TIGHT UPPER CONFIDENCE LIMITS FROM DISCRETE DATA

Consider the problem of finding an upper 1 –α confidence limit for a scalar parameter of interest ø in the presence of a nuisance parameter vector θ when the data are discrete. Approximate upper limits T may be found by approximating the relevant unknown finite sample distribution by its limiting distribution. Such approximate upper limits typically have coverage probabilities below, sometimes far below, 1 –α for certain values of (θ, ø). This paper remedies that defect by shifting the possible values t of T so that they are as small as possible subject both to the minimum coverage probability being greater than or equal to 1 –α, and to the shifted values being in the same order as the unshifted ts. The resulting upper limits are called ‘tight’. Under very weak and easily checked regularity conditions, a formula is developed for the tight upper limits.