Matrix Algebra Useful for Statistics by Shayle R. Searle

Author:Shayle R. Searle
Language: eng
Format: epub
Published: 2017-04-07T00:00:00+00:00

11.1 Estimation of β by the Method of Least Squares

Estimation of in model (11.5) requires the availability of n (n > p) observations on the response y where each observation is obtained as a result of a running an experiment using particular settings of the control variables x1, x2, …, xk. Let denote a vector consisting of the settings xui of the control variables used in the run (u = 1, 2, …, n). The subscript i keeps track of the control variables used in such a run (i = 1, 2, …, k). Correspondingly, model (11.4) can be written as

(11.6)

where yu is the observed response value at and εu is a random experimental error associated with yu (u = 1, 2, …, n). In matrix form, the totality of all the yu’s and corresponding settings of the control variables can be represented, using model (11.6), as

(11.7)

where , is an n × p matrix whose row is and . The model matrix is assumed to be of rank p, that is, has full column rank. Model (11.7) is then described as being of full rank. Additionally, we assume that and that y1, y2, …, yn are uncorrelated and have variances equal to σ2. Hence, the expected value (or mean) of in model (11.7) is equal to and its variance-covariance matrix is , where is the identity matrix of order n × n.

Under the above assumptions, the parameter vector in model (11.7) can be estimated by using the method of ordinary least squares (OLS). This is accomplished by minimizing the square of the Euclidean norm of with respect to , that is, minimizing the expression,

(11.8)

A necessary condition for to have a minimum at is that at , that is,

(11.9)

Applying formulas (9.11) and (9.17) in Sections 9.3.1 and 9.3.3, respectively, we get

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