Bias of a PCA Estimator
For the model
$mathbf{y} = gammamathbf{X}+mathbf{epsilon}$
we do a PCA regression and estimate the parameter $Vhat gamma= hatbeta = sum_{i=1}^k lambda^{-1}_i mathbf{v_i} mathbf{v_i}^T mathbf{X}^Tmathbf{y}$,
where the $mathbf{v_i} $ are the eigenvectors of $mathbf{X}^Tmathbf{X}$ and $mathbf{V}$ is the matrix where the $mathbf {v_i}$ are the columns. $lambda_i$ are the eigenvalues of $mathbf{X}^Tmathbf{X}$. k is the total number of principal components.
Now we choose an $m <k$.
We have a biased estimator:
$tildebeta = sum_{i=1}^m lambda^{-1}_i mathbf{v_i} mathbf{v_i}^T mathbf{X}^Tmathbf{y}$.
I want to calculate the bias of this estimator:
$Bias(tildebeta) = beta - mathrm{E}(sum_{i=1}^m lambda^{-1}_i mathbf{v_i} mathbf{v_i}^T mathbf{X}^Tmathbf{y}) $.
I can use that $hat beta$ is unbiased:
$Bias(tildebeta) = beta + mathrm{E}(sum_{i=1}^k lambda^{-1}_i mathbf{v_i} mathbf{v_i}^T mathbf{X}^Tmathbf{y}-sum_{i=1}^m lambda^{-1}_i mathbf{v_i} mathbf{v_i}^T mathbf{X}^Tmathbf{y})$
$ = beta + sum_{i=m+1}^k lambda^{-1}_i mathbf{v_i} mathbf{v_i}^T mathbf{X}^Tmathrm{E}(mathbf{y}) = beta + sum_{i=m+1}^k lambda^{-1}_i mathbf{v_i} mathbf{v_i}^T mathbf{X}^Tmathrm{E}(mathbf{X}mathbf{V}mathbf{beta}+epsilon)$
The error have expectation zero:
$ = beta + sum_{i=m+1}^k lambda^{-1}_i mathbf{v_i} mathbf{v_i}^T mathbf{X}^Tmathbf{X}mathbf{V}mathbf{beta}$
Am I on the right track? How do I proceed?
statistics estimation
asked Nov 25 at 14:32
PascalIv
505
add a comment |
For the model
$mathbf{y} = gammamathbf{X}+mathbf{epsilon}$
we do a PCA regression and estimate the parameter $Vhat gamma= hatbeta = sum_{i=1}^k lambda^{-1}_i mathbf{v_i} mathbf{v_i}^T mathbf{X}^Tmathbf{y}$,
where the $mathbf{v_i} $ are the eigenvectors of $mathbf{X}^Tmathbf{X}$ and $mathbf{V}$ is the matrix where the $mathbf {v_i}$ are the columns. $lambda_i$ are the eigenvalues of $mathbf{X}^Tmathbf{X}$. k is the total number of principal components.
Now we choose an $m <k$.
We have a biased estimator:
$tildebeta = sum_{i=1}^m lambda^{-1}_i mathbf{v_i} mathbf{v_i}^T mathbf{X}^Tmathbf{y}$.
I want to calculate the bias of this estimator:
$Bias(tildebeta) = beta - mathrm{E}(sum_{i=1}^m lambda^{-1}_i mathbf{v_i} mathbf{v_i}^T mathbf{X}^Tmathbf{y}) $.
I can use that $hat beta$ is unbiased:
$Bias(tildebeta) = beta + mathrm{E}(sum_{i=1}^k lambda^{-1}_i mathbf{v_i} mathbf{v_i}^T mathbf{X}^Tmathbf{y}-sum_{i=1}^m lambda^{-1}_i mathbf{v_i} mathbf{v_i}^T mathbf{X}^Tmathbf{y})$
$ = beta + sum_{i=m+1}^k lambda^{-1}_i mathbf{v_i} mathbf{v_i}^T mathbf{X}^Tmathrm{E}(mathbf{y}) = beta + sum_{i=m+1}^k lambda^{-1}_i mathbf{v_i} mathbf{v_i}^T mathbf{X}^Tmathrm{E}(mathbf{X}mathbf{V}mathbf{beta}+epsilon)$
The error have expectation zero:
$ = beta + sum_{i=m+1}^k lambda^{-1}_i mathbf{v_i} mathbf{v_i}^T mathbf{X}^Tmathbf{X}mathbf{V}mathbf{beta}$
Am I on the right track? How do I proceed?
statistics estimation
asked Nov 25 at 14:32
PascalIv
505
add a comment |
For the model
$mathbf{y} = gammamathbf{X}+mathbf{epsilon}$
we do a PCA regression and estimate the parameter $Vhat gamma= hatbeta = sum_{i=1}^k lambda^{-1}_i mathbf{v_i} mathbf{v_i}^T mathbf{X}^Tmathbf{y}$,
where the $mathbf{v_i} $ are the eigenvectors of $mathbf{X}^Tmathbf{X}$ and $mathbf{V}$ is the matrix where the $mathbf {v_i}$ are the columns. $lambda_i$ are the eigenvalues of $mathbf{X}^Tmathbf{X}$. k is the total number of principal components.
Now we choose an $m <k$.
We have a biased estimator:
$tildebeta = sum_{i=1}^m lambda^{-1}_i mathbf{v_i} mathbf{v_i}^T mathbf{X}^Tmathbf{y}$.
I want to calculate the bias of this estimator:
$Bias(tildebeta) = beta - mathrm{E}(sum_{i=1}^m lambda^{-1}_i mathbf{v_i} mathbf{v_i}^T mathbf{X}^Tmathbf{y}) $.
I can use that $hat beta$ is unbiased:
$Bias(tildebeta) = beta + mathrm{E}(sum_{i=1}^k lambda^{-1}_i mathbf{v_i} mathbf{v_i}^T mathbf{X}^Tmathbf{y}-sum_{i=1}^m lambda^{-1}_i mathbf{v_i} mathbf{v_i}^T mathbf{X}^Tmathbf{y})$
$ = beta + sum_{i=m+1}^k lambda^{-1}_i mathbf{v_i} mathbf{v_i}^T mathbf{X}^Tmathrm{E}(mathbf{y}) = beta + sum_{i=m+1}^k lambda^{-1}_i mathbf{v_i} mathbf{v_i}^T mathbf{X}^Tmathrm{E}(mathbf{X}mathbf{V}mathbf{beta}+epsilon)$
The error have expectation zero:
$ = beta + sum_{i=m+1}^k lambda^{-1}_i mathbf{v_i} mathbf{v_i}^T mathbf{X}^Tmathbf{X}mathbf{V}mathbf{beta}$
Am I on the right track? How do I proceed?
statistics estimation
asked Nov 25 at 14:32
PascalIv
505
For the model
$mathbf{y} = gammamathbf{X}+mathbf{epsilon}$
we do a PCA regression and estimate the parameter $Vhat gamma= hatbeta = sum_{i=1}^k lambda^{-1}_i mathbf{v_i} mathbf{v_i}^T mathbf{X}^Tmathbf{y}$,
where the $mathbf{v_i} $ are the eigenvectors of $mathbf{X}^Tmathbf{X}$ and $mathbf{V}$ is the matrix where the $mathbf {v_i}$ are the columns. $lambda_i$ are the eigenvalues of $mathbf{X}^Tmathbf{X}$. k is the total number of principal components.
Now we choose an $m <k$.
We have a biased estimator:
$tildebeta = sum_{i=1}^m lambda^{-1}_i mathbf{v_i} mathbf{v_i}^T mathbf{X}^Tmathbf{y}$.
I want to calculate the bias of this estimator:
$Bias(tildebeta) = beta - mathrm{E}(sum_{i=1}^m lambda^{-1}_i mathbf{v_i} mathbf{v_i}^T mathbf{X}^Tmathbf{y}) $.
I can use that $hat beta$ is unbiased:
$Bias(tildebeta) = beta + mathrm{E}(sum_{i=1}^k lambda^{-1}_i mathbf{v_i} mathbf{v_i}^T mathbf{X}^Tmathbf{y}-sum_{i=1}^m lambda^{-1}_i mathbf{v_i} mathbf{v_i}^T mathbf{X}^Tmathbf{y})$
$ = beta + sum_{i=m+1}^k lambda^{-1}_i mathbf{v_i} mathbf{v_i}^T mathbf{X}^Tmathrm{E}(mathbf{y}) = beta + sum_{i=m+1}^k lambda^{-1}_i mathbf{v_i} mathbf{v_i}^T mathbf{X}^Tmathrm{E}(mathbf{X}mathbf{V}mathbf{beta}+epsilon)$
The error have expectation zero:
$ = beta + sum_{i=m+1}^k lambda^{-1}_i mathbf{v_i} mathbf{v_i}^T mathbf{X}^Tmathbf{X}mathbf{V}mathbf{beta}$
Am I on the right track? How do I proceed?
statistics estimation
statistics estimation
asked Nov 25 at 14:32
PascalIv
505
asked Nov 25 at 14:32
PascalIv
505
asked Nov 25 at 14:32
PascalIv
505
asked Nov 25 at 14:32
PascalIv
505
asked Nov 25 at 14:32
PascalIv
505
505
add a comment |
add a comment |
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