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previous estimate can be repeated until, hopefully, the sequence of estimates converges. See Weisberg (1985, Section 12.1), Myers (1990, 9), Bates and Watts (1988), and Seber and Wild (1989). The least-squares procedure in the preceding paragraph is most justifiable if the errors are assumed to be independent and all to have the same variance. However, the data may more closely follow a model in which the errors have unequal variances. In the model Y; = g(x;, 9) + e;, let us allow the variance of e; to depend on the explanatory variables, on the parameters in 9, and on other parameters as well. A common model of this type is the one that assumes Var(e) = (J"2g(x;, 9)\ where (J" and A are unknown parameters. One can estimate the parameters by an iterative procedure similar to the one above except that weighted, rather than ordinary, linear least squares is used at each iteration. See Carroll and Ruppert (1988, 2) and Seber and Wild (1989, Section 2.8).
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Generalized Linear Models. In a generalized linear model the regression function has the form JL; = h(fJ'x), where h is a known function. The function h can be any strictly monotone (that is, either strictly increasing or strictly decreasing) differentiable function. Unless h is a linear function, this is a nonlinear regression function, but note that the explanatory variables and the parameters are interrelated in a linear way. The generalized linear model further assumes that the random variable y; has a distribution with a probability density function (or probability mass function if y; is discrete) of the form !(y;; 0;, </ = c(y;, </ exp{a(</ [YiO; - b(O)]}. This includes normal, gamma, binomial, and Poisson distributions. The parameter vector fJ is linked to the distribution of Y; by the fact that JLi = b'(O). To estimate the parameters, the maximum likelihood method is used. The linear form of the interrelationship between X; and fJ and the exponential form of !(Y;; 0;, </ allow some useful aspects of linear regression analysis to be carried over to the analysis of these more general models. See McCullagh and Nelder (1989) and Myers (1990, Section 7.6). Nonparametric Smoothing. In nonlinear regression it is assumed that the form of the regression function is known. That is, the function g in JL; = g(x;,O) is assumed known. A more general regression model is obtained by letting the form of the regression function be unknown. Such a model is especially appropriate for exploratory data analysis. Let us suppose JL; = g(x), where g is an unknown function. Estimating the function g can be viewed as "smoothing" a plot of the data. One tries to find a "smooth" function g(x) such that, in the (p + 1}-dimensional plot of the data, the points (x;, g(x)) are close to the data points (x;, y). For example, one might take g(x) to be some sort of "local average", such as the average of all values
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Hardie (1990). Such regression procedures are called nonparametric because no parametric model is assumed for the regression function, but note that they are unrelated to the nonparametric regression procedure described in 6. Estimation by local averaging does not perform as well in multiple regression as it does in simple regression. This problem led to the development of projection pursuit regression, which estimates a multiple regression function through an iterative sequence of simple regression smoothings; see Friedman and Stuetzle (1981). The same idea has been used to generalize generalized linear models to generalized additive models. A generalized linear model assumes that f.Li = h(f3'x) = h(f3 1 X i1 + ... +f3pXip)' whereas a generalized additive model assumes only that f.Li = h(gl(X i1 ) + ... +g/x ip ' where g l' ... , g p are unknown smooth functions; see Hastie and Tibshirani (1990).
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Measurement Error in the Explanatory Variables. Another situation in which the linear regression model (1.1) may be inadequate is when the explanatory variables are subject to measurement error. Suppose that (1.1) is valid but that we are not able to observe the exact values of the explanatory variables X j . Instead, due to measurement error, we observe (*) Zj = Xj + d j , where d j is a random error. Unless the measurement errors are very small, it is not safe to proceed as if Zj = X j . If we do, the estimates of the regression coefficients will be biased. In simple regression, the estimate of the slope will be biased toward O. See Draper and Smith (1981, Section 2.14) and Myers (1990, Section 7.8). Better estimates can be derived from the model defined by (1.1) and (*) together; see Fuller (1987, Section 2.2).
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