A note on the conditional approach to interval estimation in the calibration problem

In the calibration problem, the need to construct a confidence interval to estimate the unknown chi 0 arises when the null hypothesis of zero slope is rejected. Otherwise, the resulting confidence interval will be infinite to reflect the fact that the slope of the regression line may be zero. Under the condition of rejecting the hypothesis of zero slope, we study the properties of the conditional coverage rate of the calibration confidence interval. The conditional coverage rate (P1) is a function of the slope, distance between chi 0 and the mean of the trailing sample means, the sum of squares of chi, and n. When the true slope is close to 0 and chi 0 is away from means, P1 can go down to 0. On the other hand, as the power of testing zero slope reaches 1, with or without chi 0 close to means, P1 will tend to the desired nominal coverage rate. In summary, one should choose a reasonably small alpha in testing zero slope to avoid constructing a confidence interval for chi 0 when the true slope is 0. In addition, it is desirable to have high power in testing zero slope so that the resulting confidence interval will maintain the desired coverage rate when using the conditional approach in the calibration problem.