Topics on Change-Point Estimation Under Adaptive Sampling Procedures.

[摘要] The problem of locating a jump discontinuity (change-point) in a smooth parametric regression model with abounded covariate is considered. It is assumed that one can sample the covariate at different values and measurethe corresponding responses. Budget constraints dictate that a total of n such measurements can be obtained. Amultistage adaptive procedure is proposed, where at each stage an estimate of the change-point is obtained andnew points are sampled from its appropriately chosen neighborhood. It is shown that such procedures acceleratethe rate of convergence of the least squares estimate of the change-point. Further, the asymptotic distributionof the estimate is derived using empirical processes techniques. The improved efficiency of the procedure isdemonstrated using real and synthetic data. This problem is primarily motivated by applications in engineeringsystems.In many applications, one is interested in estimating the entire regression function. Two criteria based on theexpected L2 error are defined. The optimality problems are studied and solved. The first criterion focuses onallocating the available samples appropriately over the covariate domain.The procedure involves using aninitial fraction of the budget to estimate the underlying model parameters. Subsequently, the design region ispartitioned into three segments, with the middle segment defined as a fixed neighborhood around the estimatedchange-point. The objective then becomes to allocate the available samples to these three segments so as tominimize an asymptotic expected L2 error depending on model parameters. In the second criterion one isinterested in allocating the proportion of samples between the two sampling stages, where in the first stageinitial estimates of the model parameters and the change-point are obtained while in the second stage animproved estimate of the change-point is sought.Finally, applications of the obtained results are extended to the jump boundary curve detection problem for a3-dimensional response surface.The motivation for examining this problem comes primarily from geology, whereone need recover the structure of the mine surface from mineral samples.Further an adaptive qualitativemethodology is proposed and its performance is illustrated through simulated examples.

[发布机构] University of Michigan

[效力级别] Change Point Estimation

[关键词] Adaptive Sampling;Change Point Estimation;Multi-stage Procedure;Skorohod Topology;Two-stage Procedure;Zoom In;Mathematics;Science;Statistics