TIGHT UPPER CONFIDENCE LIMITS FROM DISCRETE DATA

summary 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 limitsTmay 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 valuestofTso 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 unshiftedts. The resulting upper limits are called ‘tight’. Under very weak and easily checked regularity conditions, a formula is developed for the tight upper limits.