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Just a note

For a conventional 2-tailed confidence interval, attaching Î to yields lower and upper confidence limits, L and U (and U-L=Î). One-tailed confidence limits use either the upper or lower limit - and treat it as a random variable.

For any confidence limit, the intended coverage is the proportion of intervals which are expected to enclose Θ - in other words 1 - α of them. For 1-tailed confidence limit the actual coverage by U is the proportion of intervals where U is greater than Θ, in other words P[Θ < U]. The coverage error is simply the difference between observed and expected coverage, P[Θ < U] - [1 - α]. So, if P[Θ < U] = [1 - α] there is no coverage error.

Under this model it is the location of L and U, relative to Θ, which is of importance - not the length of the interval. One-tailed confidence interval lengths, L to ∞, or −∞ to U, are liable to be extremely large. How far Θ lies inside or outside that interval is of no interest - merely whether it does or not. In other words, provided that U is greater than Θ on 100[1 - α]% of occasions, there is perfect coverage - irrespective of its mean location, or how else U is distributed, or what else Î does...