Curve estimation from observed noisy data has broad applications. Jong-Hoon Joo and Peihua Qiu consider this in “Jump Detection in a Regression Curve and Its Derivative.” Their ideas are important for applications in which the underlying regression curve may have singularities, including jumps and roofs/valleys (i.e., jumps in the first order derivative of the regression curve), at some unknown positions, representing structural changes of the related process. A number of jump-detection procedures have been proposed, most of which are based on estimation of the (one-sided) first-order derivatives of the true regression curve. Motivated by related research in image processing, this article proposes an alternative jump-detection procedure that exploits jumps in the second-order derivatives and the first-order derivatives. Theoretical justifications and numerical studies show that this jump detector works well in applications. This procedure is then extended for detecting roofs/valleys of the regression curve. A curve estimation procedure is also proposed, which can preserve possible jumps/roofs/valleys when removing noise.
Much research in multiple regression has been devoted to identification of the best subset from among a set of candidate predictors. In the case of polynomial regression, the variable selection process can be further complicated by the desire to obtain subsets that are hierarchically well-formulated, so that inclusion of a high-order effect requires inclusion of lower-order effects in the same factors. Michael J. Brusco, Douglas Steinley, and J. Dennis Cradit develop an algorithm for computing such subsets in their paper, “An Exact Algorithm for Hierarchically Well-Formulated Subsets in Second-Order Polynomial Regression.” They present a branch-and-bound algorithm for subset selection in second-order polynomial regression. They apply the new algorithm to a well-known data set from the regression literature and compare the results to those obtained from a branch-and-bound algorithm that does not impose the hierarchical constraints. The results of this comparison reveal that the hierarchical constraints yield only a small penalty in explained variation. Fortran and MATLAB implementations of the branch-and-bound algorithm are available as supplemental materials here.
The generalized Pareto distribution is widely used to model extreme values such as exceedences over thresholds in flood data. Existing methods for estimating parameters have theoretical or computational defects. In “A New and Efficient Estimation Method for the Generalized Pareto Distribution,” Jin Zhang and Michael A. Stephens propose a new estimator that is computationally easy, free from the problems observed in traditional approaches, and performs well compared to existing estimators. A numerical example involving heights of waves is used to illustrate the various methods, and tests of fit are performed to compare them.
The gamma distribution is relevant to numerous areas of application in the physical, environmental, and biological sciences. Dulal K. Bhaumik, Kush Kapur, and Robert D. Gibbons present methods for “Testing Parameters of a Gamma Distribution for Small Samples.” They derive new small sample-based tests for the shape, scale, and mean of the gamma distribution. Simulations are used to study the type I error rates and statistical power of the tests and reveal that the new tests maintain nominal type I error rates well as long as the shape parameter is not too small; even then, the results are only slightly conservative. The authors illustrate the new tests using real applications taken from engineering, medicine, and environmental science.
The issue concludes with “A Technical Note on ‘Sample Size Determination for Achieving Stability of Double Multivariate Exponentially Weighted Moving Average Controller’,” by Sheng-Tsaing Tseng and Bo-Yan Jou. This note is a sequel to a recent Technometrics article by Tseng et al. that presented an explicit formula for determining a minimum sample size (needed to construct the input-output predicted model) in such a way that the asymptotic stability of a double multivariate exponentially weighted moving average controller can be achieved with a guaranteed probability. That formula indicates that two key components of the sample size determination are the canonical correlation of input-output variables and the condition number of the covariance matrix of input variables. This note shows that the goal can be accomplished with a reduced sample size that depends on only the canonical correlation.
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