Sampling and Estimation from Finite Populations. Yves Tille

Sampling and Estimation from Finite Populations

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Proof:

2.5 Estimation of a Total

For estimating the total

the basic estimator is the expansion estimator:

This estimator was proposed by Narain (1951) and Horvitz & Thompson (1952). It is often called the Horvitz–Thompson estimator, but Narain's article precedes that of Horvitz–Thompson. It may also be referred to as the Narain or Narain–Horvitz–Thompson estimator or images ‐estimator or estimator by dilated values.

Often, one writes

but this is correct only if images , for all images If any inclusion probabilities are zero, then images is divided by 0. Of course, if an inclusion probability is zero, the corresponding unit is never selected in the sample.

In this estimator, the values taken by the variable are weighted by the inverse of their inclusion probabilities. The inverse of images is often denoted by images and can be interpreted as the number of units that unit images represents in the population. The value images is the weight assigned to unit images The expansion estimator can also be written as

Result 2.4

A necessary and sufficient condition for the expansion estimator images to be unbiased is that images for all images .

Proof:

Since

the bias of the estimator is

The bias is zero if and only if images for all images

If some inclusion probabilities are zero, it is impossible to estimate images without bias. It is said that the sampling design does not cover the population or is vitiated by a coverage problem. We sometimes hear that a sample is biased, but this terminology should be avoided because bias is a property of an estimator and not of a sample. In what follows, we will consider that all the sampling designs have nonzero first‐order inclusion probabilities.

2.6 Estimation of a Mean

For estimating the mean

we can simply divide the expansion estimator of the total by the size of the population. We obtain

However, the population size is not necessarily known.

Estimator images is unbiased, but it suffers from a serious problem. If the variable of interest images is constant, in other words, images then

(2.1) equation

Indeed, there is no reason that

is equal to 1, even if the expectation of this quantity is equal to 1. The expansion estimator of the mean is thus vitiated by an error that does not depend on the variability of variable images . Even if images images remains random.

We refer to the following definition:

Definition 2.7

An estimator of the mean images is said to be linearly invariant if, for all images , when images then images

For this reason, even when the size of the population images is known, it is recommended to use the estimator of Hájek (1971) (HAJ) which consists of dividing the total by the sum of the inverses of the inclusion probabilities:

(2.2) equation

When images then images Therefore, the bad property of the expansion estimator is solved because the Hájek estimator is linearly invariant. However, images is usually biased because it is a ratio of two random variables. In some cases, such as simple random sampling without replacement with fixed sample size, the Hájek ratio is equal to the expansion estimator.

Sampling and Estimation from Finite Populations. Yves Tille