##### Department of Mathematics,

University of California San Diego

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### Math 288 - Statistics

## Irene Gijbels

#### Katholieke Universiteit van Leuven, Belgium

## Local polynomial fitting and smooth and non-smooth curve estimation

##### Abstract:

In regression analysis the interest is in exploring the relationship between a response variable and some explanatory variable(s). Without knowledge about an appropriate parametric relationship between the variables, one often relies on nonparametric methods. Local polynomial fitting leads to estimators of the regression function and its derivatives up to a certain order. We very briefly discuss the basic properties of these estimators when the regression function is smooth. In particular we pay attention to the behaviour of the estimators in boundary regions. This local modelling technique is applicable in a variety of applications. When the regression function is non-smooth, e.g. discontinuous, the estimates are inconsistent in the non-smooth points. We briefly discuss some available nonparametric methods. We discuss, among others, a nonparametric estimation method that searches for compromising between the properties of jump-preserving and smoothing. The method chooses, in each point, among three estimates: a local linear estimate using only data to the left of the point, a local linear estimate based on only data to the right, and a local linear estimate using data in a two-sided neighborhood around the point. The data-driven choice among the three estimates is made by comparing, in some appropriate way, the weighted residual mean squares of the three fits. This results into a consistent estimate. We establish asymptotic properties of the estimator, and illustrate its performance via simulations and examples.

Host: A. Delaigle

### March 22, 2006

### 12:00 PM

### AP&M 2402

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