Creation European Article Number 13 in Java MORE GENERAL METHODS OF REGRESSION
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standardized if the sum of the squares of its coefficients is equal to 1.) Among all standardized linear combinations of the explanatory variables that are orthogonal to the first principal component, the second principal component is the one showing the most variability. Continuing in this way, p principal components can be constructed. Sometimes, by looking at the sizes and signs of the coefficients in the linear combination, a meaningful interpretation can be given to a principal component. Explanatory variables that are collinear with one another will typically occur together with large coefficients in a single principal component. In principal components regression, the first several principal components are taken as a new set of explanatory variables and least-squares estimates are calculated. See Gunst and Mason (1980, Section 10.1) and Myers (1990, Section 8.4). Maximum Likelihood Estimation. If one is willing to assume that the error population has a particular kind of distribution, maximum likelihood (ML) estimation can be applied. The error distribution is usually assumed to be continuous, which means that it can be described in terms of a probability density function (p.d.f.). The exact error distribution is almost always unknown to us, but suppose we are willing to assume that the distribution is contained in a known family of distributions with p.dJ.'s f(e; 0') indexed by the parameter 0'. (More generally, the family could be indexed by a vector of parameters.) Then the p.d.f. of the sample of observed response variables is the product f( YI - f3' X I; 0') ... f( Y - f3' X n; 0'). Regarded as a function of n f3 and 0', this is the likelihood function. The maximum likelihood estimates of f3 and 0' are chosen to maximize the likelihood function. If the family of possible error distributions is the family of normal distributions with mean 0, then the ML estimates of the regression coefficients coincide with the least-squares estimates; the ML estimate of 0' is (..; (n - p - 1) In )cT LS . For the family of Laplace (double exponential) distributions with mean 0, the ML estimates of the regression coefficients coincide with the LAD estimates.
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For some sets of regression data a linear regression model may not be appropriate. The techniques mentioned in 2 may fail to produce a satisfactory model. Or theory about the process that generated the data may suggest a different type of model. Such data require more general models. A very general model is Yi = J.Li + e i , where J.Li is the expectation of the random variable Yi and where e i is defined to be the difference (or "error")
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Yi - fJ-i Different kinds of models are obtained by making different assumptions about the structure of the expectations fJ-i and about the distribution of
the errors e i The linear regression model assumes that fJ-; is a linear function (J'x; of the explanatory variables and that the random variables ei are independent of one another and have the same distribution. Weighted Least Squares. For some regression data sets it may be valid to assume that fJ-; = f3'x; but invalid to assume that the errors all come from the same population. Suppose that the errors are independent but have unequal standard deviations that are proportional to one of the explanatory variables, say, XI' so that Var(e) = O' 2X;I. Or more generally, suppose Var(e) = O' 2 v; for known positive quantities Vi. The weighted least-squares estimate PWLS is the value of f3 that minimizes E( y; - f3' xY Iv i This becomes the ordinary least-squares procedure in the special case when all the v;'s are equal. See Weisberg (1985, 4) and Myers (1990, Section 7.1). Generalized Least Squares. Continue to assume that the expectation of
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y; is a linear function of the explanatory variables. In vector notation, y = X{J + e. In ordinary least-squares regression the vector of errors is
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assumed to have the variance-covariance matrix Cov(e) = 0'2/. In weighted least squares it is assumed that Cov(e) = O' 2 V, where V is a diagonal matrix with positive diagonal entries. More generally, suppose Cov(e) = O' 2 V, where V is any invertible variance-covariance matrix. This allows the errors to be correlated as well as to have unequal variances. The generalized least-squares estimate of the vector of regression coefficients is PGLS = (X'V-IX)-I X'V-1y. When V = I, this reduces to formula (3.8). See Myers (1990, Section 7.1) and Seber (1977, Section 3.6). Nonlinear Regression. A linear regression function is not always suitable. Consider a regression function of the general form fJ-i = g(x;, 0), where g is a known function that is not necessarily linear and 0 is a vector of unknown parameters. (In nonlinear regression no convenience is gained by including 1 as the first component of Xi and so we use the more natural notation X; = (Xii' ' X;p).) The least-squares estimate 8LS is the value of 0 that minimizes E[y; - g(x i , oW. When g is a nonlinear function, there is generally no explicit formula for 8LS ; it must be computed iteratively. One possible iterative algorithm is the following. First pick an initial estimate. Using a Taylor series expansion about the initial estimate (Jet us consider only functions g that are differentiable with respect to 0), we can approximate g(x;,O) by a linear function of o. Thus we obtain an approximating linear regression model in which an ordinary linear least-squares estimate of 0 can be computed. This process of computing an updated estimate based on a
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