Jacobian Matrix of Residuals

[zli034]

There are 2 parameters in the Gamma distribution, alpha and beta. If sample 500 of the Gamma random variable, there unbiased mean and variance can be estimated by the sample moments.

If it is also interested to estimate the variance and covariance of the parameters, alpha and beta; Jacobian matrix of residuals has to be defined, Jr. There fore the covariance matrix is:

inverse(transpose(Jr)residual)sample variance

I want to know about the calculation of the Jacobian matrix of residuals.

 

[Stephen Tashi]

If it is also interested to estimate the variance and covariance of the parameters, alpha and beta

Do you mean "the variance and covariance of the estimators of the parameters"? The parameters themselves are constant, they don't have a variance.

 

[zli034]

Do you mean "the variance and covariance of the estimators of the parameters"? The parameters themselves are constant, they don't have a variance.

Yes, I should stated more clearly. How to do the covariance of the estimators? I use too much simulation methods, this kind exact formulation I did not work with before.

 

[Stephen Tashi]

I don't understand enough about this problem to know which Jacobian we are talking about or how it would be used.

I did notice that the following PDF asserts the independence of some estimators for the gamma distribution:

http://www.google.com/url?sa=t&sour...mNGzAg&usg=AFQjCNF6xk8IBwfv-zAm-hnoNyVcEXCAPA

I haven't found the papers by Hwang and Hu that it cites. Do you know if they are online somewhere?

 

[blue_raver22]

The Jacobian matrix of residuals is a mathematical tool used in the estimation of parameters in a statistical model. In this case, the model is the Gamma distribution with two parameters, alpha and beta. The Jacobian matrix is used to calculate the variance and covariance of these parameters.

To understand the calculation of the Jacobian matrix, we first need to understand what residuals are. Residuals are the differences between the observed values and the predicted values from a statistical model. In this case, the observed values are the 500 samples of the Gamma random variable and the predicted values are the estimated mean and variance using the sample moments.

The Jacobian matrix is a matrix of partial derivatives that relates the residuals to the parameters of the model. In simpler terms, it shows how the residuals change with respect to changes in the parameters. In this case, the Jacobian matrix of residuals, Jr, will have two rows (one for each parameter) and 500 columns (one for each sample).

To calculate Jr, we need to take the partial derivatives of the residuals with respect to each parameter. This can be done using mathematical techniques such as the chain rule or by using software programs. Once Jr is calculated, we can use it to calculate the covariance matrix of the parameters, which can then be used to estimate the variance and covariance of alpha and beta.

In summary, the Jacobian matrix of residuals is an important tool in parameter estimation for statistical models. It allows us to calculate the covariance matrix of parameters, which is essential in understanding the relationships between different parameters and their uncertainties.