Page 601 - Probability and Statistical Inference

578 13. Sample Size Determination: Two-Stage Procedures

From this table, some features are immediately noticed. First, the val-
ues have always exceeded the target confidence coefficient 0.95 except in
one case. In the case m = 10, C = 50 we have = .9458 with its estimated
standard error .0037. Since k is very large, an approximate 95% confidence
interval for the true coverage probability (p) in this case can be taken to be
which amounts to (.9336, .9531). Note that the target coverage
probability, that is 0.95, lies between the two confidence bounds 0.9336 and
0.9531.
Table 13.2.1. Moderate Sample Performance of Stein’s
Two-Stage Procedure (13.2.4)−(13.2.5) with α = .05:
5000 Replications

d C s( ) s( )
m = 5, z = 1.96, t = 2.776
.025 4,.025
1.789 30 62.32 0.825 0.9882 0.0041
1.386 50 103.03 1.023 0.9788 0.0043
0.980 100 199.20 1.765 0.9867 0.0041

m = 10, z = 1.96, t = 2.262
.025 9,.025
1.789 30 40.48 0.264 0.9577 0.0029
1.386 50 67.22 0.416 0.9458 0.0037
0.980 100 135.16 0.915 0.9524 0.0032

Next, we observe that the values are sizably larger than the correspond-
ing C values, a feature which validates the fact that the Stein procedure over-
samples significantly. But, the extent of over-sampling falls substantially when
we choose m = 10 instead of m = 5. Compared with C, over-sampling amounts
to approximately 100% and 33% respectively when m = 5 and m = 10. We
may add that the ratio approximately reduces to 2.006 and
1.332 respectively when m = 5 and m = 10. Since and E(N)
≈ the amount of over-sampling we have experienced
should be quite expected. More elaborate exercises with computer simula-
tions are left as class projects. !
The Theorem 13.2.2 shows that the two-stage procedure of Stein solved
a fundamental problem of statistical inference which could not be tackled by
any fixed-sample-size methodology.
On the other hand, there are several important issues one can raise in
this context. The choice of m, the pilot sample size, plays a crucial role in
the performance of the two-stage estimation procedure (13.2.4)-(13.2.5).