Estimation of location of discontinuity in a density

Statistical Simulation

“…Following the approach of Goria (1972) and Le Cam and Yang (2000, p. 103), we show that the log likelihood ratio is Locally Asymptotically Quadratic (LAQ) for local variations in the parameter that converge to zero at rate δ N = N -1/α This is sufficient for the argument of Goria (1972, Chapter 2) that N 1/α is the best attainable rate of convergence.…”

“…The MASS estimator in this application can be interpreted as an alternative to the kernel-smoothing methods developed by Ionides (2001) It is easy to characterize the asymptotic distributions of some conventional estimators for location, the mean, the median, and the average of the extremes. Following Goria (1972), we will show that none of these converge at the best attainable rate. The mean is consistent and asymptotically normal for θ at a N 1/2 rate, by conventional application of a central limit theorem.…”

“…The following paragraphs outline a simple class of irregular location parameter problems where MLE are not welldefined, but it is possible to characterize features of good estimators and show that MASS estimation can yield estimators that perform well. Our analysis draws on the analysis of non-regular estimation problems by Goria (1972), Akahira and Takeuchi (1995), and Le Cam and Yang (2000). The MASS estimator in this application can be interpreted as an alternative to the kernel-smoothing methods developed by Ionides (2001) It is easy to characterize the asymptotic distributions of some conventional estimators for location, the mean, the median, and the average of the extremes.…”