Quadratic forms in Normal Variates Under Ridge Regression. By: Abdul-Mordy Hamed Azzam | ||||
| The Egyptian Statistical Journal | ||||
| Article 2, Volume 41, Issue 2, December 1997, Page 95-109 | ||||
| Document Type: Original Article | ||||
| DOI: 10.21608/esju.1997.314642 | ||||
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| Abstract | ||||
| This paper concentrates on studying the quadratic forms in normal variates which appear when testing linear statistical hypothesis under ridge regression with positive non-stochastic biased factors k1, k2, …, kp . Except for the correction factor nȳ2, it is shown that all other quadratic forms are not independent and do not follow central or non-central x2 distributions. Hence the classical F statistics are irrelevant under ridge. The results of Hoerl and Kennard (1990) are obtained as special cases when the biased factors are all positive and equal. Moreover, the classical ordinary least squares results are also obtained as special cases when all the biased factors are set to zero. | ||||
| Keywords | ||||
| Ordinary Least Squares (OLS); Ordinary Ridge (OR); Generalized Ridge (GR); Sum of Squares of Regression under OLS (SSRols); Sum of Squares of Errors under OLS (SSEols); Sum of Squares of Interaction under OLS (SSIols); Sum of Squares of Regression under Ridge (SSRr); Sum of Squares of Errors under Ridge (SSEr); Sum of Squares of Interaction under Ridge (SSIr); Mean Square Error (MSE); Orthogonal Projection Operator (OPO); Range space of a matrix (Ṟ(.)]; Null space of a matrix [Ṉ(.)]; trace of a matrix (tr(.)]; Canonical Parametrization | ||||
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