We　consider　the　statistical　inference　for　right-censored　data　when　censoring　indicators　are　missing　but　nonignorable,　and　propose　an　adjusted　imputation　product-limit　estimator.　The　proposed　estimator　is　shown　to　be　consistent　and　converges　to　a　Gaussian　process.　Furthermore,　we　develop　an　empirical　process-based　testing　method　to　check　the　MAR　(missing　at　random)　mechanism,　and　establish　asymptotic　properties　for　the　proposed　test　statistic.　To　determine　the　critical　value　of　the　test,　a　consistent　model-based　bootstrap　method　is　suggested.　We　conduct　simulation　studies　to　evaluate　the　numerical　performance　of　the　proposed　method　and　compare　it　with　existing　methods.　We　also　analyze　a　real　data　set　from　a　breast　cancer　study　for　an　illustration.