Question

Let the population have N=7 units, with {(unit,value)} = {(A,-1),(B,+1),(C,-2),(D,+3),(E,-4),(F,+5),(G,-6)}. The design is as follows: first choose A or B at random; if A then choose from {C,D} at random, if B then choose from {E,F,G} at random. 1) Find the first-order inclusion probabilities (note that the sample size n is fixed at 2).

Verify (show numerically for this example) that the Horvitz-Thompson estimator is unbiased for the population total. (Hint: find the probability of each sample and the value of the HTE for that sample, multiply and sum.) Repeat your calculations with (G,-106) instead of (G,-6). Is the HTE still unbiased? Comment.

Answer 1

P(A is included)=1/2 becausein the desingn ,we first choose randomly between A and B.

Similarly, P(B is included)=1/2.

Now, if C or D is included, then A must be included. Similarly, if either of E,F or G is included, then B must be included.

Now, P(C is included)=P(A is included)*P(C is included|A is included)=(1/2)*(1/2)=1/4

Similarly, P(D is included)=1/4

Now, P(E is included)=P(E is included)*P(E is included|B is included)=(1/2)*(1/3)=1/6

Similarly, P(F ios included)=P(G is included)=1/6

Note that, A and B will only be included in the first draw and the remaining can only be included at the second draw.