On Hinkley's estimator: Inference about the change point

Maximum likelihood method is applied to estimate the change point of a distribution function associated with a sequence of independent random elements. Fluctuation theory of random walks is applied to show exact expressions for the limiting distribution of the maximum likelihood estimator of a change point. The derived expressions are computationally accessible in the sense that one may compute the exact asymptotic distribution of the change point through an algorithmic approach on the basis of the expressions derived. In showing this, a new formula for the ultimate maximum, the maximum of the sequence of partial maxima of a random walk is exhibited. Assuming that the underlying process is Gaussian, asymptotic distribution for the maximum likelihood estimate of the change point is obtained.