Maximum likelihood estimation of a change-point in the distribution of independent random variables: General multiparameter case

In a sequence ofn independent random variables the pdf changes fromf(x, 0) tof(x, 0 + [delta]vn-1) after the firstn[lambda] variables. The problem is to estimate[lambda] [set membership, variant] (0, 1 ), where 0 and [delta] are unknownd-dim parameters andvn --> [infinity] slower thann1/2. Letn denote the maximum likelihood estimator (mle) of[lambda]. Analyzing the local behavior of the likelihood function near the true parameter values it is shown under regularity conditions that ifnn2(- [lambda]) is bounded in probability asn --> [infinity], then it converges in law to the timeT([delta]j[delta])1/2 at which a two-sided Brownian motion (B.M.) with drift1/2([delta]'J[delta])1/2[short parallel]t[short parallel]on(-[infinity], [infinity]) attains its a.s. unique minimum, whereJ denotes the Fisher-information matrix. This generalizes the result for small change in mean of univariate normal random variables obtained by Bhattacharya and Brockwell (1976,Z. Warsch. Verw. Gebiete37, 51-75) who also derived the distribution ofT[mu] for[mu] > 0. For the general case an alternative estimator is constructed by a three-step procedure which is shown to have the above asymptotic distribution. In the important case of multiparameter exponential families, the construction of this estimator is considerably simplified.