Regression and likelihood (springer.1999)(305s)

Local Regression and Likelihood Clive Loader Springer [...]... astronomy, actuarial science and economics Modern areas of application include numerical analysis (Lancaster and Salkauskas 1986), sociology (Wu and Tuma 1990), economics (Cowden 1962; Shiskin, Young and Musgrave 1967; Kenny and Durbin 1982), chemometrics (Savitzky and Golay 1964, Wang, Isaksson and Kowalski 1994), computer graphics (McLain 1974) and machine learning (Atkeson, Moore and Schaal 1997) Despite... methods of local averaging were considered by Daniell (1946), Bartlett (1950), Grenander and Rosenblatt (1953), Blackman and Tukey (1958), Parzen (1961) and others Local polynomial methods were applied to this problem by Daniels (1962) 1.4 Modern Local Regression 11 1.4 Modern Local Regression The importance of local regression and smoothing methods is demonstrated by the number of different fields in which... the second derivative And so on Hastie and Loader (1993) contains an extensive discussion of these issues An alternative theoretical treatment of local regression is to view the method as an extension of kernel methods and attempt to extend the theory of kernel methods This treatment has become popular in recent years, for example in Wand and Jones (1995) and to some extent in Fan and Gijbels (1996) The... spline and penalized likelihood methods were introduced by Whitaker (1923) and Henderson (1924a) In modern literature there are several distinct smoothing approaches using splines; references include Wahba (1990), Friedman (1991), Dierckx (1993), Green and Silverman (1994), Eilers and Marx (1996) and Stone, Hansen, Kooperberg and Truong (1997) Orthogonal series methods such as wavelets (Donoho and Johnstone... definition of local regression to multiple predictors is straightforward; we require a multivariate weight function and multivariate local polynomials This was considered by McLain (1974) and Stone (1982) Statistical methodology and visualization for multivariate fitting was developed by Cleveland and Devlin (1988) and the associated loess method With two predictor variables, the local regression model... Local Regression Much work remains to be done to make local regression useful in practice There are several components of the local fit that must be specified: the bandwidth, the degree of local polynomial, the weight function and the fitting criterion 2.2.1 Bandwidth The bandwidth h(x) has a critical effect on the local regression fit If h(x) is too small, insufficient data fall within the smoothing window, and. .. as an extension of parametric regression methods, and is accompanied by an elegant finite sample theory of linear estimation that builds on theoretical results for parametric regression The work was initialized in some of the papers mentioned above and in the early work of Henderson The theory was significantly developed in the book by Katkovnik (1985), and by Cleveland and Devlin (1988) Linear estimation... kernel in Table 1 of M¨ller (1984) 2 Local Regression Methods This chapter introduces the basic ideas of local regression and develops important methodology and theory Section 2.1 introduces the local regression method Sections 2.2 and 2.3 discuss, in a mostly nontechnical manner, statistical modeling issues Section 2.2 introduces the bias-variance tradeoff and the effect of changing smoothing parameters... Local Regression Estimate 0.6 0.8 1.0 1.2 E FIGURE 2.1 Local regression: Smoothing windows (bottom); local least squares fits (solid curves) and estimates µ(x) (big circles) ˆ The local regression procedure is illustrated in Figure 2.1 The ethanol dataset, measuring exhaust emissions of a single cylinder engine, is originally from Brinkman (1981) and has been studied extensively by Cleveland (1993) and. .. usually has x ≈ xw , and there is little differ¯ ence between local constant and local linear fitting A local linear estimate exhibits bias if the mean function has high curvature 2.1.1 Interpreting the Local Regression Estimate In studies of linear regression, one often focuses on the regression coefficients One assumes the model being fitted is correct and asks questions 2.1 The Local Regression Estimate . 248 C c-locfit 251 C. 1 Installation 251 C. 1.1 Windows 95, 98 and NT 251 C. 1.2 UNIX 251 C. 2 Using c- locfit 252 C. 2.1 Data in c- locfit 253 C. 3 Fitting with c- locfit 255 C. 4 Prediction 256 C. 5 Some. theoretical work and in applications. Most chapters contain distinct sections introducing methodology, computing and practice, and theoretical results. The methodological and practice sections should. applies to any model and software; not just local regression and loc- fit!). On the other hand, asymptotic methods for boundary correction re- ceive no coverage, since local regression provides