Sampling and Estimation from Finite Populations. Yves Tille

Sampling and Estimation from Finite Populations

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of inclusion probabilities is

Define also the symmetric matrix:

and the variance–covariance matrix

Matrix images is a variance–covariance matrix which is therefore semi‐definite positive.

Result 2.1

The sum of the inclusion probabilities is equal to the expected sample size.

Proof:

If the sample size is fixed, then images is not random. In this case, the sum of the inclusion probabilities is exactly equal to the sample size.

Result 2.2

If the random sample is of fixed sample size, then

Proof:

Let images be a column vector consisting of images ones and images a column vector consisting of images zeros, the sample size can be written as images . We directly have

and

If the sample is of fixed sample size, the sum of all rows and all columns of images is zero. Therefore, matrix images is singular. Its rank is then less than or equal to images .

Example 2.1

Let images be a population of size images = 3. The sampling design is defined as follows:

and images for all other samples. The random sample is of fixed sample size images .

The sum of the inclusion probabilities is equal to the sample size. Indeed,

The joint inclusion probabilities are

Therefore, the matrices are

and

We find that the sums of all the rows and all the columns of images are null because the sampling design is of fixed sample size images

2.4 Parameter Estimation

A parameter or a function of interest images is a function of the values taken by one or more variables in the population. A statistic is a function of the data observed in the random sample images .

Definition 2.4

An estimator images is a statistic used to estimate a parameter.

If images denotes the value taken by the estimator on sample images , the expectation of the estimator is

Definition 2.5

An estimator images is said to be unbiased if images , for all images , where images

Definition 2.6

The bias of an estimator images is the difference between its expectation and the parameter it estimates:

From the expectation, we can define the variance of the estimator:

and the mean squared error (MSE):

Result 2.3

The mean squared error is the sum of the variance and the square of the bias:

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