Certain tests of constant failure rate versus failure rate increasing on the average are unbiased when complete samples of observations are available, as pointed out by Bickel and Doksum in the Annals of Mathematical Statistics (1968). In the present note, unbiasedness is proved when incomplete samples of failure data are available. A similar result is obtained for monotone tests of constant versus increasing failure rate. Finally, a table of percentiles is given to facilitate application of the total time on test statistic for testing constant failure rate versus failure rate increasing on the average.