# Information matrix for exponential distribution (using covariates)

(Reposting after changing some things and the title: standard error for glm: R and theory arent agreeing)

I was trying to work out by hand the standard errors for an exponential glm (gamma distn with dispersion=1) with a log link ($\log\mu=\beta_0 + \beta_1 x$).

I'm getting some contradictions. by hand I do this:

exponential has prob function $f = (1/\mu) \exp(-y/\mu)$, so log-likelihood is $\ell = -\sum \log\mu + (y/\mu)$. That means $d\ell/d\mu = \sum (y-\mu)/\mu^2$.

Also:

• $\partial\ell/\partial\beta_0 = \sum (y-\mu)/\mu$ and
• $\partial\ell/\partial\beta_0 = \sum x(y-\mu)/\mu$.

so using the chain rule the information matrix bits are:

• $\partial^2\ell/\partial\beta_0^2 = \sum y/\mu$
• $\partial^2\ell/\partial\beta_1^2 = \sum x^2y/\mu$
• $\partial^2\ell/\partial\beta_0\beta_1 = \sum xy/\mu$

I tried to see if that worked in a numerical example in R.

in R:

data(trees) # comes with R

y <- trees$Volume x <- trees$Girth

### in r:
mm <- glm(y ~ x, family=Gamma(link=log))
InfMat.R <- solve(summary(mm, dispersion=1)$cov.unscaled) ### by hand lb0b0 <- sum( y / fitted(mm)) lb0b1 <- sum( y * x / fitted(mm)) lb1b1 <- sum( y * x^2 / fitted(mm)) InfMat.me <- array( dim=c(2,2)) InfMat.me[1,1] <- lb0b0 InfMat.me[1, 2] <- InfMat.me[2, 1] <- lb0b1 InfMat.me[2, 2] <- lb1b1 InfMat.R # what r gives me InfMat.me # when I do it ny hand  the two information matrices are so close to being the same: > InfMat.R (Intercept) x (Intercept) 31.0 410.70 x 410.7 5736.55 > InfMat.me [,1] [,2] [1,] 31.0 410.700 [2,] 410.7 5718.696  So it is only element (2,2) that differs. and the rest are right. It seems odd that I can get all the bits right except one. Can anyone explain what I'm doing wrong? • Are your derivations assuming the$x\$ is centered? – AdamO Nov 27 '17 at 23:00
• Please don't repost modifications of your questions--that just litters our site with duplicates. See our help center for more information about how this site works. – whuber Nov 27 '17 at 23:19
• how does centering change things? i didn't center in the hand calculations. but wouldn't that change all the bits in the info matrix? why is only (2,2) different? – Wiggin Peters Nov 27 '17 at 23:25