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Statistics and Probability Letters
journal homepage: www.elsevier.com/locate/stapro
Simultaneous testing for the successive differences of exponential
location parameters under heteroscedasticity
Vishal Maurya ∗ , Anju Goyal, Amar Nath Gill
Department of Statistics, Panjab University, Chandigarh 160 014, India
article
info
Article history:
Received 1 March 2011
Received in revised form 16 May 2011
Accepted 19 May 2011
Available online 27 May 2011
Keywords:
Two-parameter exponential distribution
Two-stage procedure
One-stage procedure
Bonferroni inequality
Heteroscedasticity
abstract
In this paper, the design-oriented two-stage and data-analysis one-stage multiple
comparison procedures for successive comparisons of exponential location parameters
under heteroscedasticity are proposed. One-sided and two-sided simultaneous confidence
intervals are also given. We also extend these simultaneous confidence intervals for
successive differences to a larger class of contrasts of the location parameters. Upper limits
of critical values are obtained using the recent techniques given in Lam [Lam, K., 1987.
Subset selection of normal populations under heteroscedasticity. In: Proceedings of the
Second International Advanced Seminar/Workshop on Inference Procedures Associated
with Statistical Ranking and Selection, Sydney, Australia; Lam, K., 1988. An improved
two-stage selection procedure. Communications in Statistics Simulation and Computation.
17 (3), 995–1006]. These approximate critical values are shown to have better results
than the approximate critical values using the Bonferroni inequality developed in
this paper. Finally, the application of the proposed procedures is illustrated with an
example.
© 2011 Elsevier B.V. All rights reserved.
1. Introduction
Consider k (≥ 3) independent exponential populations π1 , . . . , πk such that the observations from populations πi follow
distribution with probability density function (pdf),
f (x | µi , θi ) =
1 −((x−µi )/θi )
e
I[µi ,∞) (x),
θi
θi > 0,
where IA (·) is the indicator function of event A and µi (θi ) is the location (scale) parameter, i = 1, . . . , k. It is assumed that
location parameters µ′ s satisfy the simple ordering µ1 ≤ · · · ≤ µk with at least one strict inequality.
The exponential distribution in reliability, life testing and medical sciences is as important as the normal distribution in
sampling theory and agricultural statistics. In dose–response experiments, the exponential distribution E (µ, θ ) is generally
used to model the effective duration of a drug, where the location parameter µ is referred to as the guaranteed effective
duration and the scale parameter is called the mean effective duration in addition to µ. Also, in biological and epidemiological
studies the location parameter µ is called the latency period (time elapsed between the first exposure to an agent and the
appearance of the symptoms) and the scale parameter θ is termed as the mean duration of a disease in addition to the latency
period. In reliability and engineering, the location parameter is called the guaranteed mean life and scale parameter is called
the mean life in addition to guaranteed life. Many practical situations where it is known a priori that µ1 ≤ · · · ≤ µk , arise in
∗
Corresponding author. Tel.: +91 22 28494590; fax: +91 9316113341.
E-mail addresses: [email protected], [email protected] (V. Maurya).
0167-7152/$ – see front matter © 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.spl.2011.05.010
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Statistics and Probability Letters 81 (2011) 1507–1517
V. Maurya et al. / Statistics and Probability Letters 81 (2011) 1507–1517
dose–response experiments when the k treatments consist of monotonically increasing levels of the dose of a certain drug,
and it is postulated that the guaranteed effective period is a nondecreasing function of the dose level.
When the k scale parameters are equal i.e. θ1 = · · · = θk = θ , Chen (1982) and Dhawan and Gill (1997) discussed
test procedures for testing the null hypothesis H0 : µ1 = · · · = µk against the simple ordered alternative H1 : µ1 ≤
· · · ≤ µk with at least one strict inequality, and inverted the test statistic to obtain simultaneous confidence intervals
for the differences µj − µi , 1 ≤ i < j ≤ k of exponential location parameters. In the literature, Marcus (1976), Hayter
(1990) and Hayter and Liu (1996) and references cited therein have addressed the related problems of testing homogeneity
against simple ordered alternative; other references cited are Barlow et al. (1972), Robertson et al. (1988), Hochberg and
Tamhane (1987). William (1977), and Marcus (1982) considered simultaneous one-sided confidence intervals for the class
of monotone contrasts. However, an experimenter may want to look directly at only the successive differences between
the treatment effects, namely the set of differences µ2 − µ1 , µ3 − µ2 . . . µk − µk−1 . Lee and Spurrier (1995) presented a
new procedure for making successive comparisons between ordered treatment effects of normal populations and presented
required critical points for k (k ≤ 6). Later, Liu et al. (2000) provided critical points, using recursive integration technique,
for larger values of k and discussed how the procedure of Lee and Spurrier (1995) can be extended to provide information on
larger sets of contrasts of the treatment effects. All these authors have addressed the problem of testing the homogeneity of
location parameters against simple ordered alternative under either normal or exponential probability models by assuming
the homogeneity of scale parameters.
Recently, Singh et al. (2006) provided a procedure for successive comparisons of exponential location parameters
by assuming the equality of scale parameters. In this article, procedure proposed by Singh et al. (2006) for successive
comparisons of exponential location parameters is extended to the situation when scale parameters θ1 , . . . , θk are unknown
and possibly unequal. The layout of the article is as follows.
The two-stage multiple comparison procedures for successive comparisons of exponential location parameters under
heteroscedasticity using Lam’s (1987, 1988) technique are proposed in Section 2. In Section 3, the approximate critical
values obtained using Bonferroni inequality are developed. In this section, we also compared the approximate critical values
obtained by using Lam’s (1987, 1988) technique with the approximate critical values obtained by using Bonferroni inequality
and found that the latter approximate critical values are more conservative compared with the approximate critical values
obtained using the Lam’s (1987, 1988) techniques. Therefore, the Lam’s (1987, 1988) techniques are recommended to be
exploited for the two-stage multiple comparison procedures for successive comparisons of exponential location parameters
under heteroscedasticity. When the additional sample for the second stage may not be available due to the experimental
budget shortage or other factors in an experiment, a one-stage multiple comparison procedure is proposed in Section 4. In the
same section we have also discussed how the one-sided and two-sided simultaneous confidence intervals can be extended
to a larger class of contrasts of the location parameters. In Section 5, an example of survival days of patients with inoperable
lung cancer who were subjected to a standard chemotherapeutic agent is used to illustrate the proposed procedures. Finally,
a simulation study of the performance of one-stage and two-stage procedures for different parameter configurations is given
in Section 6. The results show that all simulated coverage rates are higher than the nominal confidence coefficients.
2. Two-stage one-sided and two-sided simultaneous confidence intervals for the successive differences µi+1 − µi , i =
1, . . . , k − 1 using Lam’s (1987, 1988) technique.
A two-stage sampling procedure is described as follows. Take an initial sample Xi1 , . . . , Xim of size m (≥ 2) from πi . Let
Xi = min(Xi1 , . . . , Xim ) be a suitable estimator of µi , based on a sample of size m for i = 1, . . . , k. Define
Ni = max{m, [Si /c ] + 1},
i = 1, . . . , k,
(2.1)
where Si =
j=1 (Xij − Xi )/(m − 1), i = 1, . . . , k. The value of c, in (2.1), is an arbitrary positive constant to be chosen to
control (to be discussed later) the width of the confidence intervals for µi+1 − µi and [x] denote the largest integer smaller
than or equal to x.
When Ni > m, take Ni − m additional observations Xi,m+1 , . . . , Xi,Ni from πi and the sample values are denoted by
Xi1 , . . . , Xim , Xi,m+1 , . . . , Xi,Ni for πi . Let
∑m

X̃i,Ni = X̃i =
min(Xi , Xi,m+1 , . . . , Xi,Ni )
Xi
when Ni > m
when Ni = m,
(2.2)
be the minimum value of the combined sample from πi , i = 1, . . . , k. Hereafter P0 (A) represents the probability of the event
1
A when all µ1 , . . . , µk are equal. Let F2−,2m
−2 (1 − α) be the 100(1 − α)th percentile of F distribution with (2, 2m − 2)df . We
now propose the one-sided and two-sided simultaneous confidence intervals for the successive differences µi+1 − µi , i =
1, . . . , k − 1, in following theorem using the two-stage procedure.
Theorem 1. For a given 0 < α < 1,
1
1/(k−1)
(a) P (µi+1 − µi ≥ X̃i+1 − X̃i − cuk,m,α , i = 1, . . . , k − 1) ≥ 1 − α if uk,m,α = F2−,2m
).
−2 ((1 − α)
Thus (X̃i+1 − X̃i − cuk,m,α , ∞) is a set of lower one-sided confidence intervals for µi+1 − µi with confidence coefficient 1 − α .
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1508
1509
1
1/k
(b) P (X̃i+1 − X̃i − c vk,m,α ≤ µi+1 − µi ≤ X̃i+1 − X̃i + c vk,m,α , i = 1, . . . , k − 1) ≥ 1 − α if vk,m,α = F2−,2m
−2 ((1 − α) ).
Thus (X̃i+1 − X̃i − c vk,m,α , X̃i+1 − X̃i + c vk,m,α ) is a set of simultaneous two-sided confidence intervals for µi+1 − µi with
confidence coefficient 1 − α .
The recent techniques given in Lam (1987, 1988) are described in the following lemma:
Lemma 2. Suppose X and Y are two random variables, a and b are two positive constants, then
[aX ≥ bY − d max(a, b)] ⊇ [X ≥ −d, Y ≤ d and X ≥ Y − d].
To prove Theorem 1, we will need the following distributional results (from Lam and Ng (1990)):
(D1) 2(m − 1)Si /θi ; i = 1, . . . , k follows a chi-square distribution with 2m − 2 df.
(D2) For fixed ni ≥ m, ni (X̃i,ni − µi )/θi is obtained as standard exponential distribution.
(D3) Ni (X̃i,Ni − µi )/θi = Wi is obtained as standard exponential distribution.
(D4) Si /θi and Ni (X̃i,Ni − µi )/θi are stochastically independent.
(D5) Ni (X̃i,Ni − µi )/Si = Wi∗ is distributed as an F distribution with (2, 2m − 2) df.
Proof of Theorem 1. For (a) we have


P µi+1 − µi ≥ X̃i+1 − X̃i − cuk,m,α , i = 1, . . . , k − 1


= P X̃i − µi ≥ X̃i+1 − µi+1 − cuk,m,α , i = 1, . . . , k − 1


Si+1 ∗
Si ∗
Wi ≥
Wi+1 − cuk,m,α , i = 1, . . . , k − 1
=P
Ni
Ni+1




Si Si+1
Si ∗
Si+1 ∗
≥P
Wi ≥
Wi+1 − max
,
uk,m,α , i = 1, . . . , k − 1 .
Ni
Ni+1
Ni Ni+1
The above inequality holds because Ni ≥ Si /c for all i = 1, . . . , k. Therefore we have
c ≥ max(Si /Ni , Si+1 /Ni+1 )
∞
∫
∫
∞
···
=

P
s k =0
s1 =0
si
Ni
∗
Wi ≥
s i +1
Ni+1
Wi∗+1

− max
si
,
s i +1
Ni Ni+1


uk,m,α , i = 1, . . . , k − 1 g (s1 , . . . , sk )ds1 . . . dsk ,
where g (·) is a joint probability density function (pdf) of S1 , . . . , Sk . Letting a =
si
Ni
,b =
si+1
Ni+1
, X = Wi∗ , Y = Wi∗+1 and
applying Lemma 2, we have
∫
∞
≥
∫
∞
P (Wi∗ ≥ −uk,m,α , Wi∗+1 ≤ uk,m,α , Wi∗ ≥ Wi∗+1 − uk,m,α , i = 1, . . . , k − 1)g (s1 , . . . , sk )ds1 . . . dsk
···
s1 =0
sk =0
= P (Wi∗+1 ≤ uk,m,α , i = 1, . . . , k − 1)
= P (F2,2m−2 ≤ uk,m,α )(k−1) = 1 − α,
where Wi∗+1 = F2,2m−2 is a random variable having a F distribution with (2, 2m − 2)df from (D5). So we have uk,m,α =
1
1/(k−1)
F2−,2m
). The proof is thus obtained.
−2 ((1 − α)
For (b) we have,
P (X̃i+1 − X̃i − c vk,m,α ≤ µi+1 − µi ≤ X̃i+1 − X̃i + c vk,m,α , i = 1, . . . , k − 1)



µi+1 − µi ≤ X̃i+1 − X̃i + c vk,m,α , i = 1, . . . , k − 1
= P X̃i+1 − X̃i − c vk,m,α ≤ µi+1 − µi


 Si+1
Si+1 ∗
Si ∗
Si ∗
Wi ≥
Wi+1 − c vk,m,α
Wi∗+1 ≥
Wi − c vk,m,α , i = 1, . . . , k − 1
=P
Ni
Ni+1
Ni+1
Ni


∫ ∞
∫ ∞ 
si ∗
si+1 ∗
si si+1
≥
···
P
Wi ≥
Wi+1 − max
,
vk,m,α
s1 =0
 s i +1
Ni+1
sk =0
Wi∗+1 ≥
Ni
si
Ni
Ni+1
Wi∗ − max

Ni Ni+1
si
,
si+1
Ni Ni+1


vk,m,α , i = 1, . . . , k − 1 g (s1 , . . . , sk )ds1 . . . dsk .
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V. Maurya et al. / Statistics and Probability Letters 81 (2011) 1507–1517
V. Maurya et al. / Statistics and Probability Letters 81 (2011) 1507–1517
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1510
Using Lemma 2, we have,
∫
∞
≥
∞
∫
P Wi∗ ≥ −vk,m,α , Wi∗+1 ≤ vk,m,α , Wi∗ ≥ Wi∗+1 − vk,m,α

···
sk =0
s1 =0


≥ −vk,m,α , Wi∗ ≤ vk,m,α , Wi∗+1 ≥ Wi∗ − vk,m,α , i = 1, . . . , k − 1 g (s1 , . . . , sk )ds1 . . . dsk


≤ vk,m,α ,
Wi∗ ≤ vk,m,α , i = 1, . . . , k − 1
Wi∗+1

= P Wi∗+1
= P (F2,2m−2 ≤ vk,m,α )k = 1 − α.
The proof is thus obtained.
Remark 1. When the unequal scale parameters are known, the unbiased estimator Si of θi is replaced by θi throughout
Theorem 1 and the statistic Wi∗ which is distributed as a F distribution with (2, 2m − 2)df is replaced by the statistic
Wi which is distributed as a standard exponential distribution from (D3). Therefore, the approximate critical values are
uk,m,α = − ln(1 − (1 − α)1/(k−1) ) and vk,m,α = − ln(1 − (1 − α)1/k ) when the scale parameters are known.
3. Two-stage one-sided and two-sided simultaneous confidence intervals for the successive differences µi+1 − µi , i =
1, . . . , k − 1 using Bonferroni inequality
In this section, the Bonferroni inequality is used to obtain the approximate critical values for the two-stage procedure
for one-sided and two-sided comparisons of the successive differences of the location parameters under heteroscedasticity.
The Bonferroni inequality is described as the following lemma:
Lemma 3 (Bonferroni Inequality, See Halperin et al. (1955)). Let A1 , . . . , Ak be any k events, then
1−
k
−

P (Āi ) ≤ P
k


≤1−
Ai
i=1
i=1
k
−
P (Āi ) +
i =1
k−1 −
k
−
P (Āi Āj ),
i=1 j=i+1
where Āi represents the complement set of Ai .
We now propose the one-sided and two-sided confidence intervals for µi+1 − µi , i = 1, . . . , k − 1, in the following
theorem using the two-stage procedure.
Theorem 4. For a given 0 < α < 1,
1
(a) P (µi+1 − µi ≥ X̃i+1 − X̃i − cu∗k,m,α , i = 1, . . . , k − 1) ≥ 1 − α if u∗k,m,α = F2−,2m
−2 ((k − 1 − α)/(k − 1)). Thus
∗
(X̃i+1 − X̃i − cuk,m,α , ∞) it is a set of lower one-sided confidence intervals for µi+1 − µi with confidence coefficient 1 − α .
1
(b) P (X̃i+1 −X̃i −c vk∗,m,α ≤ µi+1 −µi ≤ X̃i+1 −X̃i +c vk∗,m,α , i = 1, . . . , k−1) ≥ 1−α if vk∗,m,α = F2−,2m
−2 ((2k − 2 −α)/(2k − 2)).
∗
∗
Thus (X̃i+1 − X̃i − c vk,m,α , X̃i+1 − X̃i + c vk,m,α ) is a set of simultaneous two-sided confidence intervals for µi+1 − µi with
confidence coefficient 1 − α .
Proof of Theorem 4. For (a), we let event Ai = (µi+1 − µi ≥ X̃i+1 − X̃i − cu∗k,m,α ); i = 1, . . . , k − 1.
P (Ai ) = P (µi+1 − µi ≥ X̃i+1 − X̃i − cu∗k,m,α )

=P
θi+1
Ni+1
Wi+1 ≤
θi
Ni
Wi + cu∗k,m,α


cu∗k,m,α
θi /Ni
= P Wi+1 ≤
Wi +
θi+1 /Ni+1
θi+1 /Ni+1


∫ ∞ 
cu∗k,m,α
θi /Ni
wi −
e−wi dwi
=
1 − exp −
θi+1 /Ni+1
θi+1 /Ni+1
w i =0

 −1



cu∗k,m,α
θi /Ni
exp −
,
= 1− 1+
θi+1 /Ni+1
θi+1 /Ni+1

the last equality holds by the use of the moment generating function of a standard exponential distribution.

≥

1 − exp −
cu∗k,m,α
θi+1 /Ni+1




u∗k,m,α
≥ 1 − exp −
θi+1 /Si+1
(3.1)
=

1 − exp −

= P χ22 ≤
1511

u∗k,m,α
2
((2m − 2)/χ2m
−2 )

2u∗k,m,α
2
((2m − 2)/χ2m
−2 )
,
2
where χ22 and χ2m
−2 are the random variables having chi-square distribution with 2 and 2m − 2df respectively.

=P
χ22 /2
≤ u∗k,m,α
2
χ2m
−2 /(2m − 2)

= P (F2,2m−2 ≤ u∗k,m,α ).
(3.2)
By Lemma 3 and (3.2) we have
P (µi+1 − µi ≥ X̃i+1 − X̃i − cu∗k,m,α , i = 1, . . . , k − 1) ≥ 1 − (k − 1)(1 − P (F2,2m−2 ≤ u∗k,m,α )) = 1 − α.
1
Therefore, we have u∗k,m,α = F2−,2m
−2 ((k − 1 − α)/(k − 1)).
For (b), we let event Ai = (X̃i+1 − X̃i − c vk∗,m,α ≤ µi+1 − µi ≤ X̃i+1 − X̃i + c vk∗,m,α ), i = 1, . . . , k − 1.
Then,
P (Ai ) = P (X̃i+1 − X̃i − c vk∗,m,α ≤ µi+1 − µi ≤ X̃i+1 − X̃i + c vk∗,m,α )

=P
θi+1
Ni+1

=P
θi
Ni
θi
Wi+1 − c vk∗,m,α ≤
Wi ≤
θi+1
Ni+1
Ni
Wi ≤
Wi+1 + c vk∗,m,α
θi+1
Ni+1

Wi+1 + c vk∗,m,α

−P
θi
Ni
Wi ≤
θi+1
Ni+1

Wi+1 − c vk∗,m,α

θi

θi+1
+P
Wi ≥
Wi+1 − c vk∗,m,α − 1
Ni
Ni+1
Ni
Ni+1




θi+1 /Ni+1
θi /Ni
∗
∗
= P W1 ≤
Wi+1 + c vk,m,α + P Wi+1 ≤
Wi + c vk,m,α − 1.
θi /Ni
θi+1 /Ni+1

=P
Wi ≤
θi+1
θi
Wi+1 + c vk∗,m,α


Since both Wi and Wi+1 are standard exponentially distributed random variable, therefore from (3.1) and (3.2) we have
≥ P (F2,2m−2 ≤ vk∗,m,α ) + P (F2,2m−2 ≤ vk∗,m,α ) − 1
= 2P (F2,2m−2 ≤ vk∗,m,α ) − 1.
By Lemma 3, we have
P (X̃i+1 − X̃i − c vk∗,m,α ≤ µi+1 − µi ≤ X̃i+1 − X̃i + c vk∗,m,α , i = 1, . . . , k − 1)
≥ 1 − 2(k − 1)(1 − P (F2,2m−2 ≤ u∗k,m,α )) = 1 − α.
1
Therefore, we have u∗k,m,α = F2−,2m
−2 ((2k − 2 − α)/(2k − 2)). The proof is thus obtained.
To compare the approximate critical values using Bonferroni inequality and the approximate critical values using the
recent techniques given in Lam (1987, 1988), it is found that (k − 1 − α)/(k − 1) > (1 − α)1/(k−1) for all 0 < α < 1 and
k ≥ 3. So we have u∗k,m,α > uk,m,α . Also, it is found that (2k − 2 − α)/(2k − 2) > (1 − α)1/(k) for all 0 < α < 1 and k ≥ 3. So
we have vk∗,m,α > vk,m,α . The values of the critical constants uk,m,α ,vk,m,α (using techniques given in Lam (1987, 1988)) and
u∗k,m,α , vk∗,m,α (using Bonferroni inequality) are presented in Table 1 for selected configurations of k, m and α (mentioned in
the Table 1). It can also be observed from the Table 1 that u∗k,m,α > uk,m,α and vk∗,m,α > vk,m,α . Therefore, the approximate
critical values in Theorem 1 are recommended for the implementation of two-stage procedure for one-sided and two-sided
comparisons of the successive differences of the location parameters under heteroscedasticity.
4. One-stage multiple comparison procedure
When the additional sample for the second stage may not be available due to the experimental budget shortage or
other factors in an experiment, the two-stage procedure proposed in Section 2 cannot be used when scale parameters are
unknown and possibly unequal. Therefore, we proposed one-stage procedure for one-sided and two-sided comparisons of
the successive differences of the location parameters under heteroscedasticity as follows:
Take one-stage sample Xi1 , . . . , Xim of size m(≥ 2) from πi . Let Xi = min(Xi1 , . . . , Xim ) be a suitable estimator of µi based
on a sample of size m for i = 1, . . . , k.
Define
d = max (Si /m),
1≤i≤k
(4.1)
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V. Maurya et al. / Statistics and Probability Letters 81 (2011) 1507–1517

V. Maurya et al. / Statistics and Probability Letters 81 (2011) 1507–1517
Table 1
Approximate critical values of uk,m,α , vk,m,α , u∗k,m,α and vk∗,m,α .
K
m
α = 0.05
uk,m,α
α = 0.025
vk,m,α
u•k,m,α
vk•,m,α
uk,m,α
α = 0.01
vk,m,α
u•k,m,α
vk•,m,α
uk,m,α
vk,m,α
u•k,m,α
vk•,m,α
3
2
3
4
5
6
7
8
9
10
15
20
25
30
35
38.49
10.56
7.216
6.027
5.430
5.072
4.835
4.666
4.540
4.204
4.056
3.973
3.919
3.882
57.989
13.361
8.678
7.085
6.301
5.838
5.533
5.318
5.158
4.733
4.548
4.444
4.378
4.332
39.000
10.649
7.260
6.059
5.456
5.096
4.857
4.687
4.560
4.221
4.071
3.987
3.934
3.896
79.000
15.889
9.927
7.963
7.011
6.455
6.091
5.835
5.645
5.145
4.929
4.808
4.730
4.677
78.497
15.832
9.899
7.944
6.996
6.442
6.079
5.824
5.635
5.137
4.921
4.800
4.723
4.670
117.994
19.817
11.756
9.211
8.004
7.307
6.855
6.539
6.306
5.696
5.434
5.288
5.195
5.131
79.000
15.889
9.927
7.963
7.011
6.455
6.091
5.835
5.645
5.145
4.929
4.808
4.730
4.677
159.000
23.298
13.287
10.226
8.797
7.980
7.453
7.087
6.818
6.117
5.818
5.652
5.546
5.474
198.499
26.249
14.529
11.033
9.420
8.504
7.916
7.509
7.210
6.437
6.107
5.926
5.810
5.731
297.998
32.583
17.061
12.633
10.635
9.515
8.804
8.314
7.956
7.036
6.648
6.434
6.299
6.206
199.000
26.284
14.544
11.042
9.427
8.510
7.922
7.514
7.215
6.440
6.111
5.929
5.813
5.733
399.000
38.000
19.104
13.889
11.572
10.287
9.475
8.918
8.513
7.478
7.044
6.806
6.655
6.552
4
2
3
4
5
6
7
8
9
10
15
20
25
30
35
57.98
13.36
8.678
7.085
6.301
5.838
5.533
5.318
5.158
4.733
4.548
4.444
4.378
4.332
77.484
15.718
9.844
7.906
6.965
6.415
6.055
5.802
5.614
5.119
4.904
4.785
4.708
4.655
59.000
13.492
8.745
7.133
6.340
5.872
5.564
5.346
5.185
4.756
4.569
4.464
4.397
4.351
119.000
19.909
11.797
9.239
8.026
7.325
6.872
6.554
6.320
5.708
5.445
5.298
5.205
5.141
117.994
19.817
11.756
9.211
8.004
7.307
6.855
6.539
6.306
5.696
5.434
5.288
5.195
5.131
157.492
23.179
13.235
10.193
8.771
7.958
7.434
7.069
6.801
6.103
5.805
5.640
5.535
5.463
119.000
19.909
11.797
9.239
8.026
7.325
6.872
6.554
6.320
5.708
5.445
5.298
5.205
5.141
239.000
28.984
15.643
11.744
9.963
8.957
8.315
7.871
7.547
6.708
6.353
6.157
6.033
5.947
297.998
32.583
17.061
12.633
10.635
9.515
8.804
8.314
7.956
7.036
6.648
6.434
6.299
6.206
397.497
37.925
19.076
13.872
11.560
10.276
9.466
8.910
8.506
7.472
7.039
6.801
6.651
6.547
299.000
32.641
17.083
12.647
10.646
9.524
8.811
8.320
7.962
7.041
6.652
6.439
6.303
6.210
599.000
46.990
22.303
15.797
12.972
11.425
10.457
9.797
9.320
8.109
7.606
7.331
7.157
7.038
5
2
3
4
5
6
7
8
9
10
15
20
25
30
35
77.48
15.71
9.844
7.906
6.965
6.415
6.055
5.802
5.614
5.119
4.904
4.785
4.708
4.655
96.979
17.797
10.830
8.585
7.508
6.883
6.476
6.190
5.979
5.425
5.185
5.052
4.967
4.908
79.000
15.889
9.927
7.963
7.011
6.455
6.091
5.835
5.645
5.145
4.929
4.808
4.730
4.677
159.000
23.298
13.287
10.226
8.797
7.980
7.453
7.087
6.818
6.117
5.818
5.652
5.546
5.474
157.492
23.179
13.235
10.193
8.771
7.958
7.434
7.069
6.801
6.103
5.805
5.640
5.535
5.463
196.990
26.142
14.485
11.004
9.398
8.485
7.900
7.494
7.197
6.426
6.097
5.916
5.801
5.722
159.000
23.298
13.287
10.226
8.797
7.980
7.453
7.087
6.818
6.117
5.818
5.652
5.546
5.474
319.000
33.777
17.520
12.918
10.849
9.692
8.958
8.453
8.084
7.138
6.740
6.521
6.382
6.287
397.497
37.925
19.076
13.872
11.560
10.276
9.466
8.910
8.506
7.472
7.039
6.801
6.651
6.547
496.996
42.632
20.779
14.896
12.315
10.892
9.999
9.388
8.945
7.817
7.346
7.088
6.926
6.814
399.000
38.000
19.104
13.889
11.572
10.287
9.475
8.918
8.513
7.478
7.044
6.806
6.655
6.552
799.000
54.569
24.850
17.273
14.037
12.281
11.190
10.449
9.915
8.568
8.011
7.708
7.518
7.387
6
2
3
4
5
6
7
8
9
10
15
20
25
30
35
96.97
17.79
10.83
8.585
7.508
6.883
6.476
6.190
5.979
5.425
5.185
5.052
4.967
4.908
116.47
19.677
11.693
9.169
7.971
7.278
6.829
6.516
6.284
5.678
5.417
5.272
5.180
5.116
99.000
18.000
10.925
8.649
7.559
6.927
6.515
6.226
6.013
5.453
5.211
5.077
4.991
4.932
199.000
26.284
14.544
11.042
9.427
8.510
7.922
7.514
7.215
6.440
6.111
5.929
5.813
5.733
196.990
26.142
14.485
11.004
9.398
8.485
7.900
7.494
7.197
6.426
6.097
5.916
5.801
5.722
236.488
28.821
15.578
11.703
9.931
8.931
8.292
7.851
7.527
6.693
6.339
6.144
6.020
5.935
199.000
26.284
14.544
11.042
9.427
8.510
7.922
7.514
7.215
6.440
6.111
5.929
5.813
5.733
399.000
38.000
19.104
13.889
11.572
10.287
9.475
8.918
8.513
7.478
7.044
6.806
6.655
6.552
496.996
42.632
20.779
14.896
12.315
10.892
9.999
9.388
8.945
7.817
7.346
7.088
6.926
6.814
596.495
46.887
22.268
15.776
12.957
11.413
10.447
9.788
9.312
8.102
7.600
7.325
7.152
7.033
499.000
42.721
20.811
14.915
12.329
10.904
10.008
9.396
8.953
7.823
7.351
7.093
6.931
6.819
999.000
61.246
27.000
18.494
14.905
12.974
11.779
10.971
10.390
8.931
8.331
8.005
7.800
7.660
where Si =
j=1 (Xij − Xi )/(m − 1). We now propose one-sided and two-sided confidence intervals for µi+1 −µi , i = 1, . . . , k
in the following theorem using the one-stage procedure.
∑m
Theorem 5. For a given 0 < α < 1,
1
1/(k−1)
(a) P (µi+1 −µi ≥ Xi+1 − Xi − dqk,m,α , i = 1, . . . k − 1) ≥ 1 −α if qk,m,α = F2−,2m
). Thus (Xi+1 − Xi − dqk,m,α , ∞)
−2 ((1 −α)
is a set of lower one-sided confidence intervals for µi+1 − µi with confidence coefficient 1 − α .
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1512
1513
1
1/k
(b) P (Xi+1 − Xi − drk,m,α ≤ µi+1 − µi ≤ Xi+1 − Xi + drk,m,α , i = 1, . . . k − 1) ≥ 1 − α if rk,m,α = F2−,2m
−2 ((1 − α) ).
Thus (Xi+1 − Xi − drk,m,α , Xi+1 − Xi + drk,m,α ) is a set of simultaneous two-sided confidence intervals for µi+1 − µi with
confidence coefficient 1 − α .
To prove Theorem 5, we will need the following distributional results (from Roussas (1997)):
(E1)
(E2)
(E3)
(E4)
2(m − 1)Si /θi ; i = 1, . . . , k follows a chi-square distribution with 2m − 2df .
m(Xi − µi )/θi = Ti are obtained as standard exponential distribution.
Si /θi and m(Xi − µi )/θi are stochastically independent.
m(Xi − µi )/Si = Ti∗ are distributed as an F distribution with (2, 2m − 2)df .
Proof of Theorem 5. Using the above results and Lemma 2, the proof of Theorem 5 is on the similar lines as of Theorem 1
by replacing c with d. Using Lemma 1 of Liu et al. (2000), the simultaneous one-sided and two-sided confidence intervals given in Theorem 5
∑k
using one-stage procedure, can be extended to a larger set of contrasts { i=1 li µi : (l1 , . . . , lk ) ∈ Ψk }, where Ψk =
∑
∑
{(l1 , . . . , lk ) ∈ ℜk : kj=1 lj = 0 and ij=1 lj ≤ 0, 1 ≤ i ≤ k − 1}.
∑i
∑k
∑k
∑k−1
Let ai = − j=1 lj =
j=i+1 lj , 1 ≤ i ≤ k − 1, where (l1 , . . . , lk ) ∈ Ψk . Then
i=1 li µi =
i=1 ai (µi+1 − µi ) and each
ai ≥ 0, 1 ≤ i ≤ k − 1. Now, from part (a) of Theorem 5 we obtain
1 − α ≤ P (ai (µi+1 − µi ) ≥ ai (Xi+1 − Xi − dqk,m,α ), i = 1, . . . k − 1)

=P
k−1
−
ai (µi+1 − µi ) ≥
i =1

=P
k
−
i =1
k−1
−
ai (Xi+1 − Xi ) − dqk,m,α
i=1
l i µi ≥
k
−
li Xi − dqk,m,α
k−1
−

ai
i=1
k−1 −
k
−

lj
.
i=1 j=i+1
i =1
Therefore, one-stage one-sided simultaneous confidence intervals for all contrasts with overall confidence level of 1 − α is
k
−

l i µi ∈
i =1
k
−

k−1
−
li Xi − dqk,m,α
(li+1 + · · · + lk ), ∞ ,
i=1
for all (l1 , . . . , lk ) ∈ Ψk .
i =1
Similarly, using Lemma 2 given in Liu et al. (2000), our one-stage two-sided simultaneous confidence intervals given in
part(b) of Theorem 5 can be extended to a larger set of contrasts. The simultaneous two-sided confidence intervals for all
contrasts with overall confidence level of 1 − α using one-stage procedure is
k
−

l i µi ∈
i =1
k
−
li Xi − drk,m,α
i=1
where Ψk∗ = {(l1 , . . . , lk ) ∈ ℜk :
k−1
−
|li+1 + · · · + lk | ,
i=1
i=1
∑k
j =1 l j
k
−
li Xi + drk,m,α
k−1
−

|li+1 + · · · + lk | ,
for all (l1 , . . . , lk ) ∈ Ψk∗ ,
i =1
= 0}.
Similarly, the simultaneous two-stage one-sided and two-sided confidence intervals given in Theorem 1 can also be
extended to a larger set of contrasts.
Remark 2. (a) The set of one-sided simultaneous confidence intervals given in part (a) of Theorems 1 and 5 are not based
on the assumption that µi ’s follow simple ordering. However, if there is prior information about the ordering µi ’s i.e.
of location parameters, then this information may be used to improve the one-sided confidence intervals. A one-sided
confidence interval with a lower limit less than zero will be non-informative and may be truncated at zero.
(b) When the unequal scale parameters are known, the approximate critical values for one-stage procedure are qk,m,α =
− ln(1 − (1 − α)1/(k−1) ) and rk,m,α = − ln(1 − (1 − α)1/k ).
(c) Since uk,m,α = qk,m,α and vk,m,α = rk,m,α , the one-stage procedure for one-sided and two-sided comparisons of the
successive differences of the location parameters using Lam (1987, 1988) is expected to perform better than using the
Bonferroni inequality.
5. Example
The data set given in Hill et al. (1988), representing the survival days of patients with inoperable lung cancer who were
subjected to a standard chemotherapeutic agent, is used to illustrate the two-stage and one-stage procedure proposed in
Sections 2 and 3 respectively. The patients are divided into the following four categories depending on the histological
type of their tumour: squamous, small, adeno, and large. The data are a part of a larger data set collected by the Veterans
Administrative Lung Cancer Study Group in the USA. Since the data given in Hill et al. (1988) have unequal sample sizes, an
initial random sample of size m = 9 survival times was taken from each group in the first stage. The data are given in the
Table 2.
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V. Maurya et al. / Statistics and Probability Letters 81 (2011) 1507–1517
V. Maurya et al. / Statistics and Probability Letters 81 (2011) 1507–1517
www.amarestan.com
1514
Table 2
Survival days of patients.
Squamous
Small
72
10
81
110
100
42
8
25
11
30
13
23
16
21
18
20
27
31
Adeno
Large
8
92
35
117
132
12
162
3
95
177
162
553
200
156
182
143
105
103
Table 3
The required statistics and critical values.
Statistics
Squamous
Small
Adeno
Large
Xi
Si
d
8
48.375
11.862
13
10.250
3
78.265
103
106.750
α
uk,m,α (= qk,m,α )
vk,m,α (= rk,m,α )
0.050
0.025
0.010
5.318
6.539
8.314
5.802
7.069
8.910
As discussed in Hill et al. (1988), the data in the four categories may be assumed to be random samples from the
distributions F (x − µ1 /θ1 ), F (x − µ2 /θ2 ), F (x − µ3 /θ3 ) and F (x − µ4 /θ4 ), respectively, where the location parameters
follow the monotonic trend given by µ1 ≤ µ2 ≤ µ3 ≤ µ4 .
To test the validity of the two-parameter exponential model, we use the approach given by Hsieh (1986) (see also Lawless
(1982)). Let Xi(j) be the jth order statistic of the ith sample, i = 1, . . . , k; j = 1, . . . , m. We define
Zij = Xi(j+1) − Xi(1) = Xi(j+1) − Xi ,
i = 1, . . . , k; j = 1, . . . , m − 1.
If Xi(j) , j = 1, . . . , m, are the order statistics of a random sample of size m obtained from a two-parameter exponential
distribution with location parameter µi and scale parameter θi , then {Zij : j = 1, . . . , m − 1} can be regarded as
order statistics of a random sample of size m − 1 from a one-parameter exponential distribution with scale parameter
θi , i = 1, . . . , k. Exponentiality of each sample is then tested using the scale-free test of Gail and Gastwirth (1978). The
Gail–Gastwirth test statistic is
m−2
∑
Gi =
{j(m − j − 1)(Zi,j+1 − Zij )}
j =1
(m − 2)
,
m−1
∑
i = 1, . . . , k.
Zij
j =1
For the four-group data set given in Table 2, G1 = 0.496, G2 = 0.328, G3 = 0.438 and G4 = 0.588. The critical value for
rejecting the exponentiality at the 5% level, obtained from Table 2 of Gail and Gastwirth (1978) with m = 9, is Gi < 0.301
and Gi > 0.699, i = 1, . . . , 4. The hypothesis of exponentiality is not rejected at the 5% level of significance for all the four
samples since none of the computed Gail–Gastwirth statistic values fall in the critical region.
Assuming two-parameter exponential distributions, we test for the equality of the scale parameters using Bartlett’s
approximate test as shown in Hsieh (1986). The test statistic is

Λ=
2k(m − 1) ln(S /2) − 2(m − 1)
k
−
i=1
where Si =
∑m
j =1
(Xij − Xi )/(m − 1) =
∑m−1
j=1

ln(Si /2)
1+
k+1
−1
6k(m − 1)
Zij /(m − 1), i = 1, . . . , k and S =
,
∑k
i=1
Si /k. An approximate α level test rejects
the hypothesis of homogeneity of the scale parameters if Λ > χα2 (k − 1). For the data given in Table 2, we obtain Λ = 18.745
and χ02.05 (3) = 7.815. Therefore, the assumption of equal scale parameters is rejected at the 5% level of significance.
Thus the design-oriented two-stage procedures and data-analysis one-stage procedures using Lam’s (1987, 1988)
technique proposed in this article can be applied for testing the significance of successive differences of the location
parameters of the four populations. The required statistics and critical values of uk,m,α (= qk,m,α ) and vk,m,α (= rk,m,α )for
α = 5%, 2.5% and 1% are summarized in Table 3.
Let L = 137.646, 167.704 and 211.380 be the required lengths specified by the user for the two-stage two-sided
confidence intervals given in part (b) of Theorem 1 for confidence coefficient 1 − α = 0.95, 0.975 and 0.99 respectively.
1515
Table 4
The two-stage (and one-stage) one-sided lower confidence intervals for successive differences of the location parameters.
Parameters
(X̃i+1 − X̃i − cuk,m,α , ∞)
1 − α = 0.95
1 − α = 0.975
1 − α = 0.99
µ2 − µ1
µ3 − µ2
µ4 − µ3
(−58.082, ∞)
(−73.082, ∞)
(36.918, ∞)
(−72.566, ∞)
(−87.566, ∞)
(22.434, ∞)
(−93.620, ∞)
(−108.620, ∞)
(1.379, ∞)
Table 5
The two-stage (and one-stage) two-sided confidence intervals for successive differences of the location parameters.
Parameters
µ2 − µ1
µ3 − µ2
µ4 − µ3
(X̃i+1 − X̃i − c vk,m,α , X̃i+1 − X̃i + c vk,m,α )
1 − α = 0.95
1 − α = 0.975
1 − α = 0.99
(−63.823, 73.823)
(−78.823, 58.823)
(31.177, 168.823)
(−78.852, 88.852)
(−93.852, 73.852)
(16.148, 183.852)
(−100.690, 110.690)
(−115.690, 95.690)
(−5.690, 205.690)
Table 6
The two-stage one-sided lower confidence intervals for successive differences of the location parameters.
Parameters
(X̃i+1 − X̃i − cuk,m,α , ∞)
1 − α = 0.95
1 − α = 0.975
1 − α = 0.99
µ2 − µ1
µ3 − µ2
µ4 − µ3
(−59.880, ∞)
(−74.880, ∞)
(35.120, ∞)
(−74.776, ∞)
(−89.775, ∞)
(20.224, ∞)
(−96.431, ∞)
(−111.431, ∞)
(−1.431, ∞)
Table 7
The two-stage two-sided confidence intervals for successive differences of the location parameters.
Parameters
(X̃i+1 − X̃i − c vk,m,α , X̃i+1 − X̃i + c vk,m,α )
1 − α = 0.95
1 − α = 0.975
1 − α = 0.99
µ2 − µ1
µ3 − µ2
µ4 − µ3
(−65.784, 75.784)
(−80.784, 60.784)
(29.216, 170.784)
(−81.242, 91.242)
(−96.242, 76.242)
(13.758, 186.242)
(−103.702, 113.702)
(−118.702, 98.702)
(−8.702, 208.702)
Then we have c = L/(2uk,m,α ) = 11.862. We use c to determine the total sample size for two-stage procedure and end up
with the equal total sample size (N1 , N2 , N3 , N4 ) = (9, 9, 9, 9) by using (2.1). From part (a) of Theorem 1, we can obtain the
two-stage one-sided lower confidence bounds with confidence coefficients 0.95, 0.975 and 0.99 given in Table 4. Since all
the lower confidence bounds for µ4 − µ3 are greater than zero, we can conclude that µ4 > µ3 with confidence coefficients
0.95, 0.975 and 0.99. Using part (b) of Theorem 1, the two-stage two-sided confidence bounds with confidence coefficients
0.95, 0.975 and 0.99 are given in Table 5. In comparison with the one-sided intervals it is noticed that while upper bounds
on the successive differences are obtained, the inference that µ4 > µ3 is no longer valid at confidence coefficient 0.99.
However, since the two-sided simultaneous confidence intervals for µ4 − µ3 do not contain zero, we can conclude that
µ4 > µ3 with confidence coefficients 0.95 and 0.975.
For the one-stage procedure, we use the same total sample size m = 9 for each group and end up with the value
d = 11.862 which is the same as the value of c for two-stage intervals. Based on the same sample values given in Table 2,
the one-stage one-sided and two-sided confidence bounds with confidence coefficients 0.95, 0.975 and 0.99 are identical
to the two-stage procedures and they are given in Tables 4 and 5. Let L∗ be the length of one-stage two-sided confidence
intervals. Hence it can concluded from the above results that if c = d then we have L = L∗ and the two-stage procedure and
one-stage procedure have the same overall sample size, except for a rounding error in sample size definitions (2.1).
On the other hand, if the user takes the length ratio for the two-stage procedure to the one-stage procedure as L/L∗ =
1.0285. Then we have L = 141.568, 172.484 and 217.404 for confidence coefficients 0.95, 0.975 and 0.99 respectively and
c = L/(2uk,m,α ) = 12.20. By the use of Eq. (2.1) the required overall sample size for two-stage procedure can be determined
as (N1 , N2 , N3 , N4 ) = (9, 9, 9, 9). Based on the same sample values under the same sample sizes, the two-stage one-sided
and two-sided confidence bounds with confidence coefficients 0.95, 0.975 and 0.99 are presented in Tables 6 and 7.
Comparing Tables 5 and 7, we have same conclusion for two-stage and one-stage procedure. Further, comparing Tables 4
and 6 we have almost same conclusion for two-stage and one-stage procedure except that the inference µ4 > µ3 cannot be
made for the confidence coefficient 0.99 as the lower limit of the one-sided confidence interval is less than zero. Hence, it
can be concluded that if c > d then we have L > L∗ and there is no need to draw the second stage sample for the two-stage
procedure. Under the same total sample size m, the one-stage procedure has shorter confidence length than the two-stage
procedure. Hence, the one-stage procedure is recommended. Also if min(S1 /m, . . . , Sk /m) > c then we have L∗ > L and
the overall sample size of the one-stage procedure is smaller than that of two-stage procedure. All other situations, the
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V. Maurya et al. / Statistics and Probability Letters 81 (2011) 1507–1517
V. Maurya et al. / Statistics and Probability Letters 81 (2011) 1507–1517
Table 8
The coverage rates of lower and two-sided confidence intervals under structure of scale parameters (θ1 , θ2 , θ3 , θ4 ) = (1.0, 1.0, 1.0, 1.0) for 1 − α = 0.95.
m
L
Lower
Two-sided
Ratio
One-stage
Two-stage
One-stage
Two-stage
10
1.5
1.0
0.5
0.1
0.998
0.998
0.995
0.988
0.999
0.991
0.974
0.970
0.997
0.996
0.992
0.982
0.999
0.991
0.972
0.967
1.035
1.242
2.300
11.283
20
1.5
1.0
0.5
0.1
0.995
0.995
0.995
0.989
1.000
1.000
0.988
0.973
0.992
0.992
0.992
0.982
1.000
1.000
0.987
0.969
1.000
1.000
1.092
4.931
30
1.5
1.0
0.5
0.1
0.993
0.994
0.994
0.989
1.000
1.000
0.999
0.973
0.989
0.990
0.990
0.983
1.000
1.000
0.999
0.969
1.000
1.000
1.000
3.157
Table 9
The coverage rates of lower and two-sided confidence intervals under structure of scale parameters (θ1 , θ2 , θ3 , θ4 ) = (1.0, 1.2, 1.3, 1.4) for 1 − α = 0.95.
m
L
Lower
Two-sided
Ratio
One-stage
Two-stage
One-stage
Two-stage
10
1.5
1.0
0.5
0.1
0.997
0.995
0.992
0.988
0.994
0.984
0.972
0.969
0.996
0.994
0.991
0.985
0.996
0.985
0.969
0.965
1.114
1.461
2.799
13.808
20
1.5
1.0
0.5
0.1
0.994
0.994
0.993
0.988
1.000
0.999
0.980
0.973
0.993
0.993
0.991
0.985
1.000
0.999
0.980
0.968
1.000
1.001
1.263
6.033
30
1.5
1.0
0.5
0.1
0.992
0.992
0.991
0.988
1.000
1.000
0.993
0.973
0.990
0.990
0.990
0.985
1.000
1.000
0.994
0.970
1.000
1.000
1.011
3.864
one-stage procedure could be better than, worse than, or not much different from that of the two-stage procedure depending
on the actual sample data and the true scale parameters.
6. Simulation study
A simulation study of the proposed one-sided and two-sided confidence intervals for µi+1 − µi , i = 1, . . . , k − 1 given
in Theorems 1 and 5 is investigated based on 100,000 simulation runs in this section. For the given confidence lengths of
two-stage procedures L = 1.5, 1.0, 0.5, 0.1 we can obtained the value of c = L/(2uk,m,α ) and thus the required overall
sample size for two-stage procedure can be determined by the use of Eq. (2.1). For the plausibility of comparison of two
∑k
procedures, we take the sample size for each population in the one-stage procedure as [ i=1 Ni /k], where [x] stands for the
greatest integer less than and equal to x. The coverage rates of the proposed one-sided and two-sided confidence intervals
for 1 − α = 0.95, m = 10, 20, 30 and the given two-stage confidence length L = 1.5, 1.0, 0.5, 0.1 are listed in Tables 8
and 9 for various structures of scale parameters (θ1 , θ2 , θ3 , θ4 ) = (1.0, 1.0, 1.0, 1.0), (1.0, 1.2, 1.3, 1.4) respectively. The
sample ratios (denoted as ratio) are defined as the average of the ratios of the required total sample size for the twostage SCI over the total initial sample size after 100,000 simulation runs and they are listed in Tables 8 and 9 following
by the coverage rates. From the tables we can see that all simulated coverage rates are higher than the nominal confidence
coefficients.
It can also be seen that both procedures are quite conservative. For two-stage procedure, the required sample ratio is
larger for smaller m with fixed L under various structures of scale parameters. When the sample ratio approaches to 1, the
overall sample size for two-stage procedure is approximately equal to the initial sample size m and also equal to the sample
size for one-stage procedure for each population. Under this condition, the one-stage procedure has coverage rates closer
to the nominal confidence coefficients than the two-stage procedure. That is, the one-stage procedure is less conservative
than the two-stage procedure when the sample ratio is approaching to 1.
Acknowledgements
The authors are grateful to the editor’s and referees’ comments and suggestions which improved this article.
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