Multiple linear regression is a widely used statistical method. Its application, especially in the sciences, social sciences, and economics assists administrators in evaluating programs and planners in predicting future situations. The method is so common that most institutions have in their computer operation some standard programs to deal with the calculations. These traditional approaches use the method of least squares and yield an unbiased estimate of the parameters. The general linear model used is Y = Xβ+ e, where E(e) = 0, E(ee`) = σ2In and X is (n x p) and full rank. The least squares estimate of the unknown parameter vector β is then given by β = (X'X)-1X̀Y. This approach, however, often produces unsatisfactory (or even inaccurate) results if the data vectors are ill-conditioned. Such ill-conditioning is a result of non-orthogonal data vectors and inter-correlation of response variables that are unfortunately quite common in all fields.In recent years it has become obvious that for these applications the unbiased estimate is not necessarily the best over-all in terms of mean square error. A biased estimate may actually be of more value in analysis and prediction. Ridge estimators are biased estimators that have proved useful in these cases. In their basic form β(k) = [(X'X) + kI]-1 X́Y, they differ from the least squares estimator in that they have a small positive constant added to the diagonal elements of the X́X matrix.This thesis will first deal with the situations in which the least squares approach is not adequate and the cases where the ridge estimate contributes to a usable solution. The significant work which has been done in the field will be surveyed and the main problem of determining an appropriate constant k for the ridge estimate will be considered.
| Identifer | oai:union.ndltd.org:BSU/oai:cardinalscholar.bsu.edu:handle/181986 |
| Date | January 1979 |
| Creators | Bulmahn, Barbara J. |
| Contributors | Ali, Mir M. |
| Source Sets | Ball State University |
| Detected Language | English |
| Format | iv, 104 leaves ; 28 cm. |
| Source | Virtual Press |
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