Nonanticipating estimation applied to sequential analysis and changepoint detection

Abstract

Suppose a process yields independent observations whose distributions belong to a family parameterized by θ ∈ Θ. When the process is in control, the observations are i.i.d. with a known parameter value θ0. When the process is out of control, the parameter changes. We apply an idea of Robbins and Siegmund [Proc. Sixth Berkeley Symp. Math. Statist. Probab. 4 (1972) 37-41] to construct a class of sequential tests and detection schemes whereby the unknown post-change parameters are estimated. This approach is especially useful in situations where the parametric space is intricate and mixture-type rules are operationally or conceptually difficult to formulate. We exemplify our approach by applying it to the problem of detecting a change in the shape parameter of a Gamma distribution, in both a univariate and a multivariate setting.

Guaranteed maximum likelihood splitting tests of a linear regression model

“…. , Z n ) as the MLEs of β 10 , θ 1 , we obtain an analogous conclusion regarding the asymptotic result for α [15]. This illustrates the closeness of the true significance level to its upper bound (in the inequality of Proposition 2.1) for these cases.…”

“…Remark 2.1 Note that the estimation is similar to the one in d 1ij , which was used in the sequential approach of Robbins and Siegmund [14], Lorden and Pollak [15], and Gurevich and Vexler [5]. The estimation is intended to preserve a Martingale structure of likelihood ratios kn under H 0 .…”

“…On the basis of this idea, they produced an upper bound for the level of significance. The same method was used by Yakir et al [4], Gurevich and Vexler [5] and Lorden and Pollak [15]. Although our model does not consider identical distributed observations with known distribution before the change, we use a Martingale structure to obtain a guaranteed non-asymptotic upper bound for the significance levels of our tests.…”

Estimation of Change-point and Post-change Parameters after Adaptive Sequential CUSUM Test in an Exponential Family

“…However, explicit forms are typically difficult to obtain. The third, considered in this paper, is to use the adaptive CUSUM procedure by estimating the change-point and the post-change parameter adaptively (Draglin, 1990;Wu, 2005Wu, , 2015Lorden & Pollak, 2005. The advantage for this approach is the change-point and post-change parameters are easily identified after the detection as the basic form of the CUSUM procedure is kept.…”

“…Capizzi and Mascrotto (2003) proposed an adaptive EWMA procedure. An adaptive Shiryayev-Roberts procedure using the adaptive estimators is considered in Lorden and Pollak (2005). Yashchin (1995) and Jiang et al (2008) used the EWMA as the adaptive post-change mean estimator.…”