Yves Tille - Sampling and Estimation from Finite Populations

and

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

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

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

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 are null because the sampling design is of fixed sample size

A parameter or a function of interest 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 .

An estimator is a statistic used to estimate a parameter.

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

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

The bias of an estimator 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):

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

For estimating the total

the basic estimator is the expansion estimator: