Two　novel　adaptive　filtering　algorithms　based　on　the　mixed　square/fourth　error　criterion　are　proposed　for　solving　sparse　system　identification　problems.　Motivated　by　the　fact　that　the　proportionate　update　scheme　can　enhance　the　tracking　ability　of　the　system,　we　develop　a　proportionate　least　mean　square/fourth　(PLMS/F)　algorithm　in　this　paper.　Combining　the　proportionate　update　scheme　and　the　LMS/F　algorithm,　the　proposed　PLMS/F　algorithm　shows　superiority　for　non-Gaussian　noise　environments.　Moreover,　to　further　improve　the　performance　of　the　PLMS/F　algorithm　in　the　noisy　input　cases,　a　bias-compensated　PLMS/F　algorithm　is　developed　by　incorporating　an　unbiased　criterion　to　compensate　the　bias　caused　by　input　noises.　Simulation　results　in　the　context　of　the　sparse　system　identification　framework　demonstrate　that　the　proposed　PLMS/F　and　bias-compensated　PLMS/F　algorithms　can　achieve　excellent　identification　performance　in　terms　of　steady-state　misalignment　and　convergence　speed　under　noisy　input　and　non-Gaussian　output　noise　environments.　©　2018　John　Wiley　&　Sons,　Ltd.