Comput Stat (2014) 29:1083–1094 DOI 10.1007/s00180-014-0481-6 ORIGINAL PAPER
Comparing exponential location parameters with several controls under heteroscedasticity A. Malekzadeh · M. Kharrati-Kopaei · S. M. Sadooghi-Alvandi
Received: 9 September 2013 / Accepted: 15 January 2014 / Published online: 2 February 2014 © Springer-Verlag Berlin Heidelberg 2014
Abstract Suppose that random samples are taken from k treatment groups and l control groups, where the observations in each group have a two-parameter exponential distribution. We consider the problem of constructing simultaneous confidence intervals for the differences between location parameters of the treatment groups and the control groups when the scale parameters may be unequal. Using the parametric bootstrap approach, we develop a new multiple comparisons procedure when the scale parameters and sample sizes are possibly unequal. We then present a simulation study in which we compare the performance of our proposed procedure with two other procedures. The results of our simulations indicate that our proposed procedure performs better than other procedures. The usefulness of our proposed procedure is illustrated with an example. Keywords Multiple comparisons · Simultaneous confidence intervals · Parametric bootstrap · Coverage probability · Monte Carlo method · Simulation Mathmetics Subject Classification
62F25 · 62F40
A. Malekzadeh · M. Kharrati-Kopaei (B) · S. M. Sadooghi-Alvandi Department of Statistics, Shiraz University, Shiraz, Iran e-mail:
[email protected];
[email protected] A. Malekzadeh e-mail:
[email protected];
[email protected] S. M. Sadooghi-Alvandi e-mail:
[email protected];
[email protected]
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1 Introduction Suppose that random samples are taken from k treatment groups and l control groups, where the observations in each group have a two-parameter exponential distribution. Multiple comparisons of several exponential distributions have applications in engineering, life testing, and medical sciences. In reliability analysis, the location parameter is known as the guaranteed mean life and the scale parameter is known as the mean life. In dose-response analysis, the location and scale parameters are known as the guaranteed and mean effective duration, respectively. For more detailed applications of the exponential distribution, see Johnson and Kotz (1970), Zelen (1966), Lawless and Singhal (1980), and Maurya et al. (2011a); Maurya et al. (2011b). We use the notation Exp (ξ, σ ) to denote the exponential distribution with location parameter ξ ∈ (−∞, +∞), scale parameter σ ∈ (0, +∞), and density function x −ξ 1 I[ξ,∞) (x) , f (x; ξ, σ ) = exp − σ σ where I A (.) is the indicator function of event A. Suppose that the observations in the ith treatment group have a two-parameter exponential distribution Exp ξiT , σiT and the observations in the jth control group have a two-parameter exponential distribution C C Exp ξ j , σ j . We consider the problem of constructing simultaneous confidence intervals (SCIs) for ξiT − ξ Cj , i = 1, . . ., k, j = 1, . . ., l, under heteroscedasticity, i.e., when the scale parameters may be unequal. Singh and Abebe (2009) proposed a multiple comparison procedure when the sample sizes and scale parameters are equal; see also Ng et al. (1993). A procedure for multiple comparisons with a control under heteroscedasticity was first proposed by Lam and Ng (1990). However, their method is based on a two-stage sampling procedure. For single-stage sampling, Wu et al. (2010) proposed multiple comparisons with a control under heteroscedasticity when the sample sizes are equal. Maurya et al. (2011a) extended these procedures to comparisons with more than one control. The procedures proposed by these authors are based on Lam’s (1987, 1988) technique. However, using this technique leads to SCIs that are too conservative; see Wu et al. (2010) and Maurya et al. (2011a). More recently, Karrati-Kopaei et al. (2013) proposed simultaneous fiducial generalized confidence intervals (SFGCIs) for the successive differences of the location parameters of several exponential distributions and indicated how their proposed procedure can be modified to obtain SFGCIs for comparisons of the location parameters with one or several controls. In this paper, we use the parametric bootstrap approach to obtain a new multiple comparisons procedure when the scale parameters and sample sizes are possibly unequal. The parametric bootstrap approach has been successfully used in other contexts, to solve inference problems (mainly testing problems) involving nuisance parameters; see, e.g. Krishnamoorthy et al. (2007), Ma and Tian (2009), Tian et al. (2009), Krishnamoorthy and Lu (2010), Li et al. (2011), Xu et al. (2012), and SadooghiAlvandi and Jafari (2013).
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The organization of this paper is as follows: In Sect. 2, we present parametric bootstrap SCI (PBSCIs) when the scale parameters and sample sizes are possibly unequal. In Sect. 3, we present a simulation study in which we compare the coverage probability of our proposed procedure with the procedures proposed by Maurya et al. (2011a) and Karrati-Kopaei et al. (2013). The results of our simulation study indicate that our proposed PBSCIs have coverage probabilities close to the nominal confidence coefficient and, consequently, the resulting confidence intervals are typically shorter. In Sect. 4, we illustrate our proposed procedure with an example. 2 Parametric bootstrap simultaneous confidence intervals sample of sizes m i from the ith treatLet X i = X i1 , . . . , Xim i denote a random ment group and Y j = Y j1 , . . . , Y jn j denote a random sample of sizes n j from the jth control group (i = 1, . . ., k, j = 1, . . ., l), and let ξ T = ξ1T , . . . , ξkT , ξ C = C ξ1 , . . . , ξlC , σ T = σ1T , . . . , σkT , and σ C = σ1C , . . . , σlC denote the vectors of the location and scale parameters of the treatment and control groups. Let X i(1) and Y j (1) denote the first order statistics of X i and Y j , respectively, and let
SX i
m
nj
t=1
t=1
i 1 1 = (X it − X i(1) ) and SY j = (Y jt − Y j (1) ) mi − 1 nj − 1
denote the unbiased estimators of σiT and σ jC , respectively. It is well-known that X i(1) ∼ Exp ξiT , σiT /m i and Y j(1) ∼ Exp ξ Cj , σ jC /n j ,
(2.1)
independently of SX i ∼
σiT χ2 (2m i − 2) 2m i −2
and
SY j ∼
σ jC 2n j − 2
2 χ2n , j −2
(2.2)
where χr2 denotes a chi-square distribution with r degrees of freedom. Preliminaries. Suppose that the values of σiT ’s and σ jC ’s were known. Let X C T i(1) − Y j (1) − ξi − ξ j , T∗ = max √ 1≤i≤k,1≤ j≤l γi j where
γi j = Var X i(1) − Y j (1) =
σiT mi
2 +
σ jC nj
(2.3)
2 .
(2.4)
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Then from (2.1), it follows T ∗ is a pivotal quantity, and 100(1-α) % two-sided SCIs for ξiT − ξ Cj are √ X i(1) − Y j (1) ± qα∗ γi j , i = 1, . . . , k;
j = 1, . . . , l,
(2.5)
where qα∗ denotes the (1 − α)th quantile of the distribution of T ∗ . (The approximate value of qα∗ may be obtained by Monte Carlo methods.) For the realistic case when the values of σiT ’s and σ jC ’s are unknown, we propose to replace their values with their estimates and base SCIs on the random quantity X C T i(1) − Y j (1) − ξi − ξ j , T = max 1≤i≤k,1≤ j≤l Vi j where Vi j =
(m i − 1)S X2 i m i3
+
(n j − 1)SY2 j
(2.6)
(2.7)
n 3j
is the minimum variance unbiased estimator of γi j . We propose to approximate the quantiles of T using a parametric bootstrap (PB) approach. As noted in the Introduction, the PB approach has been successfully used in various other contexts. Parametric bootstrap SCI (PBSCI). The PB distribution of T is obtained by finding its sampling distribution when the values of the nuisance parameters σiT and σ jC are replaced with the observed values of their estimates sxi and syj . It follows from (2.1) and (2.2) that
X i(1) − ξiT ∼ Exp 0, σiT /m i , Y j(1) − ξ Cj ∼ Exp 0, σ jC /n j ,
and that the PB pivotal variable corresponding to T is TB =
max
1≤i≤k,1≤ j≤l
B Vi Bj , X i(1) − Y jB(1)
(2.8)
where
Vi Bj =
2 (m i − 1) S XBi m i3
+
2 n j − 1 SYBj n 3j
,
with B ∼ sxi /m i Exp (0, 1) and Y jB(1) ∼ s y j /n j Exp(0, 1), X i(1) sy j sxi 2 χ2m χ2 SXBi ∼ and SYBj ∼ . i −2 (2n j − 2) 2n j −2 (2m i − 2)
123
(2.9)
Comparing exponential location parameters
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Then the distribution of T B provides the PB approximation to the distribution of T and 100(1 − α) % two-sided PBSCIs for ξiT − ξ Cj are X i(1) − Y j (1) ± qαB Vi j , i = 1, . . . , k;
j = 1, . . . , l
(2.10)
where qαB denotes the (1 − α)th quantile of the distribution of T B . Computational procedure. The value of qαB can be estimated by the following Monte Carlo procedure: (i) Calculate sxi and syj (the observed values of SX i and SY j ) from the B , Y B , S B , S B , using (2.9) [by generating data. (ii) Generate a realization of X i(1) Xi Yj j (1) values from the standard exponential distributions Exp(0, 1)and chi-square distribu2 2 and χ2n ], and calculate the value of T B , as given by (2.8). (iii) Repeat tions χ2m i −2 j −2 step (ii) a large number of times, N , to obtain N realizations of T B . From these N values of T B , obtain the empirical distribution of T B and calculate its empirical (1 − α)th quantile as an estimate of qαB . Remark The PB method can also be used for constructing SCIs for all pair-wise comparisons of the location parameters of several exponential distributions under heteroscedasticity (details are omitted). The following theorem shows that the PBSCIs in (2.10) have correct coverage probability asymptotically. l k Theorem 1 Let n = i=1 m i + j=1 n j and suppose that n → ∞ such that m i /n → u i ∈ (0, 1) and n j /n → v j ∈ (0, 1), for i = 1, . . ., k, j = 1, . . ., l. Then Pr ξiT −ξ Cj ∈ X i(1) −Y j (1) ± qαB Vi j , all i=1, . . . , k, j=1, . . . , l → 1 − α. Proof First, note that Pr
ξiT − ξ Cj ∈ X i(1) − Y j (1) ± qαB Vi j , all i = 1, . . . , k; j = 1, . . . , l ⎧ ⎫ X C T ⎨ ⎬ i(1) − Y j (1) − ξi − ξ j ≤ q B = Pr T ≤ q B . = Pr max α α ⎩1≤i≤k,1≤ j≤l ⎭ Vi j
Now, let X (1) = X 1(1) , . . . , X k(1) and Y(1) = (Y1(1) , . . . , Yl(1) ). Then it is easily shown that X (1) − ξ T D W ∗ , as n → ∞ → n Y (1) − ξ C Z∗ D
→ denotes convergence in distribution and the components of W ∗ = ∗ where ∗ ∗ ∗ ∗ exponential variW1 , . . . , Wk and Z = Z 1 , . . . , Z l are mutually independent C ∗ T ∗ ables, with Wi ∼ Exp 0, σi /u i and Z j ∼ Exp 0, σ j /v j . But it is easily ver-
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2 ified, using (2.2), that E(SX i ) = σiT and Var(SX i ) = 2 σiT /(2m i − 2). Therefore, using Chebyshev’s inequality, it follows that S X i converges in probability to σiT (i = 1, . . ., k). Similarly, SY j converges in probability to σ jC ( j = 1, . . . , l). Hence, D
using Slutsky’s theorem (see Ferguson 1996, p. 6), it follows that T → T∞ , as n → ∞, where Wi∗ − Z ∗j T∞ = max 2 . 1≤i≤k,1≤ j≤l T 2 C σi /u i + σ j /v j D
Now we show that T B → T∞ , as n → ∞. First note that SXi P χ2 → σiT (2m i − 2) 2m i −2
and
SY j (2n j − 2)
P
2 χ2n → σ jC , as n → ∞, j −2
P
where → denotes convergence in probability. Hence n
2
Vi Bj
n 2 (m i − 1) B 2 n 2 n j − 1 B 2 SX i + SY j = m i3 n 3j 2 2 P → σiT /u i + σ jC /v j , as n → ∞.
(indepenNow let W1 , . . ., Wk and Z 1 , . . ., Z l be independent exponential variables dent of X i j and Yi j ), with Wi ∼ Exp 0, σiT /u i and Z j ∼ Exp 0, σ jC /v j . Note that SY j SXi B T B X i(1) = u i /σi Wi and Y j(1) = v j /σ jC Z j . mi nj Then B B Vi Bj X i(1) − Y j(1) 1≤i≤k,1≤ j≤l B = max − Y jB(1) n 2 Vi Bj n X i(1) 1≤i≤k,1≤ j≤l
nu S
Sy j nv j i xi B 2 V = max − n W Z i j ij 1≤i≤k,1≤ j≤l mi nj σiT σ jC 2 T 2 C D σi u i + σ j /v j → max , as n → ∞ Wi − Z j 1≤i≤k,1≤ j≤l
TB =
max
D
Thus T B → T∞ as n → ∞. Note that since the limiting distribution of T B is continuous, we have qαB → qα , as n → ∞, where qc is the (1 − α)th quantile of the
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Comparing exponential location parameters
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! distribution of T∞ . Therefore, as n → ∞, Pr T ≤ qαB → Pr {T∞ ≤ qα } = 1 − α. This completes the proof.
3 Simulation results In this section, we present the results of our simulation studies in which we examine the coverage probabilities of our proposed parametric bootstrap procedure and compare its performance with the method proposed by Maurya et al. (2011a) and also with the simultaneous fiducial generalized confidence intervals proposed by Karrati-Kopaei et al. (2013). We use the abbreviations PBSCI, M-SCI, and SFGCI to refer to these three methods. We note that Maurya et al. (2011a) compared M-SCI with the Bonferroni SCIs and concluded that M-SCI method performs better than the Bonferroni method. Following Maurya et al. (2011a), in our simulations we considered the case of k = 3 treatments and l = 2 controls. Note that it is easily verified that the coverage probabilities of these methods do not depend on the values of the location parameters ξ T and ξ C . Therefore, without loss of generality, in our simulations we assumed that ξ T = 0 and ξ C = 0. We first briefly review the methods SFGCI and M-SCI and then present our simulation results. SFGCI. This procedure is based on the concept of fiducial generalized pivotal quantities (FGPQ); see Hannig et al. (2006a, b) and Chang and Huang (2009). Let X i∗ and Y ∗j denote independent copies of X i and Y j , and let Ri j = X i(1) −
SY j ∗ SX i ∗ X i(1) − ξiT − Y j(1) + Y j(1) − ξ Cj . SX ∗i SY ∗j
Then, it is easily verified that Ri j are simultaneous fiducial generalized pivotal quantities for ξiT − ξ Cj , for i = 1, . . ., k and j = 1, . . ., l. Now, let
T
F
X i(1) − Y j (1) − Ri j = max , 1≤i≤k;1≤ j≤l Vi j
where Vi j ’s are the same as in (2.7). Then 100(1−α) % two-sided SFGCIs for ξiT −ξ Cj are X i(1) − Y j (1) ± qαF Vi j , i = 1, . . . , k, j = 1, . . . , l, where qαF is the(1-α)th quantile of the conditional distribution of T F given the observed values of X i ’s and Y j ’s. (The conditional distribution of T F does not depend on any of the parameters and depends on the observed values of X i ’s and Y j ’s only through sxi and syj .) The value of qαF is easily obtained by a Monte Carlo procedure similar to the computational procedure given in Sect. 2. For details of the SFGCI procedure, see Karrati-Kopaei et al. (2013).
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Table 1 The coverage probabilities (CP) and average volumes (AV) of 95 % SCIs, when σ T = (1, 1, 1) and σ C = (1, 1)
Method
Sample sizes 10
20
30
CP
AV
CP
AV
CP
AV
PBSCI
0.9625
2.177
0.9596
0.0142
0.9558
0.00092
SFGCI
0.9790
4.764
0.9694
0.0208
0.9628
0.00118
M-SCI
0.9978
42.620
0.9937
0.1181
0.9897
0.00549
M-SCI. In this procedure, it is assumed that all the sample sizes are equal. Let m denote the common sample size; i.e., in our notation, m i = n j = m, for i = 1, . . ., k and j = 1, . . ., l. The two-sided conservative SCIs for ξiT − ξ Cj , i = 1, . . ., k, j = 1, . . ., l, are −1 (1 − α)1/(k+l) , X i(1) − Y j (1) ± d F2,2m−2 where d = max
max S X i /m , max SY j /m
1≤i≤k
1≤ j≤l
−1 and F2,2m−2 (x) is the xth quantile
of an F distribution with 2 and 2m − 2 degrees of freedom. For details, see Maurya et al. (2011a). Simulation procedure. We used the following procedure to estimate the true coverage probabilities: I Generate X i and Y j , random samples of sizes m i and n j from exponential distribu T T C C tions Exp ξi , σi and Exp ξ j , σ j (i = 1, . . ., k, j = 1, . . ., l) and calculate the observed values of X i(1) , SX i , Y j (1) , and SY j . II Calculate the value of qαB using the computational procedure given in Sect. 2, with N = 1,000,000. Also, calculate qαF using a similar computational procedure; see Karrati-Kopaei et al. (2013). III For each method, construct two-sided SCIs and record whether or not all the values of ξiT − ξ Cj , i = 1, . . ., k, j = 1, . . ., l, fall in their corresponding SCIs. IV Repeat steps I–III, a large number of times, M = 10,000. Then, for each method, the fraction of times that all ξiT − ξ Cj (i = 1, . . ., k, j = 1, . . ., l) are in their corresponding SCIs provides an estimate of the true coverage probability. Simulation results I. We first considered the two cases considered by Maurya et al. (2011a), with k = 3, l = 2, and sample sizes equal to m, with m = 10, 20 and 30 (as small, moderate, and large sample sizes). As noted before, without loss of generality, we assumed that ξ T = 0 and ξ C = 0. We also considered two cases for the scale parameters: (i) σ T = (1, 1, 1), σ C = (1, 1) and (ii) σ T = (1, 1.1, 1.3), σ C = (1.2, 1.4). The results of simulations for these two cases with N = 1, 000, 000, M = 10, 000, and 1 − α = 0.95 are shown in Tables 1 and 2, respectively. It is seen from Tables 1 and 2 that, compared to the M-SCI method, the coverage probabilities of our proposed PBSCI method are closer to the nominal confidence
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Comparing exponential location parameters Table 2 The coverage probabilities (CP) and average volumes (AV) of 95 % SCIs, when σ T = (1, 1.1, 1.3) and σ C = (1.2, 1.4)
Table 3 The coverage probabilities (CP) and average volumes (AV) of 95 % SCIs, when σ T = (1, 1.1, 1.3) and σ C = (0.6, 1.2)
Method
1091 Sample sizes 10 CP
20 AV
30
CP
AV
CP
AV
PBSCI
0.9650
7.413
0.9587
0.0487
0.9549
0.0031
SFGCI
0.9784
24.412
0.9696
0.1096
0.9648
0.0063
M-SCI
0.9976
164.553
0.9934
0.4675
0.9899
0.0229
Method
Sample sizes 10 CP
20 AV
CP
30 AV
CP
AV
PBSCI
0.9548
4.165
0.9535
0.0298
0.9525
0.0020
SFGCI
0.9726
8.301
0.9648
0.0409
0.9567
0.0025
M-SCI
0.9972
99.740
0.9928
0.2943
0.9904
0.0142
coefficient and, consequently, the resulting confidence intervals are typically shorter (as reflected in the average volume). The SFGCI method also performs better than the M-SCI method, but not as well as the PBSCI method. Simulation results II. In the cases considered by Maurya et al. (2011a), the scale parameters are not much different. But if the scale parameters are more unequal, then the differences between the methods become more apparent, as can be seen from Table 3. Simulation results III. In M-SCI method, it is assumed that the sample sizes are equal to m (i.e., m i = n j = m, for i = 1, . . ., k and j = 1, . . ., l); but note that the PBSCI and SFGCI methods can be used with unequal sample sizes. In view of this, we decided to compare the PBSCI and SFGCI methods when the sample sizes are unequal. As before, we took k = 3 and l = 2. The results of simulation for different values of (m 1 , m 2 , m 3 ) and (n 1 , n 2 ), when σ T = (1, 1.1, 1.3) and σ C = (0.6, 1.2), are shown in Table 4. It is seen from Table 4 that, compared to the SFGCI method, the coverage probabilities of our proposed PBSCI method are closer to the nominal confidence level.
4 Illustrative example The data given in Table 5, taken from Maurya et al. (2011a), concern the effectiveness of four drugs (two test drugs and two control drugs) used in the treatment of Leukemia, as measured by the duration of remission time.
123
1092 Table 4 The coverage probabilities (CP) and average volumes (AV) of 95 % SCIs for the PBSCI and SFGCI methods, when σ T = (1, 1.1, 1.3) and σ C = (0.6, 1.2) and the sample sizes are unequal
Table 5 Remission duration by four drugs
A. Malekzadeh et al. Sample sizes
Method PBSCI
SFGCI
(m 1 , m 2 , m 3 )
(n 1 , n 2 )
CP
AV
CP
AV
(10, 10, 10)
(10, 30)
0.9498
2.1984
0.9658
3.5154
(10, 10, 10)
(30, 10)
0.9472
7.8477
0.9573
10.7208
(10, 10, 10)
(30, 30)
0.9464
2.6389
0.9559
3.2741
(30, 10, 10)
(10, 10)
0.9477
2.6611
0.9659
4.8440
(30, 30, 10)
(10, 10)
0.9508
0.9515
0.9653
1.5565
(30, 30, 30)
(10, 10)
0.9486
0.2654
0.9594
0.3821
(10, 10, 30)
(10, 10)
0.9557
1.5337
0.9713
2.9800
(10, 30, 30)
(10, 10)
0.9539
0.6395
0.9670
1.0936
(30, 10, 10)
(30, 10)
0.9431
2.0294
0.9554
2.6781
(30, 10, 10)
(10, 30)
0.9452
0.5022
0.9614
0.8369
(10, 10, 30)
(10, 30)
0.9495
0.3180
0.9645
0.5547
(10, 10, 30)
(30, 10)
0.9504
1.6573
0.9617
2.2796
Test drug 1
Test drug 2
Control drug 1
Control drug 2
1.034
5.115
2.214
4.158
2.344
4.498
4.976
4.025
1.266
4.617
8.154
5.170
1.563
4.651
2.686
11.909
1.169
4.533
2.271
4.912
4.118
4.513
3.139
4.629
1.013
7.641
2.214
3.955
1.509
5.971
4.480
6.735
1.109
12.130
8.847
3.140
1.965
4.699
2.239
12.446
5.136
4.914
3.473
8.777
1.533
17.169
2.761
6.321
1.716
5.497
2.833
3.256 8.250
2.778
11.332
2.381
2.546
18.922
3.548
3.759
2.626
13.712
2.414
5.205
3.413
6.309
2.832
3.071
1.929
10.086
5.551
3.147
2.061
9.293
3.376
9.773
2.951
11.787
2.968
10.218
To obtain the value of qαB and qαF , we used the procedure given in Sect. 2 with N = 1,000,000 runs. The estimated values of qαB and qαF for α = 0.05, 0.025, and 0.01 are shown in Table 6.
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Comparing exponential location parameters Table 6 The estimated values of qαB and qαF for two-sided PBSCIs and SFGCIs
Table 7 Two-sided PBSCIs, SFGCIs, and M-SCIs for comparing the tests drugs with the control drugs
1093
α
qαB
qαF
0.050
3.65
3.86
0.025
4.40
4.66
0.010
5.39
5.76
Parameters α = 0.050
α = 0.025
α = 0.010
PBSCIs ξ1T − ξ1C (−1.552, −0.851) (−1.623, −0.779) (−1.718, −0.684) (1.352, 3.216) (1.141, 3.427) ξ2T − ξ1C (1.509, 3.059) ξ1T − ξ2C (−2.675, −1.441) (−2.799, −1.317) (−2.967 −1.149) ξ2T − ξ2C (0.501, 2.354) Volume 2.48
(0.313, 2.541)
(0.061, 2.793)
5.20
11.74
SFGCIs ξ1T − ξ1C (−1.571, −0.831) (−1.648, −0.754) (−1.754, −0.648) ξ2T − ξ1C (1.465, 3.103) (1.296, 3.272) (1.062, 3.506) ξ1T − ξ2C (−2.709,−1.407) (−2.844, −1.272) (−3.030, −1.086) ξ2T − ξ2C (0.448, 2.406)
(0.246, 2.608)
(−0.034, 2.888)
Volume
6.56
15.35
3.09
M-SCIs ξ1T − ξ1C (−2.200, −0.202) (−2.384, −0.018) (−2.635, 0.233) ξ2T − ξ1C (1.285, 3.283) (1.101, 3.467) (0.850, 3.718) ξ1T − ξ2C (−3.057, −1.059) (−3.241, −0.875) (−3.492, −0.624) ξ2T − ξ2C (0.428, 2.426) (0.244, 2.610) (−0.007, 2.861) Volume
15.94
31.34
67.66
The corresponding two-sided PBSCIs for comparing the test drugs with the control drugs are shown in Table 7. For comparisons, the SCIs obtained by Maurya et al. (2011a) and FGPQ method are also shown in this table. It is seen that the PBSCI procedure results in shorter intervals (the volumes of MSCIs are about 6 times the volumes of PBSCIs). Note, in particular, that the conclusions based on 99 % PBSCIs are different from the conclusions based on 99 % M-SCIs and SFGCIs: The 99 % PBSCIs indicate that test drug 2 is better than control drugs 1 and 2, and test drug1 is worse than control drugs 1 and 2; whereas the 99 % M-SCIs indicate only that test drug 2 is better than control drug 1, and test drug1 is worse than control drug 2; and the 99 % SFGCIs do not detect any difference between test drug 2 and control drug 2. Therefore, in this example our proposed PBSCI procedure performs better than M-SCI and SFGCI methods. Acknowledgments We are grateful to two reviewers for their valuable comments and suggestions. This work was supported by the Research Council of Shiraz University.
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