Page 609 - Probability and Statistical Inference

586 13. Sample Size Determination: Two-Stage Procedures

(ii) Propose an appropriate Stein type two-stage stopping variable N for
which one will be able to conclude that P {µ ∈ J } ≥ 1 – α for all
θ
N
fixed θ,d, α. Here, N should be defined using the pooled sample
variance S obtained from pilot samples.
2
Pm
{Note: Analogous problems for the negative exponential populations were
discussed by Mauromaustakos (1984) and Mukhopadhyay and
Mauromoustakos (1987).}
13.2.10 (Behrens-Fisher problem) Let X , ..., X be iid N(µ , ) random
ini
i1
i
variables, n ≥ 2, i = 1, 2, and that the X ’s be independent of the X ’s. Here
i
1j
2j
we assume that µ , µ , σ , σ are all unknown and (µ , µ ) ∈ ℜ , (σ , σ ) ∈
2
1 2 1 2 1 2 1 2
+2
ℜ , σ ≠ σ . Let respectively be the sample mean and variance, i =
1 2
1, 2. With preassigned numbers α ∈ (0, 1) and d(> 0), we wish to construct
a confidence interval J for µ – µ such that (i) the length of J is 2d as well as
1
2
(ii) P {µ - µ ∈ J} ≠ 1 – a for all fixed θ = (µ , µ , σ , σ ) and d, α.
θ 1 2 1 2 1 2
(i) Let us denote a fixed-width confidence interval J = [{ – }
n
± d] for µ - µ where n = (n , n ). Find the expression for P {µ -
1
2
?
1
2
1
µ ∈ J } with fixed n;
2
n
(ii) Define h such that P{W – W ≤ h} = 1 – ½α, where W , W are iid
1
2
2
1
Student’s t variables with some fixed m ≥ 2. Let m be the pilot
m–1
sample size from both populations. We define N = max {m, <h
2
i
2
/d > + 1}, and sample the difference N – m in the second stage
i
from the i population, i = 1, 2. Now, with N = (n , n ), we propose
th
1 2
the fixed-width confidence interval for
µ – µ . Show that P {µ – µ ∈ J } ≥ 1 – α for all fixed θ = (µ , µ ,
θ
N
2
1
2
2
1
1
σ , σ ) and d, α.
1 2
{Hint: It is a hard problem. This exact and elegant solution is due to
Chapman (1950). In the Exercise 11.3.15, we cited the original papers of
Behrens (1929), Fisher (1935,1939), Creasy (1954) and Fieller (1954). Ghosh
(1975) tabulated the h values for a range of values of m and a. The analogous
two-sample problem for the negative exponential distributions was developed
in Mukhopadhyay and Hamdy (1984). Section 6.7.1 in Ghosh et al. (1997)
gave a detailed account of the related sequential literature.}
13.3.1 Prove Theorem 13.3.1 along the lines of the proof of Theorem
13.2.1.
13.3.2 Let us consider a sequence of iid observations X , X , ... from
2
1
a N(µ, σ ) population where the parameters µ ∈ ℜ and σ ∈ ℜ are both
2
+
assumed unknown. Having recorded the observations X , ..., X , the
1 n
customary point estimator for the population mean µ is the sample mean