In　multi-sensor　distributed　estimation　fusion,　local　estimation　errors　are　correlated　in　general.　Two　extreme　ways　to　handle　this　correlation　is　either　to　ignore　them　completely　or　to　have　them　fully　considered.　There　is　another　case　in　the　middle:　it　admits　the　existence　of　the　correlation,　but　does　not　know　how　large　it　is.　A　sensible　way　is　to　set　up　an　optimality　criterion　and　optimize　it　over　all　possible　such　correlations.　This　work　is　a　new　development　in　the　third　class.　First,　a　new　general　objective　function　is　introduced,　which　is　the　minimum　sum　of　statistical　distances　between　the　fused　density　and　the　local　posterior　densities.　Then　it　is　shown　that　the　new　criterion　leads　to　a　convex　optimization　problem　if　the　Kullback-Leibler　(KL)　divergence　is　used　as　the　statistical　distance　between　assumed　Gaussian　densities.　It　is　found　that　although　the　analytically　obtained　fused　estimate　using　the　new　criterion　differs　from　the　simple　convex　combination　rule　only　in　mean　squared　error　(MSE)　by　a　scaling　factor　N　(the　number　of　sensors　used),　it　is　pessimistic　semi-definite　in　MSE.　Numerical　examples　illustrate　the　effectiveness　of　the　proposed　distributed　fuser　by　comparing　with　several　widely　used　distributed　fusers.