# 8. GEOMETRICAL LIMITS FOR INFINITE POPULATIONS

The propositions and conclusions relating to geometrical boundaries for the accuracy curves of finite populations can be easily expanded and adapted to populations of infinite size. We start with the observation that when the population size N increases indefinitely, the parameter a given in (5.22) and based on expressions (5.20) and (5.21), takes the following limit values:

Population can be concave

 (8.1)

Population cannot be concave

 (8.2)

Thus, the pessimistic accuracy curve for an infinite population should be intercepting the A-axis at an accuracy value defined by (8.1) or (8.2). Furthermore and since the population is infinite, the pessimistic accuracy curves, rather than having a maximum value 1 corresponding to a complete enumeration of the population elements, should be approaching the line in an asymptotic manner. It is easy to verify that the model:

 (8.3)

satisfies the two conditions described above. The parameter is a function of n and defined as:

 (8.4)

where, as in the case of a finite population, the primary parameter and its three derivatives have been computed as:

Population can be concave

 (8.5)

Population cannot be concave

 (8.6)

Formulae for a, g and S

 (8.7) (8.8) (8.9)

For large populations the two models described by the functions (7.5) and (8.3) give approximately the same results with the latter being more optimistic to the left and more pessimistic to the right of the critical sample size . However, the model for infinite populations applies rather poorly to populations of small size since it computes pessimistic accuracy values well below those given by (7.5). Figures 8.1 and 8.2 illustrate these observations for two convex populations with sizes N=12 and N=1000. The limit curves A_(x) are described by a darker line while the thin lines correspond to the application of the function .

Fig. 8.1. The two limit curves for a convex population with N=12

Fig. 8.2. The two limit curves for a convex population with N=1000