number of times, a posterior distribution of the parameter of interest will emerge based on the

time spent on each point in the parameter space.

The Gibbs Sampler Algorithm for the Royle and Nichols (2003) Model

Step 1: Selecting the initial values for r, 2 and N,.

Iteration 1

 r ) : random number chosen from a Uniform (0,1) distribution
 So, q(1) = logit[r()]
 (1) : random number chosen from a Gamma (a,b), where a and b are the shape and scale
 parameters initially selected.
 N,() : random number chosen from a Poisson[ 1)], where i = 1, 2, ...... R sites.

Step 2: Updating the values of r, 2 and N,.

Iterationj [ranging from 2 to a large number]

 [NO) I w,., A'1), r-1)] : random number drawn according to Equation 2-10 where
 i= 1, 2, ...... R sites
 [A2)I {N,)}] : random number drawn according to equation (8). The '{}' indicates the
 entire vector of site abundances.
 [r/) | {w}, {N,)}] : random number drawn according to the proportionality relationship of

 Equation 2-9. Consequently r() --
 1+e10

 Step 2 is repeated a large number of times. Using the Equations 2-8 and 2-10 the updates

for NO) and A(J) can be made quite directly in the Gibbs sampler. However, making the updates

for r/) requires the use of the Metropolis algorithm (Gelman et al., 1995) with a Gaussian

proposal distribution since Equation 2-9 is only a proportionality relationship.

Simulation Design

 Royle and Nichols (2003) have already shown the performance of the model in varying

large sample situations and have established that the likelihood-based inference works

reasonably well for inferences about estimates of A for even low values ofr and Twhen R is 200

or greater. However, in their simulation design, they have chosen values for the true value of A