I fitted an hyperbolic distribution to my data with the `hyperbFit(mydata,hessian=TRUE)`

command (package HyperbolicDist). The hessian looks like:

`> hyperbfitmymodel$hessian hyperbPi lZeta lDelta mu hyperbPi 536.61654 -23.82800 25.62345 26153.16561 lZeta -23.82800 250.74196 -261.20570 -35.58481 lDelta 25.62345 -261.20570 272.77771 182.75927 mu 26153.16561 -35.58481 182.75927 2028904.75586 `

Now I want to calculate the variance-covariance matrix of the parameter estimates, according to this page 2:

The asymptotic covariance matrix of $hat{theta}$ is given by the

inverse of the negative of the Hessian matrix evaluated at

$hat{theta}$.

I therefore calculate:

`solve(-hyperbfitalv$hessian) `

which gives

` hyperbPi lZeta lDelta mu hyperbPi -5.113433e-03 -0.0091511819 -0.0083271877 6.650321e-05 lZeta -9.151182e-03 -1.6617499980 -1.5905496996 2.320893e-04 lDelta -8.327188e-03 -1.5905496996 -1.5261031428 2.169113e-04 mu 6.650321e-05 0.0002320893 0.0002169113 -1.365591e-06 `

This looks clearly wrong to me, because there are negative values for the variance, but a variance cannot be negative? The covariance yes, but not the variance?

EDIT: The complete output of `hyperbFit(mydata,hessian=TRUE)`

:

`Data: mydata Parameter estimates: pi zeta delta mu 0.090747 0.204827 0.002035 -0.002494 Likelihood: 756.911 Method: Nelder-Mead Convergence code: 0 Iterations: 365 `

2nd EDIT: If I use `solve(hyperbfitalv$hessian)`

I get

` hyperbPi lZeta lDelta mu hyperbPi 5.113433e-03 0.0091511819 0.0083271877 -6.650321e-05 lZeta 9.151182e-03 1.6617499980 1.5905496996 -2.320893e-04 lDelta 8.327188e-03 1.5905496996 1.5261031428 -2.169113e-04 mu -6.650321e-05 -0.0002320893 -0.0002169113 1.365591e-06 `

3rd EDIT: The output of `summary(hyperbfitalv)`

:

`Data: mydata Parameter estimates: pi zeta delta mu 0.090747 0.204827 0.002035 -0.002494 ( 0.071508) ( 0.264040) ( 0.002514) ( 0.001169) Likelihood: 756.911 Method: Nelder-Mead Convergence code: 0 Iterations: 365 `

4th EDIT: Ok, this is the hessian of pi, log(zeta), log(delta), and mu but how can I get the hessian of pi, zeta, delta and mu?

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#### Best Answer

Optim in default setting is doing minimization, see the manual:

By default optim performs minimization

So the output is already the negative hessian.

It should be further noted that:

Because the parameters in the call to the optimiser are pi, log(zeta), log(delta), and mu, the delta method is used to obtain the standard errors for zeta and delta.

Source here.

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