Sequential fixed-width confidence bands for kernel regression estimation

We consider a random design model based on independent and identically distributed (iid) pairs of observations (Xi, Yi), where the regression function m(x) is given by m(x) = E(Yi|Xi = x) with one independent variable. In a nonparametric setting the aim is to produce a reasonable approximation to the unknown function m(x) when we have no precise information about the form of the true density, f(x) of X. We describe an estimation procedure of non-parametric regression model at a given point by some appropriately constructed fixed-width (2d) confidence interval with the confidence coefficient of at least 1−. Here, d(> 0) and 2 (0, 1) are two preassigned values. Fixed-width confidence intervals are developed using both Nadaraya-Watson and local linear kernel estimators of nonparametric regression with data-driven bandwidths.

The sample size was optimized using the purely and two-stage sequential procedure together with asymptotic properties of the Nadaraya-Watson and local linear estimators. A large scale simulation study was performed to compare their coverage accuracy. The numerical results indicate that the confidence bands based on the local linear estimator have the best performance than those constructed by using Nadaraya-Watson estimator. However both estimators are shown to have asymptotically correct coverage properties.

Sample size determination for kernel regression estimation using sequential fixed-width confidence bands

We consider a random design model based on independent and identically distributed pairs of observations (Xi, Yi), where the regression function m(x) is given by m(x) = E(Yi|Xi = x) with one independent variable. In a nonparametric setting the aim is to produce a reasonable approximation to the unknown function m(x) when we have no precise information about the form of the true density, f(x) of X. We describe an estimation procedure of non-parametric regression model at a given point by some appropriately constructed fixed-width (2d) confidence interval with the confidence coefficient of at least 1−. Here, d(> 0) and 2 (0, 1) are two preassigned values. Fixed-width confidence intervals are developed using both Nadaraya-Watson and local linear kernel estimators of nonparametric regression with data-driven bandwidths. The sample size was optimized using the purely and two-stage sequential procedures together with asymptotic properties of the Nadaraya-Watson and local linear estimators. A large scale simulation study was performed to compare their coverage accuracy. The numerical results indicate that the confi dence bands based on the local linear estimator have the better performance than those constructed by using Nadaraya-Watson estimator. However both estimators are shown to have asymptotically correct coverage properties.

An analytical approach to predict web-warping and longitudinal strain in flexible roll formed sections of variable width

To reduce weight and improve passenger safety there is an increased need in the automotive industry to use Ultra High Strength Steel (UHSS) for structural and crash components. However, the application of UHSS is restricted by their limited formability and the difficulty of forming them in conventional stamping. An alternative method of manufacturing structural auto body parts from UHSS is the flexible roll forming process, which allows the manufacture of metal sheet with high strength and limited ductility into complex and weight-optimized components. One major problem in the flexible roll forming of UHSS is the web-warping defect, which is the deviation in height of the web area over the length of the profile. It has been shown that web-warping is strongly dependant to the permanent longitudinal strain formed in the flange of the part. Flexible roll forming is a continuous process with many roll stands, which makes numerical analysis extremely time intensive and computationally expensive. An analytical model of web-warping is therefore critical to improve design efficiency during the early process design stage before FEA is applied. This paper establishes for the first time an analytical model for the prediction of web-warping for the flexible roll forming of a section with variable width. The model is based on evaluating longitudinal edge strain in the flange of the part. This information is then used in combination with a simple geometrical model to investigate the relationship between web-warping and longitudinal strain with respect to process parameters.