A sequential sampling scheme in which observations are taken until the observed Fisher information exceeds a target value is considered for a restricted class of parametric families. It is proved that the theorems for fixed sample size asymptotic likelihood inference conditional on ancillary statistics are directly applicable to the sequential sampling scheme for location families. The fixed sample-size approximate ancillary statistic for two-dimensional curved exponential families is shown to be ancillary in the sequential sampling scheme. Evidence from Monte Carlo simulations suggests the applicability of the fixed sample size conditional likelihood inference theorems to these families as well. The distribution of the sequential sample size is shown to be asymptotically normal.
"Sequential Sampling Based on the Observed Fisher Information to Guarantee The Accuracy of the Maximum Likelihood Estimator." Ann. Statist. 11 (1) 68 - 77, March, 1983. https://doi.org/10.1214/aos/1176346057