Sampling and Estimation from Finite Populations. Yves Tille

of inclusion probabilities is

      Define also the symmetric matrix:

      and the variance–covariance matrix

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

      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 .

      Example 2.1

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

2.4 Parameter Estimation

      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 .

      Definition 2.4

      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

      Definition 2.5

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

      Definition 2.6

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):

      Result 2.3

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