Page 595 - Probability and Statistical Inference

572 13. Sample Size Determination: Two-Stage Procedures

(1991)], and (iii) survival analysis [Gardiner and Susarla (1991), Gardiner et.
al. (1986)].
To achieve a balanced perspective, the reader may browse through the
books of Wald (1947), Bechhofer et al. (1968), Ghosh (1970), Armitage (1973),
Sen (1981), Woodroofe (1982), Siegmund (1985), Whitehead (1983),
Mukhopadhyay and Solanky (1994), Balakrishnan and Basu (1995), and Ghosh
et al. (1997) as well as the Handbook of Sequential Analysis edited by Ghosh
and Sen (1991). The Sequential Analysis journal provides the most up-to-date
account in this important area. We hope that the cited sources will guide
interested readers to consider many other important and relevant references.
There are important reasons why we include such topics to end this book.
One point to note is that many results derived earlier come together here as
important tools and machineries needed for technical analyses. But, there is
another point to note. From previous chapters, a reader may get an impres-
sion that all reasonable statistical problems can be handled by the classical
fixed-sample-size analyses and we show otherwise.
We recall Example 7.3.2 where we showed that it was impossible to find
-1
any unbiased estimator of p when the data consisted of iid Bernoulli(p) ob-
servations X , ..., X with predetermined n. But, in Example 7.3.3 we showed
1
n
how inverse binomial sampling method could provide an unbiased estimator
of p , a parametric function which is very important in statistical applications
-1
in ecology.
In the present chapter, we touch upon two statistical problems with very
realistic goals which can not be solved if the sample size is fixed in advance,
but these problems can be solved with the help of two-stage sampling strate-
gies. The emphasis is to show that there is a wide world out there beyond the
classical fixed-sample-size analysis.

Neither a fixed-width confidence interval satisfying both requirements
in (13.1.3) nor the bounded risk point estimation problem can be
solved by any fixed-sample-size method. A Stein type two-stage
methodology provides an exact solution for both problems.

In Section 13.2, we give the construction of a fixed-width confidence
interval for the unknown mean of a normal population whose variance is
completely unknown. This is accomplished (Theorem 13.2.2) by the Stein
two-stage methodology. In Section 13.3, we briefly address a point estima-
tion problem by requiring a preassigned upper bound on the mean square
error.