Why do Statistic recommendations disappear from DB Tuning Advisor after implementing recommended indexes?
After I implement the indexes 


I highly recommend you to read Aaron's excellent blog post: 


Don't just blindly create those "missing" indexes!


There are stats automatically created with indexes, so there's no statistic to suggest when an index exists (because by definition that statistic is already there). DTA is suggesting both an index and a stat - an index is usually best, but most costly in terms of maintenance; a stat is second-best, and can help with cardinality estimates, but not with things like seeking or sorting.  Not sure why it's obvious these aren't the same stats - but it could also be that those queries have benefitted enough from the indexes you created that there is no further benefit to derive from adding stats. 



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