The MCL approximation of the observed Fisher information is
given by (1.36) or (1.38). For these data we get
If the usual asymptotics of maximum likelihood hold
(this appears to be an open research question)
would be the asymptotic variance of the MLE. The square root
of the diagonal elements (0.18, 0.039) would be the estimated sampling
error of the MLE. This estimates 
, the difference
between the (unknown) exact MLE and the true parameter value.
We also want to estimate the accuracy of the Monte Carlo calculation,
a standard error for 
, the difference
between the MCMLE and the (unknown) exact MLE. We will call this the
Monte Carlo standard error (MCSE). It can be calculated using formulas
from Geyer (1994, Section 3). If
is the asymptotic variance of the MCL estimate of the score, then
where 
is estimated by 
.
Looking at (1.32) we see that 
is a ratio. The denominator
estimates 
, and the numerator is the sample mean
of the vector time series
i = 1, 
, n. This time series has theoretical mean zero
(see Geyer, 1994) and also has a sample mean exactly zero when 
is the MCMLE. If 
is the estimated asymptotic variance of the sample
mean of this series, then A is estimated by
(Geyer, 1994, equation 20). We estimate using the method of
overlapping batch means (Meketon and Schmeiser, 1984) with a batch length
of 200.
Using all of this we obtain an estimate of the MC error variance
and the square root of the diagonal elements gives (0.019, 0.0037) for the MCSE.
One gets an approximate MCSE for MCNR by using the same formulas as for MCL except setting all the importance weights to 1/n. In our example, this gives (0.0062, 0.0012) for the MCSE. Although, this would give the correct answer for a very small Newton step, it generally underestimates the error. The MCSE using the importance weights described above should be used instead.
