Shape-Restricted Inference

AbstractDMS-0204572PI: Mary MeyerTitle: Shape-Restricted InferenceConsider the problem of estimating a function given observations with some random component. The function may be a regression function, a density, or a probability density function such as in bioassay models. Shape-restricted methods allow the practitioner to impose only qualitative restrictions on the class of functions, such as increasing, concave, or sigmoidal. Estimates may be obtained using maximum-likelihood ideas; there are many problems in inference to be solved. Goals for this proposal include developing confidence bounds for regression functions using shape restrictions, developing tests for an ANCOVA type of model with a shape-restricted covariate, time-series analysis with shape-restricted trend function, smooth shape-restricted function estimation, and developing a robust regression estimator using a shape-restricted error density.Traditional statistical methods for regression, density estimation and bioassay problems include: 1) estimating a function, 2) estimating the quality of fit, perhaps using confidence bounds, and 3) testing hypotheses about the function. The investigator wishes to develop these methods nonparametrically, that is, without imposing a parametric form for the function. Shape-restricted methods in statistics approach these estimation and inference problems with a minimum of assumptions about the functional form. For example, a growth curve may be assumed to be increasing and concave, or a probability curve might be sigmoidal- a more general assumption than the usual logistic model. A density function might be assumed to be symmetric and unimodal, in a situation where stronger assumptions like normality might not be justified. A fit to a function using fewer assumptions will have more fidelity to the data. Perhaps more importantly, these shape-restricted fits may be used to test the validity of the parametric models, or to select from several candidate parametric models.

AbstractDMS-0204572PI: Mary MeyerTitle: Shape-Restricted InferenceConsider the problem of estimating a function given observations with some random component. The function may be a regression function, a density, or a probability density function such as in bioassay models. Shape-restricted methods allow the practitioner to impose only qualitative restrictions on the class of functions, such as increasing, concave, or sigmoidal. Estimates may be obtained using maximum-likelihood ideas; there are many problems in inference to be solved. Goals for this proposal include developing confidence bounds for regression functions using shape restrictions, developing tests for an ANCOVA type of model with a shape-restricted covariate, time-series analysis with shape-restricted trend function, smooth shape-restricted function estimation, and developing a robust regression estimator using a shape-restricted error density.Traditional statistical methods for regression, density estimation and bioassay problems include: 1) estimating a function, 2) estimating the quality of fit, perhaps using confidence bounds, and 3) testing hypotheses about the function. The investigator wishes to develop these methods nonparametrically, that is, without imposing a parametric form for the function. Shape-restricted methods in statistics approach these estimation and inference problems with a minimum of assumptions about the functional form. For example, a growth curve may be assumed to be increasing and concave, or a probability curve might be sigmoidal- a more general assumption than the usual logistic model. A density function might be assumed to be symmetric and unimodal, in a situation where stronger assumptions like normality might not be justified. A fit to a function using fewer assumptions will have more fidelity to the data. Perhaps more importantly, these shape-restricted fits may be used to test the validity of the parametric models, or to select from several candidate parametric models.