Parameter Estimation Using the Likelihood-Based Approach

 For the construction of the final likelihood equation, Royle and Nichols (2003) recommend

imposing a probability model to characterize the underlying distribution of abundances. For

animals that are distributed at random, a natural candidate for modeling the abundance may be

the Poisson model (Royle, Nichols and Kery, 2005).

 The final likelihood equation by using the Poisson model for the abundance is as follows:
 R oo A'- e A k -A
 L(w r,A)= H YTCw [1-(-(1 r)kw, [(1 r)k]T- (2-2)
 1 k=0

 R is the number of sites,
 T is the number of repeated samples,
 w is the detection vector of the total number of detections from each site i, i.e. a vector of
 all the individual site-specific detections, w,.
 2 is the expected abundance at each site, also the Poisson mean.

 For the convenience of numerically maximizing the Equation 2-2, the upper limit of the

variable k is set to a very large number K. So for practical estimation of the parameters, Equation

2-3 is used.
 R K ke-11
 L(w r,A)= Zf TCw,[1- (1- r)k]w [(1- r -w) k- (2-3)
 1 k=0

Parameter Estimation Using the Bayesian Approach

 The Royle and Nichols (2003) model that uses the Poisson distribution to characterize the

abundance can be viewed as a hierarchical model of random variables as follows:

 [w, T r, N, ] binomial[T (1 r)N] (2-4)
 [N, A] poisson[A] (2-5)
 r ~ uniform[0,1] (2-6)
 A gamma[a,b] (2-7)

 Here a and b are the shape and scale parameters associated with the gamma distribution.

Relationships 2-4 and 2-5 jointly represent the likelihood function, while Relationships 2-6 and