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Consider a linear system $y_i=Wx_i+v_i$, where $x_iin R^dtimes1 sim N(0,Sigma_x)$, $v_iin R^ptimes1 sim N(0,Sigma_v)$, $y_iin R^ptimes1$ and $Win R^ptimes d$.

Now we consider a linear estimator $hatx_i=beta_1y_i+beta_0$. We know that the optimal theoretical least square solution of is given by:

$$beta^ast_1=Sigma_xW^top(WSigma_xW^top+Sigma_v)^-1$$

$$beta^ast_0=0$$

On the other hand, suppose we have a set consisting of $N$ ovservations pairs $(x_1,y_1),(x_2,y_2),..,(x_N,y_N)$. We denote $Y_N=[y_1,y_2,...,y_N]$, $X_N=[x_1,x_2,...,x_N]$ and $V_N=[v_1,v_2,...,v_N]$. the abover system and the linear estimator can be written as

$$Y_N=WX_N+V_N$$

$$hatX_N=[beta_1, beta_0]beginbmatrixY_N\1endbmatrix$$.

Therefore, the least square solution is given by :

$$[beta^astast_1, beta^astast_0]=X_N[Y^top_N, 1^top]left(beginbmatrixY_N\1endbmatrix[Y^top_N, 1^top]right)^-1$$

where $beta^astast_1$ and $beta^astast_0$ can be considered as random variables.

We know that when $Nto infty$, $beta^astast_1to beta^ast_1$.

My question is: when N is finite, can we describe the relationship between $beta^astast_1$ and $beta^ast_1$, e.g., the pdf of $beta^astast_1-beta^ast_1$, for a given $N$?

linear-algebra random-variables

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0

$begingroup$

Consider a linear system $y_i=Wx_i+v_i$, where $x_iin R^dtimes1 sim N(0,Sigma_x)$, $v_iin R^ptimes1 sim N(0,Sigma_v)$, $y_iin R^ptimes1$ and $Win R^ptimes d$.

Now we consider a linear estimator $hatx_i=beta_1y_i+beta_0$. We know that the optimal theoretical least square solution of is given by:

$$beta^ast_1=Sigma_xW^top(WSigma_xW^top+Sigma_v)^-1$$

$$beta^ast_0=0$$

On the other hand, suppose we have a set consisting of $N$ ovservations pairs $(x_1,y_1),(x_2,y_2),..,(x_N,y_N)$. We denote $Y_N=[y_1,y_2,...,y_N]$, $X_N=[x_1,x_2,...,x_N]$ and $V_N=[v_1,v_2,...,v_N]$. the abover system and the linear estimator can be written as

$$Y_N=WX_N+V_N$$

$$hatX_N=[beta_1, beta_0]beginbmatrixY_N\1endbmatrix$$.

Therefore, the least square solution is given by :

$$[beta^astast_1, beta^astast_0]=X_N[Y^top_N, 1^top]left(beginbmatrixY_N\1endbmatrix[Y^top_N, 1^top]right)^-1$$

where $beta^astast_1$ and $beta^astast_0$ can be considered as random variables.

We know that when $Nto infty$, $beta^astast_1to beta^ast_1$.

My question is: when N is finite, can we describe the relationship between $beta^astast_1$ and $beta^ast_1$, e.g., the pdf of $beta^astast_1-beta^ast_1$, for a given $N$?

linear-algebra random-variables

share|cite|improve this question

edited 15 hours ago

czk32611

asked 2 days ago

czk32611czk32611

11

New contributor

czk32611 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

add a comment | 

$begingroup$

Consider a linear system $y_i=Wx_i+v_i$, where $x_iin R^dtimes1 sim N(0,Sigma_x)$, $v_iin R^ptimes1 sim N(0,Sigma_v)$, $y_iin R^ptimes1$ and $Win R^ptimes d$.

Now we consider a linear estimator $hatx_i=beta_1y_i+beta_0$. We know that the optimal theoretical least square solution of is given by:

$$beta^ast_1=Sigma_xW^top(WSigma_xW^top+Sigma_v)^-1$$

$$beta^ast_0=0$$

On the other hand, suppose we have a set consisting of $N$ ovservations pairs $(x_1,y_1),(x_2,y_2),..,(x_N,y_N)$. We denote $Y_N=[y_1,y_2,...,y_N]$, $X_N=[x_1,x_2,...,x_N]$ and $V_N=[v_1,v_2,...,v_N]$. the abover system and the linear estimator can be written as

$$Y_N=WX_N+V_N$$

$$hatX_N=[beta_1, beta_0]beginbmatrixY_N\1endbmatrix$$.

Therefore, the least square solution is given by :

$$[beta^astast_1, beta^astast_0]=X_N[Y^top_N, 1^top]left(beginbmatrixY_N\1endbmatrix[Y^top_N, 1^top]right)^-1$$

where $beta^astast_1$ and $beta^astast_0$ can be considered as random variables.

We know that when $Nto infty$, $beta^astast_1to beta^ast_1$.

My question is: when N is finite, can we describe the relationship between $beta^astast_1$ and $beta^ast_1$, e.g., the pdf of $beta^astast_1-beta^ast_1$, for a given $N$?

linear-algebra random-variables

share|cite|improve this question

edited 15 hours ago

czk32611

asked 2 days ago

czk32611czk32611

11

New contributor

czk32611 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

share|cite|improve this question

edited 15 hours ago

czk32611

asked 2 days ago

czk32611czk32611

11

New contributor

czk32611 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|cite|improve this question

edited 15 hours ago

czk32611

asked 2 days ago

czk32611czk32611

11

New contributor

czk32611 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked 2 days ago

czk32611czk32611

11

New contributor

czk32611 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked 2 days ago

czk32611czk32611

11