# 4. CONVEX, FLAT AND CONCAVE POPULATIONS

## 4.1 Descriptive definition

Throughout the paper reference is made to three major categories of population distributions that are conventionally described as convex, flat and concave. This categorization is based on mathematical criteria but its practical meaning would be described as follows:

(a) CONVEX: population elements are clustered around their mean and the population density is decreasing near the boundaries.

(b) FLAT: Refers to random populations and populations with equidistant elements.

(c) CONCAVE: population elements are clustered around their boundaries and the population density is decreasing near the mean.

In the next sections it will be demonstrated that, overall, sampling accuracy is lower and more fluctuating when working with concave populations. Its growth pattern improves when the population is flat and keeps improving as the population becomes more convex. These observations regarding sampling accuracy are fairly well in line with experience from sample-based fishery surveys.

However, and as mentioned earlier in the introductory section, knowledge of the distribution of the target population is not conditional for setting-up geometrical constraints for the sampling accuracy. In the absence of any information regarding the population under study, the method will set-up geometrical constraints based on the pessimistic assumption that the population may be concave.

## 4.2 Mathematical definition

We will now give a mathematical definition of convex, flat and concave populations. This definition is based on a number of observations and conclusions discussed in Sections 4.3 and 5.1. For a population with size N we can always determine the "worst" possible PSA with an associated "worst" accuracy curve W(x). According to (3.2) the associated Worst Overall Mean Accuracy (WOMA) will be:

 (4.1)

Depending on the value of WOMA the three population categories are defined as follows:

(a) Population is convex if

(b) Population is flat if

(c) Population is concave if

## 4.3 Proposition concerning the Worst Overall Mean Accuracy (WOMA)

The WOMAs of all convex populations with size N are higher than the WOMA of a flat population with the same size, the latter being a lower limit when the elements of the convex populations disperse away from the population mean.

The WOMAs of all concave populations with size N are lower than the WOMA of a flat population with the same size, the latter being an upper limit when the elements of the concave populations move towards the population mean.

A thorough mathematical proof would be rather tedious and involve several computational steps not quite within the desired scope of the paper. The following descriptive approach may be considered sufficient for the purpose of the study.

We will first show that by changing the status of a population so that it becomes more concave, the resulting WOMA will be smaller.

Let us assume a population with N ranked elements and mean m. It is recalled that for any random sample with size n and sample mean mn the corresponding accuracy An will be:

 for (4.2) for (4.3)

In populations with ranked elements the WOMA is formed by sampling sequentially either from left to right or from right to left and by applying one of the expressions (4.2), (4.3).

We now select any two elements , such that and and use a small positive value d so that by transforming the selected elements to and , the ranking order of the new population remains the same. The new population will have the following properties:

(a) It will be more concave than the original one since two of its elements are closer to the extreme values 0 and 1.

(b) Since the original population mean m will also be the new population mean.

(c) All samples with size n not containing , will have the same sample mean as in the original population.

(d) Since all samples of size n containing both and will have the same sample mean as in the original population.

(e) Any left-to-right sample with size n and containing only the element will have a sample mean .

(f) Any right-to-left sample with size n and containing only the element will have a sample mean .

Let us now consider the pattern of the WOMA for the original population. This will consist of accuracy values resulting by sampling either from left to right or from right to left. By applying the same sampling pattern to the transformed population (note that this pattern is not essentially the WOMA for the new population), and by taking into account the properties listed above we observe that:

(a) for all samples with size n and not containing , , since the population means and the sample means are the same.

(b) for all samples with size n and containing both and , since the population means and the sample means are the same.

(c) For a left-to-right containing only the new accuracy will be:

(d) For a right-to-left containing only the new accuracy will be:

In other words each of the new accuracy values will be less than or equal to the original worst accuracy , which also means that .

We have thus proved that for the transformed population there exists a PSA resulting in an overall mean accuracy lower than the WOMA of the original population, which means that also the WOMA of the new population will be even lower.

It is now evident that that by repeating the transformation process described above for convex populations, there will be a point at which these populations will become flat (equidistant). This, in turn, means that the WOMA of a flat population is a lower limit for the WOMAs of all convex populations and an upper limit for the WOMAs of all concave populations.

By using the same transformation process for concave populations there will be a point at which all population elements will be clustered around the extreme values 0 and 1. The question then arises as to which population of size N and with elements 0 and 1 has the lowest possible WOMA. This question will be discussed in the next section where global boundaries will be set-up for populations with the same size N.