hessian: Hessian matrix
In maxLik: Maximum Likelihood Estimation and Related Tools

Description Usage Arguments Value Author(s) See Also Examples

This function extracts the Hessian of the objective function at optimum. The Hessian information should be supplied by the underlying optimization algorithm, possibly by an approximation.

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hessian(x, ...)
## Default S3 method:
hessian(x, ...)

`x`	an optimization result of class ‘maxim’ or ‘maxLik’
`...`	other arguments for methods

A numeric matrix, the Hessian of the model at the estimated parameter values. If the maximum is flat, the Hessian is singular. In that case you may want to invert only the non-singular part of the matrix. You may also want to fix certain parameters (see activePar).

Ott Toomet

maxLik, activePar, condiNumber

# log-likelihood for normal density
# a[1] - mean
# a[2] - standard deviation
ll <- function(a) sum(-log(a[2]) - (x - a[1])^2/(2*a[2]^2))
x <- rnorm(100) # sample from standard normal
ml <- maxLik(ll, start=c(1,1))
# ignore eventual warnings "NaNs produced in: log(x)"
summary(ml) # result should be close to c(0,1)
hessian(ml) # How the Hessian looks like
sqrt(-solve(hessian(ml))) # Note: standard deviations are on the diagonal
#
# Now run the same example while fixing a[2] = 1
mlf <- maxLik(ll, start=c(1,1), activePar=c(TRUE, FALSE))
summary(mlf) # first parameter close to 0, the second exactly 1.0
hessian(mlf)
# Note that now NA-s are in place of passive
# parameters.
# now invert only the free parameter part of the Hessian
sqrt(-solve(hessian(mlf)[activePar(mlf), activePar(mlf)]))
# gives the standard deviation for the mean

Loading required package: miscTools

Please cite the 'maxLik' package as:
Henningsen, Arne and Toomet, Ott (2011). maxLik: A package for maximum likelihood estimation in R. Computational Statistics 26(3), 443-458. DOI 10.1007/s00180-010-0217-1.

If you have questions, suggestions, or comments regarding the 'maxLik' package, please use a forum or 'tracker' at maxLik's R-Forge site:
https://r-forge.r-project.org/projects/maxlik/
--------------------------------------------
Maximum Likelihood estimation
Newton-Raphson maximisation, 6 iterations
Return code 2: successive function values within tolerance limit
Log-Likelihood: -60.20196 
2  free parameters
Estimates:
     Estimate Std. error t value Pr(> t)    
[1,]   0.0393     0.1107   0.355   0.723    
[2,]   1.1074     0.0783  14.142  <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------
              [,1]          [,2]
[1,] -81.548989783 -7.105427e-03
[2,]  -0.007105427 -1.630909e+02
          [,1]       [,2]
[1,] 0.1107365        NaN
[2,]       NaN 0.07830422
Warning message:
In sqrt(-solve(hessian(ml))) : NaNs produced
--------------------------------------------
Maximum Likelihood estimation
Newton-Raphson maximisation, 2 iterations
Return code 1: gradient close to zero
Log-Likelihood: -61.31731 
1  free parameters
Estimates:
     Estimate Std. error t value Pr(> t)
[1,]   0.0393     0.1000   0.393   0.694
[2,]   1.0000     0.0000      NA      NA
--------------------------------------------
          [,1] [,2]
[1,] -100.0018   NA
[2,]        NA   NA
           [,1]
[1,] 0.09999911