  CO isotopologue transitions are routinely observed in molecular clouds to
probe the column density of the gas, the elemental ratios of carbon and oxygen,
and to trace the kinematics of the environment. We aim at estimating the
abundances, excitation temperatures, velocity field and velocity dispersions of
the three main CO isotopologues towards a subset of the Orion B molecular
cloud. We use the Cramer Rao Bound (CRB) technique to analyze and estimate the
precision of the physical parameters in the framework of
local-thermodynamic-equilibrium excitation and radiative transfer with an
additive white Gaussian noise. We propose a maximum likelihood estimator to
infer the physical conditions from the 1-0 and 2-1 transitions of CO
isotopologues. Simulations show that this estimator is unbiased and efficient
for a common range of excitation temperatures and column densities (Tex > 6 K,
N > 1e14 - 1e15 cm-2). Contrary to the general assumptions, the different CO
isotopologues have distinct excitation temperatures, and the line intensity
ratios between different isotopologues do not accurately reflect the column
density ratios. We find mean fractional abundances that are consistent with
previous determinations towards other molecular clouds. However, significant
local deviations are inferred, not only in regions exposed to UV radiation
field but also in shielded regions. These deviations result from the
competition between selective photodissociation, chemical fractionation, and
depletion on grain surfaces. We observe that the velocity dispersion of the
C18O emission is 10% smaller than that of 13CO. The substantial gain resulting
from the simultaneous analysis of two different rotational transitions of the
same species is rigorously quantified. The CRB technique is a promising avenue
for analyzing the estimation of physical parameters from the fit of spectral
lines.
