Here we reformulate the multitaper-based simple linear regression model of Equation (409d) in standard vector/matrix notation (see, e.g., Weisberg, 2014), with the goal of verifying Equation (410b), which gives the variance of the OLS estimator
of the power-law exponent α. Let Y be a column vector containing the responses
and let X be a matrix with two columns of predictors, the first column with elements all equal to unity, and the second, with elements equal to
as defined in Equation (409e).
(a) With
being a two-dimensional column vector of regression coefficients whose second element is α, argue that
is equivalent to the model of Equation (409d), where is a column vector containing the error terms
(b) A standard result says that the OLS estimator of
(see, e.g., Weisberg, 2014). Show that the second element of βˆ is the multitaper-based OLS estimator ˆα (MT) of Equation (409e).
(c) Taking the elements of the covariance matrix Σ for to be dictated by Equation (410a); noting that
and evoking a standard result from the theory of multivariate RVs, namely, that the covariance matrix for Mis given by
show that var
is given by.