Return to Question

Suppose that $X$ is a p-dimensional random vector and $Y$ is a random scalar. Then, Dodge and Whittaker (2009) indicate that the covariance of these two variables can be formulated as a minimization problem:

\begin{equation} \text{Cov}(Y,X)^T=\arg\inf_{\alpha, \beta}\{\mathbb{E}(Y-\alpha-\beta^T\text{Var}(X)^{-1}[X-\mathbb{E}(X)])\} \end{equation}

And based on this definition of the covariance they propose a quantile covariance defined for the $\tau^{th}$ quantile as:

\begin{equation} \text{Cov}_\tau(Y,X)^T=\arg\inf_{\alpha, \beta}\{\mathbb{E}\rho_\tau(Y-\alpha-\beta^T\text{Var}(X)^{-1}[X-\mathbb{E}(X)])\} \end{equation}

where $\rho_\tau(\cdot)$ is the check function for quantile regression defined by Koenker and Basset (1978).

I am trying to understand the way this quantile covariance works, but I am having problems from the very beginning, since it is based on a definition for the covariance that I have never seen before. So my questions are:

1. How is the covariance between a random scalar and a random vector calculated if the dimensions do not match?

2. Where is this definition as an optimization problem for the covariance coming from?

3. Any insights that help understanding the quantile covariance.

References:

• Dodge, Y. and Whittaker, J. (2009). Partial quantile regression. Metrika, 70:35–57.

• Koenker, R. and Bassett, G. (1978). Regression Quantiles. Econometrica, 46(1):33–50.

Suppose that $X$ is a p-dimensional random vector and $Y$ is a random scalar. Then, Dodge and Whittaker (2009) indicate that the covariance of these two variables can be formulated as a minimization problem:

\begin{equation} \text{Cov}(Y,X)^T=\arg\inf_{\alpha, \beta}\{\mathbb{E}(Y-\alpha-\beta^T\text{Var}(X)^{-1}[X-\mathbb{E}(X)])^2\} \end{equation}

And based on this definition of the covariance they propose a quantile covariance defined for the $\tau^{th}$ quantile as:

\begin{equation} \text{Cov}_\tau(Y,X)^T=\arg\inf_{\alpha, \beta}\{\mathbb{E}\rho_\tau(Y-\alpha-\beta^T\text{Var}(X)^{-1}[X-\mathbb{E}(X)])\} \end{equation}

where $\rho_\tau(\cdot)$ is the check function for quantile regression defined by Koenker and Basset (1978).

I am trying to understand the way this quantile covariance works, but I am having problems from the very beginning, since it is based on a definition for the covariance that I have never seen before. So my questions are:

1. How is the covariance between a random scalar and a random vector calculated if the dimensions do not match?

2. Where is this definition as an optimization problem for the covariance coming from?

3. Any insights that help understanding the quantile covariance.

References:

• Dodge, Y. and Whittaker, J. (2009). Partial quantile regression. Metrika, 70:35–57.

• Koenker, R. and Bassett, G. (1978). Regression Quantiles. Econometrica, 46(1):33–50.

Suppose that $X$ is a p-dimensional random vector and $Y$ is a random scalar. Then, Dodge and Whittaker (2009) indicate that the covariance of these two variables can be formulated as a minimization problem:

\begin{equation} \text{Cov}(Y,X)^T=\arg\inf_{\alpha, \beta}\{\mathbb{E}(Y-\alpha-\beta^T\text{Var}(X)^{-1}[X-E(X)])\} \end{equation}

And based on this definition of the covariance they propose a quantile covariance defined for the $\tau^{th}$ quantile as:

\begin{equation} \text{Cov}_\tau(Y,X)^T=\arg\inf_{\alpha, \beta}\{\mathbb{E}\rho_\tau(Y-\alpha-\beta^T\text{Var}(X)^{-1}[X-E(X)])\} \end{equation}

where $\rho_\tau(\cdot)$ is the check function for quantile regression defined by Koenker and Basset (1978).

I am trying to understand the way this quantile covariance works, but I am having problems from the very beginning, since it is based on a definition for the covariance that I have never seen before. So my questions are:

1. How is the covariance between a random scalar and a random vector calculated if the dimensions do not match?

2. Where is this definition as an optimization problem for the covariance coming from?

3. Any insights that help understanding the quantile covariance.

References:

• Dodge, Y. and Whittaker, J. (2009). Partial quantile regression. Metrika, 70:35–57.

• Koenker, R. and Bassett, G. (1978). Regression Quantiles. Econometrica, 46(1):33–50.

Suppose that $X$ is a p-dimensional random vector and $Y$ is a random scalar. Then, Dodge and Whittaker (2009) indicate that the covariance of these two variables can be formulated as a minimization problem:

\begin{equation} \text{Cov}(Y,X)^T=\arg\inf_{\alpha, \beta}\{\mathbb{E}(Y-\alpha-\beta^T\text{Var}(X)^{-1}[X-\mathbb{E}(X)])\} \end{equation}

And based on this definition of the covariance they propose a quantile covariance defined for the $\tau^{th}$ quantile as:

\begin{equation} \text{Cov}_\tau(Y,X)^T=\arg\inf_{\alpha, \beta}\{\mathbb{E}\rho_\tau(Y-\alpha-\beta^T\text{Var}(X)^{-1}[X-\mathbb{E}(X)])\} \end{equation}

where $\rho_\tau(\cdot)$ is the check function for quantile regression defined by Koenker and Basset (1978).

I am trying to understand the way this quantile covariance works, but I am having problems from the very beginning, since it is based on a definition for the covariance that I have never seen before. So my questions are:

1. How is the covariance between a random scalar and a random vector calculated if the dimensions do not match?

2. Where is this definition as an optimization problem for the covariance coming from?

3. Any insights that help understanding the quantile covariance.

References:

• Dodge, Y. and Whittaker, J. (2009). Partial quantile regression. Metrika, 70:35–57.

• Koenker, R. and Bassett, G. (1978). Regression Quantiles. Econometrica, 46(1):33–50.

Suppose that $X$ is a p-dimensional random vector and $Y$ is a random scalar. Then, Dodge and Whittaker (2009) indicate that the covariance of these two variables can be formulated as a minimization problem:

\begin{equation} cov_(Y,X)^T=arginf_{\alpha, \beta}\{E(Y-\alpha-\beta^Tvar(X)^{-1}[X-E(X)])\} \end{equation}

And based on this definition of the covariance they propose a quantile covariance defined for the $\tau^{th}$ quantile as:

\begin{equation} cov_\tau(Y,X)^T=arginf_{\alpha, \beta}\{E\rho_\tau(Y-\alpha-\beta^Tvar(X)^{-1}[X-E(X)])\} \end{equation}

where $\rho_\tau(\cdot)$ is the check function for quantile regression defined by Koenker and Basset (1978).

I am trying to understand the way this quantile covariance works, but I am having problems from the very begining, since it is based on a definition for the covariance that I have never seen before. So my questions are:

1. How is the covariance between a random scalar and a random vector calculated if the dimensions do not match?

2. Where is this definition as an optimization problem for the covariance comming from?

3. Any insights that help understanding the quantile covariance.

References:

• Dodge, Y. and Whittaker, J. (2009). Partial quantile regression. Metrika, 70:35–57.

• Koenker, R. and Bassett, G. (1978). Regression Quantiles. Econometrica, 46(1):33–50.

Suppose that $X$ is a p-dimensional random vector and $Y$ is a random scalar. Then, Dodge and Whittaker (2009) indicate that the covariance of these two variables can be formulated as a minimization problem:

\begin{equation} \text{Cov}(Y,X)^T=\arg\inf_{\alpha, \beta}\{\mathbb{E}(Y-\alpha-\beta^T\text{Var}(X)^{-1}[X-E(X)])\} \end{equation}

And based on this definition of the covariance they propose a quantile covariance defined for the $\tau^{th}$ quantile as:

\begin{equation} \text{Cov}_\tau(Y,X)^T=\arg\inf_{\alpha, \beta}\{\mathbb{E}\rho_\tau(Y-\alpha-\beta^T\text{Var}(X)^{-1}[X-E(X)])\} \end{equation}

where $\rho_\tau(\cdot)$ is the check function for quantile regression defined by Koenker and Basset (1978).

I am trying to understand the way this quantile covariance works, but I am having problems from the very beginning, since it is based on a definition for the covariance that I have never seen before. So my questions are:

1. How is the covariance between a random scalar and a random vector calculated if the dimensions do not match?

2. Where is this definition as an optimization problem for the covariance coming from?

3. Any insights that help understanding the quantile covariance.

References:

• Dodge, Y. and Whittaker, J. (2009). Partial quantile regression. Metrika, 70:35–57.

• Koenker, R. and Bassett, G. (1978). Regression Quantiles. Econometrica, 46(1):33–50.