Abstract
In this paper, we consider an adaptive sequential CUSUM procedure in an exponential family where the change-point and post-change parameters are estimated adaptively. It is shown that the adaptive CUSUM procedure is efficient at the first order. The conditional biases of the estimation for the change-point and post-change parameter are studied. Comparison with the classical CUSUM procedure in the normal case is made. Nile river flow and average global temperature data sets are used for demonstration.
Highlights
Let dFθ(x) = exp(θx − c(θ))dF0(x) be a standard exponential family with c(0) = c′(0) = 0 and c′′(0) = 1
Our discussion is mainly focused on the change-point and post-change parameter estimation after detection under the exponential family model which extends the results of Wu (2005) and Lorden and Pollak (2008)
By standardizing rest of the data starting from number 69, the adapted CUSUM procedure detected the third change-point at number 97 with alarm at 103
Summary
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, 2005, 2015; Lorden & Pollak, 2005, 2008). Our discussion is mainly focused on the change-point and post-change parameter estimation after detection under the exponential family model which extends the results of Wu (2005) and Lorden and Pollak (2008). By treating the CUSUM procedure as a sequence of sequential tests, we can use the adaptive sequential tests (Robbins & Siegmund, 1974, 1975) by estimating the post-change parameters adaptively for each test. The biases for the change-point and post-change parameter estimation are studied theoretically in Section 3 by using the renewal property of the adaptive CUSUM process.
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