1. In the linear model it Yi = x1 B + Ei, i = 1, ..., N; BER, may sometimes be appropriate to assume ein (0, 0 vi), where V1, ... , Un are known positive quantities, attached to each observation. In matrix form we would write y = X + ; ~ (0,0V), EN where V is an n x n diagonal matrix with v; on the diagonal. (a) Derive expressions for the weighted least squares estimator of B, defined by n 1 BWLS arg min B (yi xB)2 = arg min(y XB)?v=l(y XB). (x - Vi (b) Give an expression for the residual sum of squares after fitting this model, and derive an unbiased estimator of o2. Hint: since the weights are known, model (1) can be linearly transformed to an ordinary least squares model. (c) If v; = v;(a) depends on a parameter a that is unknown, suggest a method for computing the estimator in (a). (d) The dataset gammaray in library(faraway) has measurements on the X-ray decay light curve of a Gamma ray burst. The column error gives an estimate of the standard error of each flux measurement. (See help(gammaray) for more details.) Fit a weighted least squares model with flux as the response and time as the explanatory variable, using weights derived from error. Compare the coefficients and their estimated standard errors from this model and an unweighted 1. In the linear model it Yi = x1 B + Ei, i = 1, ..., N; BER, may sometimes be appropriate to assume ein (0, 0 vi), where V1, ... , Un are known positive quantities, attached to each observation. In matrix form we would write y = X + ; ~ (0,0V), EN where V is an n x n diagonal matrix with v; on the diagonal. (a) Derive expressions for the weighted least squares estimator of B, defined by n 1 BWLS arg min B (yi xB)2 = arg min(y XB)?v=l(y XB). (x - Vi (b) Give an expression for the residual sum of squares after fitting this model, and derive an unbiased estimator of o2. Hint: since the weights are known, model (1) can be linearly transformed to an ordinary least squares model. (c) If v; = v;(a) depends on a parameter a that is unknown, suggest a method for computing the estimator in (a). (d) The dataset gammaray in library(faraway) has measurements on the X-ray decay light curve of a Gamma ray burst. The column error gives an estimate of the standard error of each flux measurement. (See help(gammaray) for more details.) Fit a weighted least squares model with flux as the response and time as the explanatory variable, using weights derived from error. Compare the coefficients and their estimated standard errors from this model and an unweighted