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B-splines are attractive for nonparametric modelling, but choosing the optimal number and positions of knots is a complex task. Equidistant knots can be used, but their small and discrete number allows only limited control over smoothness and fit. We propose to use a relatively large number of knots and a difference penalty on coefficients of adjacent B-splines. We show connections to the familiar spline penalty on the integral of the squared second derivative. A short overview of B-splines, of their construction and of penalized likelihood is presented. We discuss properties of penalized B-splines and propose various criteria for the choice of an optimal penalty parameter. Nonparametric logistic regression, density estimation and scatterplot smoothing are used as examples. Some details of the computations are presented.
A recent trend in the industrial applications of robust parameter design is to consider complex systems which are called "systems with dynamic characteristics" in Taguchi's terminology or signal-response systems in this paper. This potentially important tool in quality engineering lacks a solid basis on which to build a rigorous body of theory and methodology. The purpose of this paper is to provide such a basis. We classify signal-response systems into two broad types: measurement systems and multiple target systems. Three issues are then of fundamental importance. First, a proper performance measure needs to be chosen for system optimization, and this choice depends on the type of system. Taguchi's dynamic signal-to-noise ratio is shown to be appropriate for certain measurement systems but not for multiple target systems. Second, there are two strategies for modeling and analyzing data: performance measure modeling and response function modeling. Finally, the proper design of such experiments should take into account the modeling and analysis strategy. The proposed methodology is illustrated with a real experiment on injection molding.
In any sequential medical experiment on a cohort of human beings, there is an ethical imperative to provide the best possible medical care for the individual patient. This ethical imperative may be compromised if a randomization scheme involving 50-50 allocation is used as accruing evidence begins to favor (albeit not yet conclusively) one experimental therapy over another. Adaptive designs have long been proposed to remedy this situation. An adaptive design seeks to skew assignment probabilities to favor the treatment performing best thus far in the study, proportionately to the magnitude of the treatment effect.
Current researchers in adaptive designs are attempting to provide physicians with a wide choice of design options, and to address practical and ethical concerns within a rigorous mathematical framework. This paper focuses on several broad families of designs, including urn models, random walk rules and other rules. Numerous examples are given along with applications, dose-response studies, clinical trials for efficacy and combined toxicity-efficacy studies.
Ted Harris was born January 11, 1919, in Philadelphia, Pennsylvania. He grew up in Dallas, Texas, attended Southern Methodist University for two years and completed his undergraduate studies and some graduate work at the University of Texas at Austin. During World War II he served as a weather officer in England in the Army Air Force. He received his Ph.D. in 1947 from Princeton under Sam Wilks. From 1947 to 1966 he was a member of the mathematics department at The Rand Corporation in Santa Monica, California; he headed the department from 1959 to 1965. From 1966 to 1989 he was Professor of Mathematics and Electrical Engineering at the University of Southern California. Since 1989 he has been Professor Emeritus and Lecturer. In 1988 he was elected to the National Academy of Sciences, and in 1989 he received an honorary doctorate from Chalmers Institute of Technology, Sweden. He received an Albert S. Raubenheimer Distinguished Faculty Award in 1985 and a Distinguished Emeritus Award in 1990 from USC.