today and cant get anything to work from that point. Any thoughts from the community?

I am comparing the due date to the publish date, I am also comparing a blank publish date to the due date and cross comparing it to today. Here is what I have so far. I may have it solved, I am validating the data to make sure I did it correct.

=IF(AND(ISBLANK([@[Published Date]]), [@[Due Date]]>=TODAY()), "Not Due", IF(AND(ISBLANK([@[Published Date]]), [@[Due Date]]<=TODAY()), "Past Due", IF([@[Published Date]]<=[@[Due Date]], "On Time", IF([@[Published Date]]>=[@[Due Date]], "Past Due", ))))

It looks like John's solution works as well as the one I suggested earlier. Thanks for everyone's help getting this put together.