Estimation of the time of attainment of a threshold with control of the probability of overestimation

Consider a random variable whose mean value depends on time such as the mean germination rate of seeds stored in a seed bank. Interest lies in the estimation of the time at which the mean of the random variable attains a known fixed value when overestimation is considered to be more costly than underestimation. We assume the random variable under study has a mean value that is a linear function of time with i.i.d. normal errors. Estimates based on sequential sampling carried out at prespecified sampling times tend to be more costly than those based on estimated inspection times since frequent sampling increases sampling cost and infrequent sampling could lead to a high probability of stopping past the desired time and a large average distance past the desired time. Among three procedures based on estimated inspection times a procedure based on inspection times determined exactly after updating the information performed better than the other two in terms of a lower sample size requirement. The probability of stopping past the desired time was slightly inflated for this procedure but the average distance from the desired time of those estimates which exceeded it was very small. The existing procedure adopted for the current formulation and setup controlled the probability of stopping late at the desired level and performed better for small values of the slope. Compared to a procedure based on specific constants selected to control the probability of stopping late at the desired level for each slope, the sample size requirement of the existing procedure was higher, ranging from 616% to 2493%. This higher sample size requirement did not lead to an appreciable advantage when the slope is not small.