Applied Survey Methods: A Statistical Perspective (Wiley by Jelke Bethlehem

By Jelke Bethlehem

By Jelke Bethlehem

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Extra info for Applied Survey Methods: A Statistical Perspective (Wiley Series in Survey Methodology)

Example text

AN) be a sample from a population U. Suppose a sampling design p has been used with first-order inclusion expectations pk (k ¼ 1, 2, . , N) and second-order inclusion expectations pkl (k, l ¼ 1, 2, . , N). The Horvitz–Thompson estimator for the population mean of the target variable Y is now defined by yHT ¼ N 1X Yk ak : N k¼1 pk ð2:38Þ 39 ESTIMATION The Horvitz–Thompson estimator is an unbiased estimator provided that pk > 0 for all k. Since by definition pk ¼ E(ak), the expected value of this estimator is equal to EðyHT Þ ¼ N N 1X Yk 1X Yk  Eðak Þ ¼ pk ¼ Y: N k¼1 pk N k¼1 pk ð2:39Þ If all pk> are positive, the variance of the Horvitz–Thompson estimator is equal to VðyHT Þ ¼ N X N 1 X Yk Yl ðpkl Àpk pl Þ : 2 N k¼1 l¼1 pk pl ð2:40Þ For samples without replacement, ak can only assume two values: ak ¼ 0 or ak ¼ 1.

The sample can be seen as a scale model of the population. The sample has the same characteristics as the population. The sample proportions are in all respects similar to population proportions. (4) Typical or ideal case(s). The sample consists of elements that are “typical” of the population. ” This meaning probably goes back to the idea of lhomme moyenne (average man) that was introduced by the Dutch/Belgian statistician Quetelet, (1835, 1846). (5) Coverage of the populations heterogeneity.

Possibly, t may depend on the sample values x1, x2, . , xn of an auxiliary variable X and of some population parameter for X. The sample mean and the sample variance are introduced as examples of statistics. 21 The sample mean of a target variable Y is equal to y ¼ n 1X yi : n i¼1 ð2:27Þ The sample mean is simply obtained by summing the sample values and dividing the result by the sample size. 22 The sample variance of a target variable Y is equal to s2 ¼ n 1 X ðyi ÀyÞ2 : nÀ1 i¼1 ð2:28Þ This statistic is obtained by subtracting the sample mean from each sample value, by summing the squared differences, and by dividing the result by n À 1.