Abstract: | In a sequence ofn independent random variables the pdf changes fromf(x, 0) tof(x, 0 + δvn−1) after the firstnλ variables. The problem is to estimateλ (0, 1 ), where 0 and δ are unknownd-dim parameters andvn → ∞ slower thann1/2. Letn denote the maximum likelihood estimator (mle) ofλ. Analyzing the local behavior of the likelihood function near the true parameter values it is shown under regularity conditions that ifnn2(− λ) is bounded in probability asn → ∞, then it converges in law to the timeT(δjδ)1/2 at which a two-sided Brownian motion (B.M.) with drift1/2(δ′Jδ)1/2ton(−∞, ∞) 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 ofTμ forμ > 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. |