Accuracy of maximum likelihood estimators

Question

Here's a test for comparing ML estimators of the lambda parameter of a Poisson distribution.

with(data.frame(x=rpois(2000, 1.5), i=LETTERS[1:20]),
     cbind(cf=tapply(x, i, mean),
           iter=optim(rep(1, length(levels(i))), function(par) 
             -sum(x * log(par[i]) - par[i]), method='BFGS')$par))

The first column shows the ML estimator obtained from the closed-form solution (for reference), while the second column shows the ML estimator obtained by maximizing a log-likelihood function using the BFGS method. Results:

    cf     iter
A 1.38 1.380054
B 1.61 1.609101
C 1.49 1.490903
D 1.47 1.468520
E 1.57 1.569831
F 1.63 1.630244
G 1.33 1.330469
H 1.63 1.630244
I 1.27 1.270003
J 1.64 1.641064
K 1.58 1.579308
L 1.54 1.540839
M 1.49 1.490903
N 1.50 1.501168
O 1.69 1.689926
P 1.52 1.520876
Q 1.48 1.479891
R 1.64 1.641064
S 1.46 1.459310
T 1.57 1.569831

It can be seen the estimators obtained with the iterative optimization method can deviate quite a lot from the correct value. Is this something to be expected or is there another (multi-dimensional) optimization technique that would produce a better approximation?

Answer 1

Answer provided by Chase:

the reltol parameter which gets passed to control() lets you adjust the threshold of the convergence. You can try playing with that if necessary.

Edit:

This is a modified version of the code now including the option reltol=.Machine$double.eps, which will give the greatest possible accuracy:

with(data.frame(x=rpois(2000, 1.5), i=LETTERS[1:20]),
     cbind(cf=tapply(x, i, mean),
           iter=optim(rep(1, length(levels(i))), function(par) 
             -sum(x * log(par[i]) - par[i]), method='BFGS',
             control=list(reltol=.Machine$double.eps))$par))

And the result is:

    cf iter
A 1.65 1.65
B 1.54 1.54
C 1.80 1.80
D 1.44 1.44
E 1.53 1.53
F 1.43 1.43
G 1.52 1.52
H 1.57 1.57
I 1.61 1.61
J 1.34 1.34
K 1.62 1.62
L 1.23 1.23
M 1.47 1.47
N 1.18 1.18
O 1.38 1.38
P 1.44 1.44
Q 1.66 1.66
R 1.46 1.46
S 1.78 1.78
T 1.52 1.52

So, the error made by the optimization algorithm (ie. the difference between cf and iter) is now reduced to zero.