By Pascal Ardilly

This booklet comprises 116 routines of sampling tools solved intimately. The routines are grouped into chapters and are preceded by means of a quick theoretical overview specifying the notation and the central effects which are worthwhile for knowing the recommendations. a few routines improve the theoretical facets of surveys, whereas others take care of extra utilized difficulties.

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**Additional info for Sampling Methods: Exercises and Solutions**

**Sample text**

Solution If we denote Ii as the indicator variable for the presence of unit i in sample S, we have 1 if i ∈ S Ii = 0 if i ∈ / S. The ﬁrst- through fourth-order inclusion probabilities are: n π1 = E(Ii ) = , i = 1, . . , N, N π2 = E(Ii Ij ) = π3 = E(Ii Ij Ik ) = n(n − 1) , j = i, N (N − 1) n(n − 1)(n − 2) , j = i, k = i, k = j, N (N − 1)(N − 2) and n(n − 1)(n − 2)(n − 3) , N (N − 1)(N − 2)(N − 3) j = i, k = i, = i, k = j, = j, = k. 21 51 The corrected variance in the sample is: s2y = 1 n−1 2 yi − Y , i∈S where 1 n Y = yi .

It remains to calculate the total number of favourable cases. It is exactly a question of the number of surjective mappings of {1, . . , m} in {1, . . , r}, which is equal to (r) r! multiplied by the Stirling number of second kind sm , which is: s(r) m = 1 r! r i=1 r m i (−1)r−i . i The Stirling number of second kind is equal to the number of ways of ﬁnding a group of m elements in r non-empty parts (see Stanley, 1997). However, (r) (r) the calculation of sm does not interest us here. Indeed, sm does not depend on N but only on m and r.

Thus, for m > 0, magnitude by (nM/N ) N= Mn Mn ≈ (1 − ∆ + ∆2 ), m M and E(N | m > 0) ≈ Mn V (1 + E(∆2 | m > 0)) = N Pr(m > 0) 1 + M M2 . 5. The estimator is then biased. The bias results from the conjunction of two elements: on the one hand, we are restricted at m > 0, and on the other hand the random variable m is in the denominator of the estimator. 16 35 approaches 1 and that V varies by 1/n M2 and therefore approaches zero. The estimator N thus appears as an interesting estimator of N . It would remain to calculate its variance.