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Shrinkage is where extreme values in a sample are “shrunk” towards a central value, like the sample mean. Shrinking data can result in: Smoothed spatial fluctuations. However, the method has many disadvantages, including: Serious errors if the population has an atypical mean. Knowing which means are “typical” and which are “atypical ...
We can think of this as a measure of accuracy - expected squared loss which turns out to be the variance of \ (\tilde {\beta}\) + the squared bias. By shrinking the estimator by a factor of a, the bias is not zero. So, it is not an unbiased estimator anymore. The variance of \ (\tilde {\beta} = 1/a^2\).
Peter Ho Shrinkage estimators October 31, 2013 Letting w= 1 aand 0 = b=(1 a), the result suggests that if we want to use an admissible linear estimator, it should be of the form (X) = w 0 + (1 w)X; w2[0;1] We call such estimators linear shrinkage estimators as they \shrink" the estimate from Xtowards 0. Intuitively, you can think of
This estimator can be viewed as a shrinkage estimator as well, but the amount of shrinkage is di erent for the di erent elements of the estimator, in a way that depends on X. 2 Collinearity and ridge regression Outside the context of Bayesian inference, the estimator ^ = (X >X+ I) 1X>y is generally called the \ridge regression estimator."
The parameter \ (\lambda\) is a tuning parameter. It modulates the importance of fit vs. shrinkage. We find an estimate \ (\hat\beta^R_\lambda\) for many values of \ (\lambda\) and then choose it by cross-validation. Fortunately, this is no more expensive than running a least-squares regression.
Bayesian Estimation and Shrinkage I Example 1: (Schools 82 vs. 46) Data: ¯y 82 = 38.76, n 82 = 5, ˆµ 82 = 42.53 ¯y 46 = 40.18, n 46 = 21, µˆ 46 = 41.31 I Note φˆ = 48.12. I For school 82, we have substantial shrinkage toward φˆ. I For school 46, we have less shrinkage toward φˆ. I We might then rank school 82 ahead of school 46 ...
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Motivation 1: shrink the observation toward a given point c. Suppose it were thought a priori likely, though not certain, that q = c. Then we might first test a hypothesis H0 : q = c and estimate q by c if H0 is accepted and by X otherwise. Any estimator having this form is called a shrinkage estimator.
