At　hybrid　analog-digital　(HAD)　transceiver,　an　improved　HAD　estimation　of　signal　parameters　via　rotational　invariance　techniques　(ESPRIT),　called　I-HAD-ESPRIT,　is　proposed　to　measure　the　direction　of　arrival　(DOA)　of　a　desired　user,　where　the　phase　ambiguity　due　to　HAD　structure　is　dealt　with　successfully.　Subsequently,　a　machine-learning　(ML)　framework　is　proposed　to　improve　the　precision　of　measuring　DOA.　Meanwhile,　we　find　that　the　probability　density　function　(PDF)　of　DOA　measurement　error　(DOAME)　can　be　approximated　as　a　Gaussian　distribution　by　the　histogram　method　in　ML.　Then,　a　slightly　large　training　data　set　(TDS)　and　a　relatively　small　real-time　set　(RTS)　of　DOA　are　formed　to　predict　the　mean　and　variance　of　DOA/DOAME　in　the　training　stage　and　real-time　stage,　respectively.　To　improve　the　precisions　of　DOA/DOAME,　three　weight　combiners　are　proposed　to　combine　the-maximum-likelihood-learning　outputs　of　TDS　and　RTS.　Using　the　mean　and　variance　of　DOA/DOAME,　their　PDFs　can　be　given　directly,　and　we　propose　a　robust　beamformer　for　directional　modulation　(DM)　transmitter　with　HAD　by　fully　exploiting　the　PDF　of　DOA/DOAME,　especially　a　robust　analog　beamformer　on　RF　chain.　Simulation　results　show　that:　(1)　the　proposed　I-HAD-ESPRIT　can　achieve　the　HAD　Cramer-Rao　lower　bound　(CRLB);　(2)　the　proposed　ML　framework　performs　much　better　than　the　corresponding　real-time　one　without　training　stage;　(3)　the　proposed　robust　DM　transmitter　can　perform　better　than　the　corresponding　non-robust　ones　in　terms　of　secrecy　rate.