Page 608 - Probability and Statistical Inference

13. Sample Size Determination: Two-Stage Procedures 585

lot observations are then utilized to estimate the scale parameter σ by T = (m
m
− 1) (X − X ). Next, define the stopping variable,
−1
i m:1
which is a positive integer valued random variable. Here, N estimates C. Re-
call that F 2,2m−2,α is the upper 100α% point of the F 2,2m−2 distribution. This
procedure is implemented along the lines of Stein’s two-stage scheme. Fi-
nally, having obtained N and X , ..., X , we construct the fixed-width confi-
1
N
dence interval J = [X − d, X ] for µ.
N
N:1
N:1
(i) Write down the corresponding basic inequality along the lines
−1
of (13.2.6). Hence, show that d F 2,2m−2,α σ = E [N] = m
µ,σ
+d F 2,2m−2,a σ;
-1
(ii) Show that X and I(N = n) are independent for all n ≥ m;
n:1
(iii) Show that Q ≡ N(X − µ)/σ is distributed as the standard ex
N:1
ponential random variable.
{Hint: From Chapter 4, recall that n(X − µ)/σ and 2(m − 1)T /σ are
n:1 m
respectively distributed as the standard exponential and random vari-
ables, and these are also independent.}
13.2.7 (Exercise 13.2.6 Continued) Show that P {µ ∈ J } ≥ 1 − α for all
µ,σ
N
fixed d, α, µ and σ. This result was proved in Ghurye (1958). {Hint: First
show that N(X − µ)/T is distributed as F 2,2m−2 and then proceed as in the
N:1
m
proof of Theorem 13.2.2.}
13.2.8 (Exercise 13.2.6 Continued) Verify the following expressions.
(i) E [X ] = µ + σE [1/N];
N:1
µ,σ
µ,σ
(ii) E [(X − µ) ] = 2σ E [1/N ];
2
2
2
µ,σ
µ,σ
N:1
(iii) P {µ ∈ J } = 1 − E [e −Nd/σ ].
µ,σ
µ,σ
N
{Hint: Use the fact that the two random variables X and I(N = n) are
n:1
independent for all n ≥ m.}
2
13.2.9 Let the random variables X , X , ... be iid N(µ,σ ), i = 1, 2, and that
i
11
12
the X ’s be independent of the X ’s. We assume that all three parameters are
1j
2j
unknown and (µ , µ , σ) ∈ ℜ × ℜ × ℜ . With preassigned d(> 0) and α ∈ (0, 1),
+
1
2
we wish to construct a 100(1 − α)% fixed-width confidence interval for µ −
1
µ (= µ, say). Having recorded X , ..., X with n ≥ 2, let us denote
2 i1 in
for i = 1, 2 and the
pooled sample variance Consider the confidence interval
for µ − µ . Let us also denote θ = (µ ,
1 2 1
µ , σ).
2
(i) Obtain the expression for P {µ ∈ J } and evaluate the optimal fixed
n
θ
sample size C, had σ been known, such that with n = (C) +1, one
has P {µ ∈ J } ≥ 1 − α;
? n