Re: transformation of covariance matrix


In general, let ti (i=1, ..., p) be the rvs for
which you have the covariance matrix.

Let sj = sj(t1, ..., tp), (j = 1,..., q) be q new
variables which are transforms of the ti's.

Then approximately,

Cov(si,sj) ~
Sum[Sum[(dsi/dtk)*(dsj/dtl)*Cov(tk,tl)]],

where the sum is over j,l = 1, ..., p.

Just calculate the partial derivatives and plug it
in.

Jack

Thank you very much for your reply, however, I don't
know how to
calculate the dsi/dtk
and dsj/dtl in your reply.

I should describe my question more clear.
In my question, alpha, beta, C are parameters
estimated from a probit
model.
alpha, beta and C, and mu and sigma are scalar, and
not variable
(vectors)
in my question. so I don't know whether it's possible
to get the
covariance
matrix of mu and sigma.

a numeric example:

The alpha, beta and C and their standard error is
following:
Estimate Std. Error
alpha -4.1437788 1.34146006
beta 6.2306286 1.89954481
c 0.2408866 0.05225879

And the covariance matrix is:
alpha beta c
alpha 1.79951508 -2.51894315 -0.031737662
beta -2.51894315 3.60827049 0.038782941
c -0.03173766 0.03878294 0.002730981

Now,
mu = 0.665066
and
sigma = 0.1604975

Now, my question is whether it is possible to get
covariance matrix of
mu and sigma from the above information.

Thanks again.

Regards,
Jinsong


The partial derivatives, which are evaluated at the respective point estimates are

dmu/dalpha = -1/beta
dmu/dbeta = alpha/(beta^2)
dmu/dc = 0
dsigma/dalpha = 0
dsigma/dbeta = -1/(beta^2)
dsigma/dc = 0
dc/dalpha = 0
dc/dbeta = 0
dc/dc = 1

If you still have a problem following this, I can work out the numerical details step by step.

Jack
.

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