The problem of locating a jump discontinuity (change-point) in a smooth parametric regression model with a bounded covariate is considered in this dissertation. It is assumed that one can sample the covariate at different values and measure the corresponding responses. Budget constraints dictate that a total of n such measurements can be obtained. A multistage adaptive procedure is proposed, where at each stage an estimate of the change point is obtained and new points are sampled from its appropriately chosen neighborhood. It is shown that such procedures accelerate the rate of convergence of the least squares estimate of the change-point. Further, the asymptotic distribution of the estimate is derived using empirical processes techniques. The improved efficiency of the procedure is demonstrated using real and synthetic data. This problem is primarily motivated by applications in engineering systems.; In many applications, one is interested in joint estimation of the parametric model and its special feature, namely the change point. Two criteria based on the expected L2 error are defined and the optimality problems are studied and solved. The first criterion focuses on allocating the available samples appropriately over the covariate domain. The procedure involves using an initial fraction of the budget to estimate the parameters of the underlying model. Subsequently, the design region is partitioned into three segments, with the middle segment defined as a fixed neighborhood around the estimated change-point. The objective then becomes to allocate the available samples to these three segments so as to minimize an asymptotic expected L2 error that depends on model parameters, the estimates from the initial fraction acting as surrogates. In the second criterion one is interested in allocating the proportion of samples between the two sampling stages, where in the first stage initial estimates of the model parameters and the change point are obtained while in the second stage an improved estimate of the change point is sought.; Finally, applications of the obtained results are extended to the jump boundary curve detection problem for a 3-dimensional response surface. The motivation for examining this problem comes primarily from geology, where one is interested in recovering the structure of the mine surface from mineral samples. Further an adaptive qualitative methodology is proposed and its performance is illustrated through simulated examples.
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