[ADMB Users] SE Estimates including process error in ADMB
dave fournier
davef at otter-rsch.com
Tue Nov 16 04:21:47 PST 2010
Chris Gast wrote:
I think this is discussed in the paper Hans and I wrote.
anyway since uhat(x) is the value of uhat which maximizes L(x,u)
It follows that
L_u(x,uhat(x))=0 for all x so differentiating wrt x you get
L_xu + L_uu uhat'(x)=0 so that
uhat'(x) = - L_uu^{-1} * L_xu
If there are n x's and m u's then uhat'(x) is an n x m matrix.
> Thanks again, Dave. Could you clarify what the general entry of the
> gradient uhat'(xhat) looks like? I had thought this would be
> dx_i/du_j, but I think these would all be 1's or 0's (since, for
> instance , Shat_1 = shat + uhat_1). It is likely that I am
> misunderstanding something.
>
>
> Thanks,
>
> Chris
>
>
>
>
> -----------------------------
> Chris Gast
> cmgast at gmail.com <mailto:cmgast at gmail.com>
>
>
> On Mon, Nov 15, 2010 at 7:00 PM, dave fournier <davef at otter-rsch.com
> <mailto:davef at otter-rsch.com>> wrote:
>
> OK,
>
> there are inly ar efew thing to play with. for notation let
>
> F(x,u) be the likelihood for x and u.
>
> let
>
> L(x) = int F(x,u) du
>
> be what we get after integrating out u either exactly or by the
> laplace approx.
>
> and let xhat uhat(xhat) be the miximizing values.
>
> We have the Hessians L_xx(xhat), and F_uu(xhat,uhat(xhat)
> and the gradients uhat'(xhat)
> Then the covariance matrix for x,u is assumed to be
>
> L_xx^{-1} L_xx^{-1} * uhat'(x)
>
> uhat'(xhat)*L_xx^{-1} F_uu^{-1} +
> uhat'(xhat)*L_xx^{-1}*uhat'(x)
>
> The idea is that u-uhat(xhat) is independent of xhat i.e the only
> correlation is between
> xhat and uhat(x)
>
> Put in transposes where necessary.
>
> For any function of x,u the covariance is computed by the delta
> method.
>
>
>
>
>
> _______________________________________________
> Users mailing list
> Users at admb-project.org <mailto:Users at admb-project.org>
> http://lists.admb-project.org/mailman/listinfo/users
>
>
More information about the Users
mailing list