Sampling and Estimation from Finite Populations. Yves Tille

Sampling and Estimation from Finite Populations - Yves Tille


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      where images denotes the empty set.

      Definition 2.1

      A sampling design without replacement images is a probability distribution on images such that

equation

      Definition 2.2

      A random sample images is a random variable whose values are the samples:

equation

      A random sample can also be defined as a discrete random vector composed of non‐negative integer variables images. The variable images represents the number of times unit images is selected in the sample. If the sample is without replacement then variable images can only take the values 0 or 1 and therefore has a Bernoulli distribution. In general, random variables images are not independent except in very special cases. The use of indicator variables images was introduced by Cornfield (1944) and greatly simplified the notation in survey sampling theory because it allows us to clearly separate the values of the variables images or images from the source of randomness images.

      Often, we try to select the sample as randomly as possible. The usual measure of randomness of a probability distribution is the entropy.

      Definition 2.3

      The entropy of a sampling design is the quantity

equation

      We suppose that images

      The sample size images is the number of units selected in the sample. We can write

equation

      When the sample size is not random, we say that the sample is of fixed sample size and we simply denote it by images.

      The variables are observed only on the units selected in the sample. A statistic images is a function of the values images that are observed on the random sample: images. This statistic takes the value images on the sample images. The expectation under the design is defined from the sampling design:

equation

      The variance operator is defined using the expectation operator:

equation

      The inclusion probability images is the probability that unit images is selected in the sample. This probability is, in theory, derived from the sampling design:

equation

      for all images. In sampling designs without replacement, the random variables images have Bernoulli distributions with parameter images There is no particular reason to select units with equal probabilities. However, it will be seen below that it is important that all inclusion probabilities be nonzero.

      The second‐order inclusion probability (or joint inclusion probability) images is the probability that units images and images are selected together in the sample:

equation

      for all images In sampling designs without replacement, when images, the second‐order inclusion probability is reduced to the first‐order inclusion probability, in other words images for all images

      The variance of the indicator variable images is denoted by

equation

      which is the variance of a Bernoulli variable. The covariances between indicators are

equation equation

      be a column vector. The


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