The problem of estimating the change-point in a sequence of independent random variables is considered. As the sample sizes before and after the change-point tend to infinity, Hinkley (1970) showed that the maximum likelihood estimate of the change-point converges in distribution to that of the change-point based on an infinite sample. Letting the amount of change in distribution approach 0, it is shown that the distribution, suitably normalized, of the maximum likelihood estimate based on an infinite sample converges to a simple one which is related to the location of the maximum for a two-sided Wiener process. Numerical results show that this simple distribution provides a good approximation to the exact distribution (with an infinite sample) in the normal case. However, it is unclear whether the approximation is good for general nonnormal cases.
"Approximating the Distribution of the Maximum Likelihood Estimate of the Change-Point in a Sequence of Independent Random Variables." Ann. Statist. 15 (3) 1321 - 1328, September, 1987. https://doi.org/10.1214/aos/1176350509