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&source=web&cd=3&sqi=2&ved=0CCwQFjAC&url=http%3A%2F%2Fwww.ism.ac.jp%2Feditsec%2Faism%2F pdf%2F054_4_0840.pdf&rct=j&q=on%20new%20moment%20estimation%20of%20parameters %20of%20the%20gamma%20distribution&ei=zOpzTrvWEsfUiALSmNGzAg&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?