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(12.9) involves only left-handed electrons (or right-handed positrons). In the relativistic limit (v == c), show that the electrons have negative helicity. The 1(1 - y5) in (12.9) automatically selects a left-handed neutrino (or a right-handed antineutrino). This V-A (vector-axial vector) structure of the weak current can be directly exposed by scattering Pe's off electrons (see Section 12.7), just as the y" structure of electromagnetism was verified by measurements of the angular distribution of e+ e- scattering. It is natural to hope that all weak interaction phenomena are described by a V-A current-current interaction with a universal coupling G. For example, /3-decay of Fig. 12.2 and p,-decay of Fig. 12.4 can be described by the amplitudes 0lL(p and

/ / CONSTRUCTION: with an open istream // / / ******************PUBLIC OPERATIONS***********************

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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

- - > Read one bit as a 0 or 1 / / int readBit ( ) / / istream & getInputStream( ) - - > Return underlying stream / / ********x*********ERRORs******r**************************

~ ne+Pe )

/ / Error checking can be done on result of getInputStream( 1

int readBit ( ) ; istream & getInputStream( private: istream & in; char buffer; int bufferpos;

const;

[ii n y"(l-

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

/ / The underlying input stream / / Buffer to store eight bits at a time / / Position in buffer for next read

A container represents a group of objects, known as its elements Some implementations such as vectors and lists are unordered; others, such as sets and maps, are ordered Some implementations allow duplicates; others do not All containers support the following operations

y5)Up][iiv~y,,(1- y5)U e ]

boo1 empty(

iterator begin(

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

iterator end(

(12.10)

returns an i t e r a t o r that represents the end marker, a position past the last element in the container

int size(

Fig. 12.4 The diagram for p. - decay: p. -

returns the number of elements in the container The most interesting of these methods are those that return an i t e r a t o r We describe the operations that can be performed by an i t e r a t o r in Section 732

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