Question

Let Y = Xβ + ε be the linear model where X be an n × p matrix with orthonormal columns (columns of X are orthogonal to each other and each column has length 1)

Let  $\hat{\beta} = (\beta_{1}, ..., \beta_{p})$ be the least-squares estimate of β, and let $\hat{\beta}^{*}} = (\beta_{1}^{*}}, ..., \beta_{p}^{*}})$ be the ridge regression estimate with tuning parameter λ.

Prove that for each j, $\hat{\beta_{j}^{*}} = \frac{\hat{}\beta _{j}}{1+\lambda }$ .

Note: The ridge regression estimate is given by: $\hat{\beta}^{*}} = (X^{T}X + \lambda I)^{-1}X^{T}Y$

The least squares estimate is given by:  $\hat{\beta}} = (X^{T}X)^{-1}X^{T}Y$

Answer 1

ei 2 Bhimatで hagi alusinv 싸innretnin.fhetnn. ht an, erm hne

Ridge regression estimate is obtained here for a particular choice of Data matrix X.