Fiducial Generalized Confidence Intervals for the Common Mean of K Normal Populations Paul Patterson, Jan Hannig, Hari Iyer Department of Statistics, Colorado State University, Fort Collins, CO 80523 Abstract Suppose Xi1 , . . . , Xini are iid N (µ, σi2 ), i = 1, . . . , k. We consider construction of a confidence interval for the common mean µ of these k populations. We propose two new confidence interval methods based on two different fiducial generalized pivotal quantities. These methods are known to yield asymptotically correct confidence intervals. Small sample performance of these methods is evaluated by Monte Carlo simulation and compared with an exact method based on R. A. Fisher’s proposal for combining P -values from independent tests. Both of the generalized confidence interval methods have satisfactory small sample coverage properties and are more efficient than the exact method. Key words: Generalized Pivots, Fiducial inference, Structural Inference, Conditional Inference, Asymptotic properties, Common mean problem.
1. INTRODUCTION Suppose Xi1 , . . . , Xini are iid N (µ, σi2 ), i = 1, . . . , k. It is of interest to construct a confidence interval for the common mean µ of these k populations. This problem has many practical applications where information about a quantity of interest is obtained from several sources with differing precisions. For instance, suppose k different laboratories are asked to measure a certain physical or chemical property associated with a certain artefact. Let the true value of the property of interest be µ. The ith laboratory reports the results of ni independent measurements. It is then of interest to combine the information from all the laboratories to obtain not only an efficient point estimate of µ but also to provide an interval estimate for it. This is a commonly occuring situation in field of metrology. This is also one of the simplest problems that comes under the heading of meta analysis. Many authors have addressed this problem from a frequentist perspective. As a matter of fact, several exact confidence sets have been suggested in the literature. See, for instance, Jordan and 0 Paul Patterson is graduate student, Department of Statistics, Colorado State University, Fort Collins, CO 80523, and Research Statistician, USFS-Rocky Mountain Research Station (
[email protected]). Jan Hannig is Assistant Professor, Department of Statistics, Colorado State University, Fort Collins, CO 80523 (E-mail:
[email protected]). Hari Iyer is Professor, Department of Statistics, Colorado State University, Fort Collins, CO 80523 (
[email protected]).
1
Krishnamoorthy (1996), Yu et. al.(1999), and Krishnamoorthy and Lu (2003). All of the previously proposed methods are based on ad-hoc constructions of appropriate pivotal quantities for µ. None of them are guaranteed to produce an actual interval and there is a positive probability of obtaining a disjoint collection of intervals as the confidence set for µ. Yu et. al.(1999) report the results of a study comparing a number of different interval estimators for µ. They concluded that a method based on R. A. Fisher’s technique for combining P -values performed at least as well as or better than other competing methods. We will refer to this method as the Fisher Method. Weerahandi (1993) introduced the concept of generalized pivotal quantities and demonstrated their use in developing confidence interval procedures in problems involving nuisance parameters. See also Weerahandi (1991, 1995) and Tsui and Weerahandi (1989) for related discussions. Iyer and Patterson (2002) provide a recipe for constructing generalized pivotal quantities. Hannig et. al (2004) extended this recipe and studied the asymptotic properties of GCIs. They also showed that GCIs can be obtained using R. A. Fisher’s fiducial principle. As an illustration of their approach they proposed two new confidence interval procedures for the common mean µ. The purpose of this paper is to compare the performance of the three methods – (1) A generalized confidence interval based on the two-stage construction of a generalized pivotal quantity, (2) a generalized confidence interval based on a general fiducial construction of a GPQ, and (3) An exact method based on Fisher’s proposal for combining P -values from independent tests. The first two methods were introduced in Hannig et. al, (2004). While the Fisher-method is exact, the two GCIs are only approximate. Analytical derivation of the actual coverage probabilities for the GCIs is intractable, so a Monte Carlo evaluation is needed for assessing their performance in small samples. Incidentally, Hannig et. al (2004) show that these GCIs asymptotically have the correct coverage. The paper is organized as follows. Section 2 introduces the notation used in this paper. In this same section the three competing confidence interval procedures for µ are described in detail. Description of our simulation study and the numerical results on coverage and length are given in Section 3. Section 4 provides a brief description of our findings.
2. The Three Confidence Interval Methods for the Common Means Let ¯i = X
Pni j=1
ni
Xij
Pni , and
Si2
=
¯ i )2 −X . ni − 1
j=1 (Xij
¯1, . . . , X ¯ k , S 2 , . . . , S 2 ) is a minimal sufficient statistic for (µ, σ 2 , . . . , σ 2 ), but it is not Clearly S = (X 1 1 k k complete. All three methods are based on these statistics. We denote an independent copy of S by ¯ ? , S1?2 , . . . , S ?2 ). A realization of S will be denoted by s = (¯ ¯ 1? , . . . , X x1 , . . . , x¯k , s21 , . . . , s2k ) S? = ( X k k and s? = (¯ x?1 , . . . , x¯?k , s?1 2 , . . . , s?k 2 ) will denote a realization of S? .
2.1
Confidence Interval for µ using a Two-Stage FGPQ
For a definition of a generalized pivotal quantity (GPQ) and discussion of associated generalized confidence intervals (GCI) the reader is referred to Weerahandi (1993). The special class of GPQs, called fiducial generalized pivotal quantities (FGPQ), the reader may consult Hannig et. al.(2004). It is customary to use the notation Rθ to denote a FGPQ for the parameter θ. 2
The two-stage FGPQ for µ developed in Hannig et. al.(2004) is given by ¯1 ¯k n X ¯? n1 X nk X 1 1 + ··· + + ··· + σ12 Rσ12 Rσk2 ? Rµ (S, S , ξ) = Rµ = Rµ|Rσ2 ,...,Rσ2 = n1 nk − n1 1 k + ··· + + ··· + Rσ12 Rσk2 σ12
¯? nk X k σk2 − µ nk 2 σk
where Rσi2 is a GPQ for σi2 for i = 1, . . . , k and is given by Rσi2 =
Si2 σi2 2 Si?
Observe that Rµ (S, S, ξ) = µ. One can also show that the conditional distribution of Rµ (S, S? , ξ) given S = s is independent of ξ. Thus Rµ (S, S? , ξ) is a FGPQ as defined by Hannig et. al.(2004). A 100(1 − α)% upper generalized confidence interval for µ is given by µ ≤ Rµ,1−α , where Rµ,γ denotes the 100γ-percentile of the conditional distribution of Rµ = Rµ (S, S? , ξ) given S = s. Let Fs (t) denote the cdf of Rµ (S, S? , ξ) given S = s. Then the coverage of the proposed generalized upper confidence bound is given by PS [µ ≤ Fs−1 (1 − α)]. This is equal to (1)
PS [PS? [Rµ (S, S? , ξ) ≤ µ|S = s] < 1 − α] .
It follows from equation (1) that the coverage probability is independent of µ and depends only the k − 1 ratios τ1 = σ2 /σ1 , . . . , τk−1 = σk /σ1 . This fact will be used in selecting parameter combinations for the simulation study. Note also that Hannig et. al.(2004) established the following result asymptotic result for Rµ . Proposition 1. (Hannig et. al (2004)) Let all n1 , . . . , nk approach infinity in such a way that rj = lim nj /(n1 + · · · + nk ) exists and 0 < rj < 1. Then a 100(1 − α)% generalized confidence interval for µ based on the FGPQ Rµ has asymptotically 100(1 − α)% frequentist coverage. This proposition will be used to check the correctness of simulation programs by monitoring the coverage probabilities as the sample sizes become large. When this is the case the coverages should be close to the nominal value of 1 − α. Given a realization s of S one can calculate Rµ,1−α using Monte Carlo simulation. For completeness we indicate how this is done. First we can express Rµ (S, S? , ξ) as ¯k ¯1 nk X n1 X + ··· + ? Rσ 2 Rσk2 Ek+1 r Rµ (S, S? , ξ) = Rµ = Rµ|Rσ2 ,...,Rσ2 = n11 − nk n1 nk 1 k + ··· + + · · · + Rσ12 Rσk2 σ12 σk2 ? where Ek+1 ∼ N (0, 1) and is independent of Rσj2 for j = 1, . . . , k. Using the pivotal relationships 2 for σj , j = 1, . . . , k, we can express Rσi2 (S, S? , ξ) as
Rσi2 (S, E? , ξ) =
(ni − 1)Si2 , Ei?
? ). Given a realization s of S, we where Ei? ∼ χ2ni −1 . We use E? to denote the vector (E1? , . . . , Ek? , Ek+1 ? ? ? generate N independenet realizations, (e1 , . . . , eN ), of E and calculate Rµ (s, e?1 , ξ), . . . , Rµ (s, e?N , ξ). These values are ordered from smallest to largest and the N (1 − α) element of the ordered sequence is used as the estimate of Rµ,1−α .
3
2.2
Confidence Interval for µ using the General Construction FGPQ
The general construction for FGPQs presented in Hannig et. al.(2004) yields a different FGPQ for µ than the two-stage FGPQ. This FGPQ is based on the invertible pivotal relationship r 2 σ i ¯i = µ + X Ei i = 1, . . . , k ni (ni − 1)Si2 = σi2 Ek+i where Ei , i = 1, . . . , 2k are jointly independent and for i = 1, . . . , k, Ei ∼ N (0, 1) and Ek+i ∼ χ2ni −1 . ¯1, . . . , X ¯ k , S 2 , . . . , S 2 ) and E = (E1 , . . . , E2k ). Let s = (¯ x1 , . . . , x¯k , s21 , . . . , s2k ) denote a Let S = (X 1 k realization of S and e be the corresponding realization of E. In addition, let σ ˆi2 = (ni − 1)s2i /ni ˜ µ (s, S∗ , ξ), denote the ML estimate of σi2 . Hannig et. al. (2004) show that the pdf of the FGPQ R obtained by using their general construction, is given by (2)
C fs (t) = µ ¶ µ ¶n /2 µ ¶n /2 . n /2 (¯ x1 − t)2 1 (¯ xk−1 − t)2 k−1 (¯ xk − t)2 k 1+ ··· 1 + 1+ 2 σ ˆ12 σ ˆk−1 σ ˆk2
where
R
∞
C −1 =
µ −∞
dt µ ¶ ¶n /2 µ ¶n /2 . n /2 (¯ xk−1 − t)2 k−1 (¯ x1 − t)2 1 (¯ xk − t)2 k ··· 1 + 1+ 1+ 2 σ ˆ12 σ ˆk−1 σ ˆk2
Rx Let Fs (x) = −∞ fs (t) dt, be the cdf of the FGPQ. Then a 100(1 − α)% upper confidence interval for µ is given by µ ≤ Fs−1 (1 − α). The actual coverage probability is equal to PS [µ ≤ Fs−1 (1 − α)]. One can show that this probability is independent of µ and depends only on the k − 1 ratios ˜ µ,1−α is the solution to equation, τ1 = σ2 /σ1 , . . . , τk−1 = σk /σ1 . If s is a realization of S, then R
R
(3)
˜ µ,1−α R
1−α=
fs (t) dt −∞
˜ µ,1−α may be determined using numerical techniques. In practice, R Hannig et. al. have studied the large sample behavior of the resulting GCI. Specifically, they prove the following proposition. Proposition 2. (Hannig et. al (2004)) Let all n1 , . . . , nk approach infinity in such a way that rj = lim nj /(n1 + · · · + nk ) exists and 0 < rj < 1. Then a 100(1 − α)% generalized confidence ˜ µ whose pdf is given in Equation (2), has asymptotically interval for µ, based on the FGPQ R 100(1 − α)% frequentist coverage.
4
2.3
Fisher’s Method ¯
Fisher’s method is based on the k pivotal statistics Ti = ni (XSii−µ) , i = 1, . . . , k. The statistic Ti is the statistic that would be used to construct a t-test for µ based on data from group i only. These statistics determine the k P -values, Z (4)
¯ −µ) ni (X i Si
Pi =
fi (t) dt, for i = 1, . . . , k,
−∞
where fi (t) is the density function for a t distribution with ni − 1 degrees of freedom. The P values arePcombined, using Fisher’s method of combining P -values, to form the pivotal quantity Q = −2 ln(Pi ) ∼ χ22k . Since Q is an increasing function of µ an exact 100(1 − α)% upper confidence bound for µ is given by ( (5)
CI =
µ : −2
X
) ln(Pi ) ≤ χ22k,1−α
i
where χ22k,1−α is the P 1 − α quantile of the chi-square distribution with 2k degrees of freedom. Using the fact that −2 ln(Pi ) is an increasing function of µ, Equation 5 can be easily solved using numerical procedures to produce an 100(1 − α)% upper confidence bound for µ.
3. Simulation Results Three sets of simulations were run, one set each for k equal 2, 3 and 10. For each k, a representative set of ni and σi values were chosen. For each combination of sample sizes ni and standard deviations σi , 4000 independent realizations of S where generated. Without loss of generality it was assumed that µ is zero and σ1 is one. For each simulated dataset an upper confidence bound was computed using each of the three methods as outlined in the previous section. A FORTRAN program was written to perform the simulations. Tables 1–5 report the empirical coverages and expected lengths for the one-sided upper confidence bounds obtained using each of the three methods. By “expected length” for an upper confidence bound we mean the expected value of the absolute distance of the upper confidence bound from the true value. A number of different scenarios were considered with varying sample sizes and variance ratios. Several scenarios with larger sample sizes were included in the study to judge how soon the asymptotics take effect. Results for the case of k = 2 populations are given in Table 1 (smaller sample sizes) and Table 2 (larger sample sizes). For k = 3 the results are given in Table 3 (smaller sample sizes) and Table 4 (larger sample sizes). Results for the case of k = 10 populations are given in Table 5.
5
Table 1. Empirical Coverages and “Expected Lengths” for Fisher’s Exact Upper Confidence Bound and Two Generalized Upper Bounds: k = 2 populations & smaller sample sizes n1
n2
σ1
σ2
3
3
1.00
0.10
3
3
1.00
0.25
3
3
1.00
0.50
3
3
1.00
1.00
3
3
1.00 10.00
3
10
1.00
0.10
3
10
1.00
0.25
3
10
1.00
1.00
3
10
1.00
4.00
3
10
1.00
10.00
10
10
1.00
0.10
10
10
1.00
0.25
10
10
1.00
0.50
10
10
1.00
1.00
10
10
1.00
10.00
Two-Stage General FGPQ 0.9377 0.1638 0.9305 0.3439 0.9280 0.5452 0.9227 0.7788 0.9307 1.6246 0.9445 0.0630 0.9370 0.1542 0.9323 0.4955 0.9365 1.1148 0.9470 1.4431 0.9500 0.0580 0.9477 0.1391 0.9475 0.2531 0.9280 0.3882 0.9467 0.5828
6
0.9445 0.1660 0.9340 0.3398 0.9213 0.5212 0.9137 0.7346 0.9375 1.6445 0.9450 0.0580 0.9415 0.1436 0.9337 0.4840 0.9305 1.0867 0.9465 1.4574 0.9500 0.0574 0.9460 0.1384 0.9465 0.2525 0.9263 0.3875 0.9467 0.5813
Fisher 0.9507 0.2152 0.9505 0.4287 0.9485 0.6660 0.9475 0.9528 0.9457 2.1390 0.9527 0.0738 0.9517 0.1696 0.9483 0.5470 0.9553 1.3120 0.9557 1.8428 0.9510 0.0694 0.9495 0.1574 0.9557 0.2748 0.9450 0.4132 0.9495 0.6973
Table 2. Empirical Coverages and “Expected Lengths” for Fisher’s Exact Upper Confidence Bound and Two Generalized Upper Bounds: k = 2 populations & larger sample sizes n1
n2
σ1
σ2
3
50
1.00
0.10
3
50
1.00
0.25
3
50
1.00
0.50
3
50
1.00
1.00
3
50
1.00
4.00
3
50
1.00
10.00
10
50
1.00
0.25
10
50
1.00
1.00
10
50
1.00
4.00
10
50
1.00
10.00
50
50
1.00
0.10
50
50
1.00
0.25
50
50
1.00
1.00
Two-Stage General FGPQ 0.9450 0.0256 0.9373 0.0634 0.9400 0.1264 0.9393 0.2476 0.9375 0.7243 0.9430 1.1505 0.9470 0.0593 0.9430 0.2228 0.9457 0.4971 0.9455 0.5562 0.9450 0.0243 0.9417 0.0580 0.9470 0.1701
7
0.9447 0.0238 0.9365 0.0598 0.9457 0.1185 0.9443 0.2343 0.9325 0.7079 0.9410 1.1267 0.9457 0.0588 0.9437 0.2216 0.9457 0.4973 0.9457 0.5564 0.9410 0.0239 0.9405 0.0576 0.9460 0.1697
Fisher 0.9473 0.0322 0.9397 0.0722 0.9517 0.1387 0.9517 0.2654 0.9517 0.7975 0.9467 1.3489 0.9520 0.0689 0.9455 0.2389 0.9477 0.5306 0.9510 0.6341 0.9483 0.0283 0.9430 0.0644 0.9490 0.1767
Table 3. Empirical Coverages and “Expected Lengths” for Fisher’s Exact Upper Confidence Bound and Two Generalized Upper Bounds: k = 3 populations & smaller sample sizes n1
n2
n3
σ1
σ2
σ3
3
3
3
1.00
1.00
1.00
3
3
3
1.00
0.50
0.50
3
3
3
1.00
0.50
2.00
3
3
3
1.00
0.25
4.00
3
3
3
1.00
0.50
10.00
3
3
3
1.00
0.10
10.00
3
10
3
1.00 0.50
2.00
3
10
3
1.00 0.25
4.00
3
10
3
1.00 0.50
10.00
3
10
3
1.00 0.10
10.00
3
10
10
1.00 0.50
0.50
7
20
8
1.00 0.25
1.00
7
20
12
1.00 1.00
1.00
7
20
12
1.00 0.10 16.00
10
10
10
1.00 1.00
1.00
10
10
10
1.00 0.50
2.00
10
10
10
1.00 0.25
4.00
10
10
10
1.00 0.50 10.00
10
10
10
1.00 0.10 10.00
Two-Stage General FGPQ 0.9055 0.6172 0.9075 0.3603 0.9190 0.5472 0.9243 0.3768 0.9303 0.5773 0.9407 0.1781 0.9285 0.3018 0.9430 0.1660 0.9333 0.2991 0.9455 0.0625 0.9323 0.2045 0.9553 0.0960 0.9375 0.2791 0.9465 0.0397 0.9380 0.3106 0.9350 0.2464 0.9537 0.1393 0.9453 0.2539 0.9523 0.0570
8
0.8962 0.5902 0.9002 0.3424 0.9160 0.5156 0.9300 0.3441 0.9293 0.5246 0.9493 0.1702 0.9345 0.2755 0.9447 0.1466 0.9415 0.2786 0.9450 0.0567 0.9365 0.1928 0.9577 0.0949 0.9390 0.2769 0.9443 0.0392 0.9345 0.3096 0.9357 0.2454 0.9547 0.1385 0.9457 0.2535 0.9500 0.0565
Fisher 0.9493 0.7640 0.9505 0.4546 0.9530 0.6948 0.9545 0.5138 0.9543 0.8051 0.9553 0.2949 0.9513 0.3432 0.9540 0.1970 0.9463 0.3685 0.9490 0.0841 0.9477 0.2211 0.9497 0.1183 0.9497 0.3020 0.9465 0.0568 0.9523 0.3380 0.9447 0.2815 0.9480 0.1774 0.9500 0.3092 0.9480 0.0806
Table 4. Empirical Coverages and “Expected Lengths” for Fisher’s Exact Upper Confidence Bound and Two Generalized Upper Bounds: k = 3 populations & larger sample sizes n1
n2
n3
σ1
σ2
σ3
10
50
10
1.00
0.25
4.00
10
50
50
1.00
1.00
1.00
10
50
50
1.00
0.50
0.50
40
30
50
1.00
0.10
10.00
40
30
50
1.00 10.00
0.10
50
50
50
1.00
1.00
1.00
50
50
50
1.00
0.50
2.00
50
50
50
1.00
0.25
4.00
50
50
50
1.00
0.10
10.00
100
80
90
1.00
0.25
0.25
100
80
90
1.00 16.00
0.20
100
100
100 1.00
1.00
Two-Stage General FGPQ 0.9477 0.0602 0.9430 0.1635 0.9470 0.0843 0.9493 0.0313 0.9495 0.0241 0.9473 0.1392 0.9500 0.1046 0.9485 0.0588 0.9497 0.0241 0.9497 0.0319 0.9483 0.0341 0.9505 0.0985
1.00
9
0.9470 0.0596 0.9430 0.1625 0.9470 0.0838 0.9470 0.0309 0.9467 0.0237 0.9470 0.1388 0.9490 0.1042 0.9473 0.0583 0.9463 0.0236 0.9483 0.0314 0.9470 0.0337 0.9497 0.0981
Fisher 0.9467 0.0788 0.9513 0.1761 0.9533 0.0934 0.9487 0.0418 0.9473 0.0329 0.9510 0.1461 0.9527 0.1175 0.9493 0.0717 0.9463 0.0321 0.9523 0.0350 0.9480 0.0439 0.9510 0.1039
Table 5. Empirical Coverages and “Expected Lengths” for Fisher’s Exact Upper Confidence Bound and Two Generalized Upper Bounds: k = 10 populations n1
n2
n3
n4
n5
n6
n7
n8
n9
n10
σ1 3 1.00
σ2 3 1.00
σ3 3 1.00
σ4 3 1.00
σ5 3 1.00
σ6 3 1.00
σ7 3 1.00
σ8 3 1.00
σ9 3 1.00
σ10 3 1.00
3 1.00
3 0.25
3 0.25
3 0.50
3 0.50
3 3 3 10.00 10.00 10.00
3 10.00
3 10.00
3 10.00
3 10.00
3 10.00
3 3 10.00 10.00
7 1.00
10 0.25
10 0.50
10 0.50
7 1.00
10 0.25
10 0.25
10 0.50
10 0.50
10 10 10 10.00 10.00 10.00
10 10.00
10 10.00
10 1.00
10 1.00
10 1.00
10 1.00
10 1.00
10 1.00
10 1.00
10 1.00
10 1.00
10 1.00
10 1.00
10 1.00
10 1.00
10 1.00
10 1.00
50 1.00
50 1.00
50 1.00
50 1.00
50 1.00
50 1.00
50 1.00
50 1.00
50 1.00
50 1.00
50 1.00
50 1.00
50 1.00
50 1.00
50 1.00
50 1.00
50 0.25
50 0.25
50 0.50
50 0.50
50 50 50 10.00 10.00 10.00
50 10.00
50 10.00
10 0.25
2-Stage
Gen FGPQ
Fisher
0.8672 0.3749
0.8580 0.3258
0.9407 0.3970
0.8920 0.2000
0.8865 0.1626
0.9480 0.3005
0.9285 0.0969
0.9325 0.0850
0.9455 0.1266
0.9363 0.0868
0.9363 0.0856
0.9490 0.1255
0.9343 0.1789
0.9355 0.1754
0.9485 0.1921
0.9405 0.1007
0.9403 0.0994
0.9510 0.1131
0.9477 0.753
0.9463 0.0749
0.9547 0.0816
0.9450 0.0371
0.9433 0.0366
0.9450 0.0510
4. Discussion of Simulation Results Fisher’s method for obtaining a confidence bound for the common mean µ is an exact method but the two proposed GCI methods are only asymptotically exact. Nevertheless we see that the small sample performance of the two GCI methods are quite satisfactory and they appear to be more efficient than Fisher’s method. We also see that the GCI using the general construction performs slightly better than the two-stage GCI in terms of confidence interval expected length. We further note that, as the sample sizes get large the coverage probabilities for both GCIs approach the nominal value of 0.95. We conclude that either of the two GCI methods may be recommended for practical applications. Computer programs for the two GCI calculations may be obtained by contacting the first named author. 10
REFERENCES Iyer, H. K. and Patterson P. L. (2002). “A Recipe for Constructing Generalized Pivot Quantities and Generalized Confidence Intervals”, Technical Report 2002/10, Department of Statistics, Colorado State University. Hannig, J., Iyer, H. K. and Patterson P. L. (2005). “Fiducial Generalized Confidence Intervals”, Technical Report 2004/12, Department of Statistics, Colorado State University. Jordan, S. M., and Krishnamoorthy, K. (1996). “Exact Confidence Intervals for the Common Mean of Several Normal Populations”, Biometrics, 52, 77-86. Krishnamoorthy, K. and Yong Lu. (2003). “Inferences on the common mean of several normal populations based on the generalized variable method”, Biometrics, 59, 237-247. Tsui, K. W. and Weerahandi, S. (1989). “Generalized P -values in Significance Testing of Hypotheses in the Presence of Nuisance Parameters”, JASA, 84, 406, 602-607. Weerahandi, S. (1991). “Testing Variance Components in Mixed Models with Generalized P values”, JASA, 86, 413, 151-153. Weerahandi, S. (1993). “Generalized Confidence Intervals”, JASA, 88, 423, 899-905. Weerahandi, S. (1995). Exact Statistical Methods for Data Analysis, Springer-Verlag. Yu Philip, L. H., Yijun Sun, and Sinha, B. K. (1999). “On Exact Confidence Intervals for the Common Mean of Several Normal Populations”, Journal of Statistical Planning and Inference, 81, 263-277.
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