This　paper　proposes　an　efficient　interval　moment　method　(IMM)　for　uncertainty　propagation　analysis　with　non-parameterized　probability-box　(p-box),　where　the　bounds　of　statistical　moments　and　cumulative　distribution　function　(CDF)　of　output　response　can　be　simultaneously　obtained.　Firstly,　two　output　response　bounds　are　defined　based　on　the　equivalent　probability　transformation,　which　converts　the　original　imprecise　uncertainty　propagation　problem　into　two　precise　uncertainty　propagation　problems.　Then,　sparse　grid　numerical　integration　(SGNI)　is　employed　to　estimate　the　statistical　moments　of　output　response　bounds.　To　improve　computational　efficiency,　a　multi-interval　efficient　global　optimization　(MI-EGO)　algorithm　is　developed　to　capture　the　minimum　and　maximum　responses　on　all　collocation　intervals　of　SGNI.　By　reconstructing　the　distributions　of　output　response　bounds　using　the　maximum　entropy　method,　the　CDF　bounds　of　output　response　can　be　acquired　accordingly.　Furthermore,　by　reusing　the　previous　functional　evaluations　in　the　interval　multiplication,　the　bounds　of　the　statistical　moments　of　output　response　can　be　estimated　by　SGNI　again.　Three　numerical　examples　are　investigated　to　verify　the　accuracy　and　efficiency　of　the　proposed　method.　©　2023,　The　Author(s),　under　exclusive　licence　to　Springer-Verlag　GmbH　Austria,　part　of　Springer　Nature.