Sampling with unequal selection probabilities

Forest Inventory lecturenotes

Category Forest Inventory lecturenotes not found

Introduction

Mostly, one speaks about random sampling with equal selection probabilities: each element of the population has the same probability to be selected. However, there are situations in which this idea of equal selection probabilities does not appear reasonable: if it is known that some elements carry much more information about the target variable, they should also have a greater chance to be selected. Stratification goes into that direction: there, the selection probabilities within the strata were the same, but could be different between strata.

Sampling with unequal selection probabilities is still random sampling, but not simple random sampling, but “random sampling with unequal selection probabilities”. These selection probabilities, of course, must be defined for each and every element of the population before sampling and none of the population elements must have a selection probability of 0.

Various sampling strategies that are important for forest inventory base upon the principle of unequal selection probabilities, including

• angle count sampling (Bitterlich sampling),

• importance sampling,

• 3 P sampling,

• randomized branch sampling.

After a general presentation of the statistical concept and estimators, these applications are addressed.

In unequal probability sampling, we distinguish two different probabilities – which actually are two different points of view on the sampling process:

The selection probability is the probability that element i is selected at one draw (selection step). The Hansen-Hurwitz estimator for sampling with replacement (that is; when the selection probabilities do not change after every draw) bases on this probability. The notation for selection probability is written as \(P_i\) or \(p_i\).

The inclusion probability refers to the probability that element i is eventually (or included) in the sample of size n. The Horvitz-Thompson estimator bases on the inclusion probability and is applicable to sampling with or without replacement. The inclusion probability is generally denoted by \(\pi\).

List sampling = PPS sampling

If sampling with unequal selection probabilities is indicated, the probabilities need to be determined for each element before sampling can start. If a size variable is available, the selection probabilities can be calculated proportional to size. This is then called PPS sampling (probability proportional to size).

Table 1. Listed sampling frame as used for „list sampling” where the selection probability is determined proportional to size. Population element List of the size variables of the population elements List of cumulative sums Assigned range 1 10 10 0 - 10 2 20 30 > 10 - 30 3 30 60 > 30 - 60 4 60 120 > 60 - 120 5 100 220 > 120 - 220

This sampling approach is also called list sampling because the selection can most easily be explained by listing the size variables and select from the cumulative sum with uniformly distributed random numbers (which perfectly simulates the unequal probability selection process). This is illustrated in Table 1: the size variables of the 5 elements are listed (not necessarily any order!) and the cumulative sums calculated. The, uniformly distributed random number is drawn between the lowest and highest possible value of that range, that is from 0 to the total sum. Assume, for example, the random number 111.11 is drawn; this falls into the range “>60 – 120” so that element 4 is selected. Obviously, the elements have then a selection probability proportional to the size variable.

Hansen-Hurwitz estimator

The Hansen-Hurwitz estimator gives the framework for all unequal probability sampling with replacement (Hansen and Hurwitz, 1943). “With replacement” means that the selection probabilities are the same for all draws; if selected elements would not be replaced (put back to the population), the selection probabilities would change after each draw for the remaining elements.

Suppose that a sample of size n is drawn with replacement and that on each draw the probability of selecting the i-th unit of the population is \(p_i\).

Then the Hansen-Hurwitz estimator of the population total is

\[\hat \tau = \frac {1}{n} \sum_{i=1}^n \frac {y_i}{p_i}\]

Here, each observation \(y_i\) is weighted by the inverse of its selection probability \(p_i\).

The parametric variance of the total is

\[var (\hat \tau) = \frac {1}{n} \sum_{i=1}^N p_i \left (\frac {y_i}{p_i} - \tau \right )^2\]

which is unbiasedly estimated from a sample size n from

\[v\hat ar (\hat \tau) = \frac {1}{n} \frac {\sum_{i=1}^n \left (\frac {y_i}{p_i} - \tau \right )^2}{n-1}\]

Hansen-Hurwitz estimator examples: 4 application examples