# Generalized Inferences about the Mean Vector of Several Multivariate Gaussian Processes

Authors: Pilar Ibarrola; Ricardo Vélez
Journal: Journal of Probability and Statistics
 (2015)

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## Abstract

We consider in this paper the problem of comparing the means of several multivariate Gaussian processes.
It is assumed that the means depend linearly on an unknown vector parameterθ and that nuisance parameters appear in the covariance matrices. More precisely, we deal with the problem of testing hypotheses, as well as obtaining confidence regions for θ. Both methods will be based on the concepts of generalized p value and generalized confidence region adapted to our context.


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## Body

## 1. Introduction

The generalizedp values to test statistical hypotheses in the presence of nuisance parameters are introduced by Tsui and Weerahandi (1989) [1], where the univariate Behrens-Fisher problem, as well as other examples, is considered in order to illustrate the usefulness of this approach. Afterwards Weerahandi (1993) [2] introduces the generalized confidence intervals.In 2004, Gamage et al. [3] developed a procedure based on the generalized p values to test the equality of the mean vectors of two multivariate normal populations with different covariance matrices. They also construct a confidence region for the means difference, using the concept of generalized confidence regions. Finally, by means of the generalized p value approach, a solution is obtained for the heteroscedastic MANOVA problem, but without reaching the desirable invariance property.In 2007, Lin et al. [4] considered the generalized inferences on the common mean vector of several multivariate normal populations. They obtained a confidence region for the common mean vector and simultaneous confidence intervals for its components. Their method is numerically compared with other existing methods, with respect to the expected area and coverage probabilities.In 2008, Xu and Wang [5] considered the problem of comparing the means of K populations with heteroscedastic variances. They provided a new generalized p value procedure for testing the equality of means, assuming that the variables are univariate and normally distributed. Numerical results show that their generalized p value test works better than a generalized F-test. We will set out our MANOVA problem as a generalization of their framework.In 2012, Zhang [6] considered the general linear hypothesis testing (GLHT) in an heteroscedastic one-way MANOVA. The multivariate Behrens-Fisher problem is a special case of GLHT.In this paper we first consider the generalized inference for the case of two continuous time Gaussian processes. Later, the results will be extended for suchN processes. In both cases, for the testing problem, the main step is constructing ageneralized test process and analyzing the associated generalized p value, proving some linear invariance properties.With respect to the construction of generalized confidence regions, one should use a generalized pivotal quantity and use the approach of multiple comparisons as in [4].Finally, in the same line of Zhang [6], we consider the general linear hypothesis testing (GLHT) as a generalization of the MANOVA, adapting the setting and method of this paper.It must be emphasized that all the references above develop these techniques for discrete univariate or multivariate models, whereas here we are concerned with a continuous time model. It is well known that when the underlying phenomenon is in essence continuous, even if it is observed at a sequence of epochsnΔt, different models may be necessary for distinct values of Δt. On the contrary a continuous time model embodies simultaneously all the statistical properties of the time series obtained for each value of Δt.


## 2. Continuous Time Generalized Tests and Confidence Regions

Let{Xt}t∈T be a k-dimensional stochastic process with distribution depending on the unknown parameter ξ=(θ,ζ), θ being the vector of parameters of interest and ζ a nuisance parameter vector. For any random vector Y, Y~ will denote its observed value.For the problem of testing a null hypothesisH0:θ≤θ0 against the alternative H1:θ>θ0, where θ0 is a given vector (the inequalities like θ1≤θ2 should be understood componentwise.), ageneralized test process is defined, following [3], as follows.Definition 1.
A generalized test processWt(X,X~,θ,ζ) is, for each t∈T, a one-dimensional function depending on {Xs}s≤t and its observed value {X~s}s≤t, as well as the parameter value ξ=(θ,ζ), satisfying the following:(1)
the distribution ofWt(X,X~,θ0,ζ) does not depend on ζ, for any fixed X~,(2)
the observed valueWt~=Wt(X~,X~,θ0,ζ) does not depend on ζ,(3)
Pθ{Wt(X,X~,θ,ζ)≥w} is nondecreasing in every component of θ, for any w∈R and any fixed X~ and ζ.Under the above conditions, the generalizedp value is defined as(1)pt=supθ≤θ0⁡PθWtX,X~,θ,ζ≥W~t=Pθ0WtX,X~,θ0,ζ≥W~t.When testing H0:θ=θ0 versus H1:θ≠θ0, condition (3) must be replaced by(3′)
Wt(X,X~,θ,ζ) is stochastically larger under H1 than under H0, for any fixed X~ and ζ.In this case, the generalizedp value is given by(2)pt=Pθ0WtX,X~,θ0,ζ≥W~t.Towards the confidential estimation ofθ, we give the following definition.Definition 2.
A generalized pivotal quantityWt(X,X~,θ,ζ) is, for each t∈T, a one-dimensional function satisfying the following: (1)
the distribution ofWt(X,X~,θ,ζ) does not depend on θ nor ζ,(2)
the observed valueWt~=Wt(X~,X~,θ,ζ) does not depend on ζ.Then, ifc1<c2 are such that (3)Pθc1≤WtX,X~,θ,ζ≤c2=1-γ,{θ∣c1≤Wt~≤c2} is a 100(1-γ)% generalized confidence region for θ.


## 3. Estimation Method

In the previous paper [7] we have considered the confidence estimation of a k-dimensional parameter θ, when observing a continuous time p-dimensional Gaussian process Xt, with covariances function B(t,t′)=σ2K(t,t′) and mean function μ(t)=A(t)θ, where K(t,t′) and A(t) are known matrices, but σ2 and θ are unknown parameters. More concrete assumptions were specified by Ibarrola and Vélez [7] and we will here suppose that they hold for all the considered processes.The estimation method ofθ, described in [7], is based on the estimator(4)θt=Σt-1θ^twithθ^t=∫0tf^t′uXudu,where f^t′ is the k×p-matrix with columns in L2,p[0,t] satisfying the equation(5)Au=∫0tKu,u′+AuA′u′f^tu′du′,∀u≤tand Σt is given by(6)Σt=∫0tf^t′uAudu.As proved in [7], {θt} does not depend on σ2 and constitutes a Gaussian process such that (7)Eθθt=θ,Covθθt,θt′=σ2σt∨t′whereσt=Σt-1-I.Consequently, θt and θt2′-θt1′ are independent if t1′<t2′≤t, while (8)Eθθt′-θθt2-θt1′=σ2σt2-σt1ift′≤t1<t2.Moreover, θt is a mean square consistent estimator of θ, since all the eigenvalues of the covariance matrix σt converge to 0.In order to estimateσ2, if t1<t2 are arbitrarily chosen so that σt1-σt2 is nonsingular, we can consider the random variable (9)η=θt2-θt1′σt1-σt2-1θt2-θt1such that η/σ2 has a χk2 distribution.


## 4. The Behrens-Fisher Problem

We first consider the case of two independent Gaussian stochastic processes,{Xt=Xt1,…,Xtp′} and {Ys=Ys1,…,Ysq′}, of dimensions p and q, respectively, and with similar characteristics. More precisely,(10)Eθ1Xt=A1tθ1,CovXt,Xt′=σ12K1t,t′,Eθ2Ys=A2sθ2,CovYs,Ys′=σ22K2s,s′,where θ1,θ2∈Rk and σ12, σ22 are unknown parameters, while A1(t), A2(s) and K1(t,t′), K2(s,s′) are given matrices of appropriate dimensions.We will focus on the Behrens-Fisher type problem of comparing the parametersθ1 and θ2 and, more concretely, our aim is to make inferences about θ1-θ2, based on the progressive observation of both processes Xt and Ys.The progressive estimatorsθt(1) and θt(2) of θ1 and θ2 can be constructed according to (4), (5), and (6). For i=1,2, Σt(i) and σt(i) will denote the characteristics defined in (6), (7) for {Xt} and {Ys}, respectively. We thus obtain the unbiased estimator of δ=θ1-θ2: (11)Tt,s=θt1-θs2which is normally distributed, satisfies E[Tt,s]=δ, and, since {θt(1)} and {θs(2)} are independent, has covariance (12)CovTt,s,Tt′,s′=σ12σt∨t′1+σ22σs∨s′2.In order to estimate σ12 and σ22, we can take (13)η1=θt21-θt11′σt11-σt21-1θt21-θt11,η2=θs22-θs12′σs12-σs22-1θs22-θs12,where t1<t2 and s1<s2 are such that σt1(1)-σt2(1) and σs1(2)-σs2(2) are nonsingular. In this way η1/σ12 and η2/σ22 are independent, χk2-distributed random variables, independent of Tt,s.


### 4.1. Generalizedp Value for the Behrens-Fisher Problem

We consider the problem of testingH0:δ=0 against the alternative H1:δ≠0 and we look for a generalized test process Wt,s(θt(1),θs(2),θ~t(1),θ~s(2),δ,σ12,σ22).The covariance matrix ofTt,s is(14)Ut,s=σ12σt1+σ22σs2and can be estimated by means of (15)Vt,s=η1σt1+η2σs2.Recall that T~t,s, η~i, and V~t,s represent the observed values, obtained when θt(1) and θs(2) are replaced with θ~t(1) and θ~s(2), and let us define (16)dt,s=V~t,s-1/2Tt,s,Zt,s=V~t,s-1/2Ut,sV~t,s-1/2-1/2dt,s.Under H0, dt,s has distribution Nk(0,V~t,s-1/2Ut,sV~t,s-1/2), so that the distribution of Zt,s is Nk(0,I). Moreover (17)Rt,s=V~t,s-1/2σ12σt1η~1η1+σ22σs2η~2η2V~t,s-1/2does not depend on δ and its distribution is independent of the parameters, since u1=η1/σ12 and u2=η2/σ22 are independent, χk2 distributed random variables, independent of Zt,s. The observed value of Rt,s is R~t,s=V~t,s-1/2Ut,sV~t,s-1/2. Finally,(18)Wt,s=Zt,s′Rt,sZt,s=Zt,s′V~t,s-1/2η~1u1σt1+η~2u2σs2V~t,s-1/2Zt,sis a one-dimensional random variable with distribution not depending on (σ12,σ22), whose observed value is W~t,s=T~t,s′V~t,s-1T~t,s, that neither depends on (σ12,σ22).UnderH1, when δ≠0, as Tt,s has distribution Nk(δ,Ut,s), the distribution of Zt,s is Nk(δ-,I), with δ-=(V~t,s-1/2Ut,sV~t,s-1/2)-1/2V~t,s-1/2δ. Note that, given u1 and u2, since Rt,s is a positive definite matrix, Wt,s is stochastically larger under H1 than under H0. So, since the conditions (1), (2), and (3) of Definition 1 are verified, we have proved the following result.Proposition 3.
W
t
,
s is a generalized test process for testing H0:δ=0 against H1:δ≠0.In order to simplify the expression ofWt,s, we will put (19)M1=M1t,s=V~t,s-1/2η~1σt1V~t,s-1/2,M2=M2t,s=V~t,s-1/2η~2σs2V~t,s-1/2which are positive definite matrices such that M1+M2=I. Then (20)Wt,s=Zt,s′M1u1+M2u2Zt,s=12kZ-t,s′M1β+I-M11-βZ-t,s,where, under H0, Z-t,s=2kZt,s/(u1+u2)1/2 has a k-multivariate Student’s t-distribution with 2k degrees of freedom and β=u1/(u1+u2) is a random variable with distributionBeta(k/2,k/2), which is independent of Z-t,s.Thus the generalizedp value of the given test: (21)pt,sM1=Pδ=0Wt,s≥W~t,s=Pδ=0Z-t,s′M1β+I-M11-βZ-t,s≥2kW~t,smay be calculated once M1 is observed.


### 4.2. Invariance Properties of the Generalizedp Value

Let us assume that the basic processes are transformed by means of(22)Xt⋆=Q1Xt,Ys⋆=Q2Ys,where Q1, Q2 are nonsingular square matrices of order p and q, respectively. The characteristics of the transformed processes are (23)Eθ1Xt⋆=A1⋆tθ1,Eθ2Yt⋆=A2⋆sθ2,CovXt⋆,Xt′⋆=σ12K1⋆t,t′,CovYs⋆,Ys′⋆=σ22K2⋆s,s′,where Ai⋆(·)=QiAi(·) and Ki⋆(·,·)=QiKi·,·Qi′.ForXt⋆, the solution f^t⋆(u) of (5) with the new characteristics will be related to f^t(u) by means of f^t⋆(u)=(Q1′)-1f^t(u). Therefore, according to (4) and (6), we get (24)θ^t1⋆=θ^t1,Σt1⋆=Σt1,so that θt(1)⋆=θt(1), σt(1)⋆=σt(1), and η1⋆=η1.The same results hold forYs⋆ yielding Tt,s⋆=Tt,s, as well as Wt,s⋆=Wt,s. In this sense the generalized test process and the corresponding generalized p value are invariant under the proposed transformations.


### 4.3. Generalized Confidence Region forδ

For any value of the unknown parameterδ, the difference Tt,sδ=Tt,s-δ has distribution Nk(0,Ut,s) and Zt,sδ=(V~t,s-1/2Ut,sV~t,s-1/2)-1/2V~t,s-1/2Tt,sδ is Nk(0,I). Hence (25)Wt,sδ=Zt,sδ′Rt,sZt,sδhas the same distribution as Wt,s under H0:δ=0, which is independent of all the parameters, while its observed value W~t,sδ does not depend on the nuisance parameters (σ12,σ22). Since conditions (1) and (2) of Definition 2 are accomplished, the following result is established.Proposition 4.
W
t
,
s
δ is a generalized pivotal quantity and {δ∈Rk∣W~t,sδ≤cγ} is a generalized confidence region for δ, whenever P{Wt,sδ≤cγ}=1-γ.According to (20), once M1 is observed, the constant cγ may be determined and the confidence region for δ is the ellipsoid in Rk:(26)R=δ∈Rk∣δ-T~t,s′V~t,s-1δ-T~t,s≤cγwhich is centered at T~t,s and with axes in the direction of each eigenvector of V~t,s of length ±cγλ, where λ is the corresponding eigenvalue. Thus the k-dimensional volume of the confidence region results in(27)πcγk/2V~t,s1/2Γ1+k/2.Simultaneous confidence intervals for thek components of δ can be obtained from the following consequence of the Cauchy-Schwarz inequality: x∈Rksatisfiesx′x≤c2if and only if|y′x|<c|y′y|1/2for ally≠0. Thus, with x=V~t,s-1/2(δ-T~t,s) and y=V~t,s1/2a, the above equivalence allows to express(28)R=δ∈Rk∣a′δ-T~t,s≤cγ1/2a′V~t,sa1/2∀a≠0and therefore(29)R⊂δ∈Rk∣δi-T~t,si≤cγ1/2V~t,si,i1/2∀i=1,…,k.This last set provides simultaneous confidence intervals for the components of δ with confidence level greater than 1-γ.


### 4.4. An Example with Simulation Results

In order to evaluate the performance of the proposed confidence region and confidence intervals, we will analyze a particular situation that allows accomplishing simulation studies.The considered problem is the case of two Wiener processes,Xt and Ys, with (30)EXt=A1taθ1,CovXt,Xt′=σ12K1t∧t′,EYs=A2sbθ2,CovYs,Ys′=σ22K2s∧s′.The estimators θt(i) defined in (4) together with their covariance matrices σt(i) have been determined in [8]: (31)θt1=2a-1at2a-1A1′K1A1-1A1K1-1∫0tua-1dXuwhile (32)σt1-1=a2t2a-12a-1A1′K1-1A1.Similar results hold for θs(2) and σs(2). We will take a=b=1, so that (33)σt1=t-1A1′K1-1A1-1,σt2=s-1A2′K2-1A2-1.Since (A1′K1-1A1)-1 and (A2′K2-1A2)-1 are positive definite symmetric matrices, there exists a nonsingular k×k matrix L that simultaneously diagonalizes them: (34)L′A1′K1-1A1-1L=I,L′A2′K2-1A2-1L=Λ=diag⁡λ1,…,λkand therefore L′V~t,sL=η~1t-1I+η~2s-1Λ.The random variableTt,s⋆=L′Tt,s has distribution Nk(δ⋆,σ12t-1I+σ22s-1Λ) with δ⋆=L′δ and the confidence region (26) may be written as(35)R=δ∈Rk∣δ⋆-T~t,s⋆′η~1t-1I+η~2s-1Λ-1δ⋆-T~t,s⋆≤cγ=L′-1δ⋆∈Rk∣δ⋆-T~t,s⋆′η~1t-1I+η~2s-1Λ-1δ⋆-T~t,s⋆≤cγ.Thus, the basic case corresponds to σt(1)=t-1I and σs(2)=s-1Λ, from which the confidence region for more general cases may be constructed. With this simple choice, the matrix M1 in (19) is the diagonal matrix(36)M1=diag⁡sη~1sη~1+tη~2λ1,…,sη~1sη~1+tη~2λkand (20) may be written as(37)Wt,s=12k∑i=1kZi21βsη~1sη~1+tη~2λi+11-βtη~2λisη~1+tη~2λi.In the following algorithms we suppose that the values of the dimension k and the variance parameters: σ12, σ22, and λ1,…,λk are given. The values of t, s, and γ are also fixed.Algorithm 5.
(
1
) Generate the independent variables u1, u2 with a χk2 distribution.
(
2
) Compute η~1=σ12u1 and η~2=σ22u2, and subsequently, for each i=1,…,k,(38)Ci=sη~1sη~1+tη~2λi,Di=tη~2λisη~1+tη~2λi.
(
3
) Fixing a large number of iterations m, for each j=1,…,m,(a)
generateZj with a k-variate t-distribution with 2k degrees of freedom;(b)
generateβj with aBeta(k/2,k/2) distribution;(c)
compute(39)Wj=12k∑i=1kZj,i2Ciβj+Di1-βj.
(
4
) Determine cγ as the 1-γ percentile of the sample (Wj)j=1m.
The obtained value ofcγ may be validated by generating a new sample (Wj′)j=1m′ in which the proportion 1-γ′ of (Wj′) less than cγ may be estimated. Table 1 shows the results obtained by means of amatlab program, using the data k=10, σ12=2, σ22=4, λi=1/i for i=1,…,10, t=25, s=50, and m=m′=1000.Table 1
Sample percentiles and its validation.
1
-
γ
c
γ
1
-
γ
′
0.95
2.7182
0.97
0.90
2.3249
0.935
0.85
1.8182
0.848The next algorithm is designed to obtain the expected volume of the generalized confidence region and the coverage probability of a given vectorδ=(δ1,…,δk).Algorithm 6.
Given a large number of iterationsm, for each j=1,…,m,(1)
generate ak-dimensional vector T~j with independent components, such that T~j,i has a distribution N(δi,σ12/t+σ22λi/s) for each i=1,…,k;(2)
generateη~1,j and η~2,j as in Algorithm 5;(3)
compute(40)Wj=∑i=1kδi-T~j,i2tsη~1,js+η~2,jλit,where the coefficients are the terms of the diagonal matrix (η~1t-1I+η~2s-1Λ)-1;(4)
use the valuecγ given in Algorithm 5 to compute the estimated coverage probability of δ as the proportion pc(δ) of Wj less than cγ;(5)
according to (27) compute the volume(41)vj=πk/2cγk/2Γ1+k/2∏i=1kη~1,jt-1+η~2,js-1λi1/2;and the estimated expected volume v-=(1/m)∑j=1mvj.
Taking for instanceδ=(1,-1,2,0,3,4,-2.,5,-3,1) and with the remaining data as before, the obtained coverage probability and estimated expected volume are shown in Table 2.
Finally an algorithm may be designed in order to simulate the simultaneous confidence intervals for the components ofδ and to estimate the joint coverage probability.Table 2
Coverage probability and expected volume of the confidence region.
1
-
γ
p
c
(
δ
)
v
-
0.95
0.982
271.85
0.90
0.947
130.45
0.85
0.891
37.30Algorithm 7.
Given a large number of iterationsm, for each j=1,…,m the following hold.(1)
For eachi=1,…,k, compute, according to the results of the previous algorithms, the extremes of the intervals Ij,i=(lj,i,Lj,i), where (42)lj,i=T~j,i-cγ1/2η~1,jt+η~2,jλis1/2,Lj,i=T~j,i+cγ1/2η~1,jt+η~2,jλis1/2.(2)
Compute the estimated joint coverage probability ofδ as the proportion pc(δ) of iterations such that δi∈Ij,i for all i=1,…,k.(3)
Compute(43)li=1m∑j=1mlj,i,Li=1m∑j=1mLj,ithat gives the expected length Li-li of the interval I-i=(li,Li).
With the same data as before and the same vectorδ, the obtained results are shown in Table 3. Let us observe that the coverage probability of the generalized confidence region pc(δ) always exceeds the confidence level and that the coverage probability of the confidence intervals is always very close to 1. Taking alternative parameter values, some other simulations have been made with similar results.Table 3
Confidence intervals and joint coverage probabilities.
1
-
γ
=
0.95
1
-
γ
=
0.90
1
-
γ
=
0.85
I
-
i
L
i
-
l
i
I
-
i
L
i
-
l
i
I
-
i
L
i
-
l
i
(
-
1.29,3.32
)
4.61
(
-
1.09,3.14
)
4.23
(
-
0.88,2.89
)
3.73
(
-
2.79,0.79
)
3.58
(
-
2.64,0.65
)
3.29
(
-
2.46,0.44
)
2.90
(
0.41,3.57
)
3.16
(
0.56,3.46
)
2.90
(
0.72,3.28
)
2.56
(
-
1.47,1.45
)
2.92
(
-
1.34,1.35
)
2.69
(
-
1.19,1.18
)
2.37
(
1.60,4.36
)
2.76
(
1.73,4.28
)
2.55
(
1.87,4.13
)
2.25
(
2.67,5.33
)
2.66
(
2.77,5.22
)
2.45
(
2.93,5.09
)
2.16
(
-
3.29
,
-
0.71
)
2.58
(
-
3.19
,
-
0.81
)
2.38
(
-
3.05
,
-
0.95
)
2.10
(
3.73,6.25
)
2.52
(
3.83,6.16
)
2.33
(
3.99,6.04
)
2.05
(
-
4.24
,
-
1.77
)
2.47
(
-
4.14
,
-
1.86
)
2.28
(
-
4.01
,
-
1.99
)
2.01
(
-
0.21,2.22
)
2.43
(
-
0.12,2.12
)
2.24
(
0.01,1.99
)
1.98
p
c
(
δ
)
=
1
p
c
(
δ
)
=
0.9997
p
c
(
δ
)
=
0.9997


## 4.1. Generalizedp Value for the Behrens-Fisher Problem

We consider the problem of testingH0:δ=0 against the alternative H1:δ≠0 and we look for a generalized test process Wt,s(θt(1),θs(2),θ~t(1),θ~s(2),δ,σ12,σ22).The covariance matrix ofTt,s is(14)Ut,s=σ12σt1+σ22σs2and can be estimated by means of (15)Vt,s=η1σt1+η2σs2.Recall that T~t,s, η~i, and V~t,s represent the observed values, obtained when θt(1) and θs(2) are replaced with θ~t(1) and θ~s(2), and let us define (16)dt,s=V~t,s-1/2Tt,s,Zt,s=V~t,s-1/2Ut,sV~t,s-1/2-1/2dt,s.Under H0, dt,s has distribution Nk(0,V~t,s-1/2Ut,sV~t,s-1/2), so that the distribution of Zt,s is Nk(0,I). Moreover (17)Rt,s=V~t,s-1/2σ12σt1η~1η1+σ22σs2η~2η2V~t,s-1/2does not depend on δ and its distribution is independent of the parameters, since u1=η1/σ12 and u2=η2/σ22 are independent, χk2 distributed random variables, independent of Zt,s. The observed value of Rt,s is R~t,s=V~t,s-1/2Ut,sV~t,s-1/2. Finally,(18)Wt,s=Zt,s′Rt,sZt,s=Zt,s′V~t,s-1/2η~1u1σt1+η~2u2σs2V~t,s-1/2Zt,sis a one-dimensional random variable with distribution not depending on (σ12,σ22), whose observed value is W~t,s=T~t,s′V~t,s-1T~t,s, that neither depends on (σ12,σ22).UnderH1, when δ≠0, as Tt,s has distribution Nk(δ,Ut,s), the distribution of Zt,s is Nk(δ-,I), with δ-=(V~t,s-1/2Ut,sV~t,s-1/2)-1/2V~t,s-1/2δ. Note that, given u1 and u2, since Rt,s is a positive definite matrix, Wt,s is stochastically larger under H1 than under H0. So, since the conditions (1), (2), and (3) of Definition 1 are verified, we have proved the following result.Proposition 3.
W
t
,
s is a generalized test process for testing H0:δ=0 against H1:δ≠0.In order to simplify the expression ofWt,s, we will put (19)M1=M1t,s=V~t,s-1/2η~1σt1V~t,s-1/2,M2=M2t,s=V~t,s-1/2η~2σs2V~t,s-1/2which are positive definite matrices such that M1+M2=I. Then (20)Wt,s=Zt,s′M1u1+M2u2Zt,s=12kZ-t,s′M1β+I-M11-βZ-t,s,where, under H0, Z-t,s=2kZt,s/(u1+u2)1/2 has a k-multivariate Student’s t-distribution with 2k degrees of freedom and β=u1/(u1+u2) is a random variable with distributionBeta(k/2,k/2), which is independent of Z-t,s.Thus the generalizedp value of the given test: (21)pt,sM1=Pδ=0Wt,s≥W~t,s=Pδ=0Z-t,s′M1β+I-M11-βZ-t,s≥2kW~t,smay be calculated once M1 is observed.


## 4.2. Invariance Properties of the Generalizedp Value

Let us assume that the basic processes are transformed by means of(22)Xt⋆=Q1Xt,Ys⋆=Q2Ys,where Q1, Q2 are nonsingular square matrices of order p and q, respectively. The characteristics of the transformed processes are (23)Eθ1Xt⋆=A1⋆tθ1,Eθ2Yt⋆=A2⋆sθ2,CovXt⋆,Xt′⋆=σ12K1⋆t,t′,CovYs⋆,Ys′⋆=σ22K2⋆s,s′,where Ai⋆(·)=QiAi(·) and Ki⋆(·,·)=QiKi·,·Qi′.ForXt⋆, the solution f^t⋆(u) of (5) with the new characteristics will be related to f^t(u) by means of f^t⋆(u)=(Q1′)-1f^t(u). Therefore, according to (4) and (6), we get (24)θ^t1⋆=θ^t1,Σt1⋆=Σt1,so that θt(1)⋆=θt(1), σt(1)⋆=σt(1), and η1⋆=η1.The same results hold forYs⋆ yielding Tt,s⋆=Tt,s, as well as Wt,s⋆=Wt,s. In this sense the generalized test process and the corresponding generalized p value are invariant under the proposed transformations.


## 4.3. Generalized Confidence Region forδ

For any value of the unknown parameterδ, the difference Tt,sδ=Tt,s-δ has distribution Nk(0,Ut,s) and Zt,sδ=(V~t,s-1/2Ut,sV~t,s-1/2)-1/2V~t,s-1/2Tt,sδ is Nk(0,I). Hence (25)Wt,sδ=Zt,sδ′Rt,sZt,sδhas the same distribution as Wt,s under H0:δ=0, which is independent of all the parameters, while its observed value W~t,sδ does not depend on the nuisance parameters (σ12,σ22). Since conditions (1) and (2) of Definition 2 are accomplished, the following result is established.Proposition 4.
W
t
,
s
δ is a generalized pivotal quantity and {δ∈Rk∣W~t,sδ≤cγ} is a generalized confidence region for δ, whenever P{Wt,sδ≤cγ}=1-γ.According to (20), once M1 is observed, the constant cγ may be determined and the confidence region for δ is the ellipsoid in Rk:(26)R=δ∈Rk∣δ-T~t,s′V~t,s-1δ-T~t,s≤cγwhich is centered at T~t,s and with axes in the direction of each eigenvector of V~t,s of length ±cγλ, where λ is the corresponding eigenvalue. Thus the k-dimensional volume of the confidence region results in(27)πcγk/2V~t,s1/2Γ1+k/2.Simultaneous confidence intervals for thek components of δ can be obtained from the following consequence of the Cauchy-Schwarz inequality: x∈Rksatisfiesx′x≤c2if and only if|y′x|<c|y′y|1/2for ally≠0. Thus, with x=V~t,s-1/2(δ-T~t,s) and y=V~t,s1/2a, the above equivalence allows to express(28)R=δ∈Rk∣a′δ-T~t,s≤cγ1/2a′V~t,sa1/2∀a≠0and therefore(29)R⊂δ∈Rk∣δi-T~t,si≤cγ1/2V~t,si,i1/2∀i=1,…,k.This last set provides simultaneous confidence intervals for the components of δ with confidence level greater than 1-γ.


## 4.4. An Example with Simulation Results

In order to evaluate the performance of the proposed confidence region and confidence intervals, we will analyze a particular situation that allows accomplishing simulation studies.The considered problem is the case of two Wiener processes,Xt and Ys, with (30)EXt=A1taθ1,CovXt,Xt′=σ12K1t∧t′,EYs=A2sbθ2,CovYs,Ys′=σ22K2s∧s′.The estimators θt(i) defined in (4) together with their covariance matrices σt(i) have been determined in [8]: (31)θt1=2a-1at2a-1A1′K1A1-1A1K1-1∫0tua-1dXuwhile (32)σt1-1=a2t2a-12a-1A1′K1-1A1.Similar results hold for θs(2) and σs(2). We will take a=b=1, so that (33)σt1=t-1A1′K1-1A1-1,σt2=s-1A2′K2-1A2-1.Since (A1′K1-1A1)-1 and (A2′K2-1A2)-1 are positive definite symmetric matrices, there exists a nonsingular k×k matrix L that simultaneously diagonalizes them: (34)L′A1′K1-1A1-1L=I,L′A2′K2-1A2-1L=Λ=diag⁡λ1,…,λkand therefore L′V~t,sL=η~1t-1I+η~2s-1Λ.The random variableTt,s⋆=L′Tt,s has distribution Nk(δ⋆,σ12t-1I+σ22s-1Λ) with δ⋆=L′δ and the confidence region (26) may be written as(35)R=δ∈Rk∣δ⋆-T~t,s⋆′η~1t-1I+η~2s-1Λ-1δ⋆-T~t,s⋆≤cγ=L′-1δ⋆∈Rk∣δ⋆-T~t,s⋆′η~1t-1I+η~2s-1Λ-1δ⋆-T~t,s⋆≤cγ.Thus, the basic case corresponds to σt(1)=t-1I and σs(2)=s-1Λ, from which the confidence region for more general cases may be constructed. With this simple choice, the matrix M1 in (19) is the diagonal matrix(36)M1=diag⁡sη~1sη~1+tη~2λ1,…,sη~1sη~1+tη~2λkand (20) may be written as(37)Wt,s=12k∑i=1kZi21βsη~1sη~1+tη~2λi+11-βtη~2λisη~1+tη~2λi.In the following algorithms we suppose that the values of the dimension k and the variance parameters: σ12, σ22, and λ1,…,λk are given. The values of t, s, and γ are also fixed.Algorithm 5.
(
1
) Generate the independent variables u1, u2 with a χk2 distribution.
(
2
) Compute η~1=σ12u1 and η~2=σ22u2, and subsequently, for each i=1,…,k,(38)Ci=sη~1sη~1+tη~2λi,Di=tη~2λisη~1+tη~2λi.
(
3
) Fixing a large number of iterations m, for each j=1,…,m,(a)
generateZj with a k-variate t-distribution with 2k degrees of freedom;(b)
generateβj with aBeta(k/2,k/2) distribution;(c)
compute(39)Wj=12k∑i=1kZj,i2Ciβj+Di1-βj.
(
4
) Determine cγ as the 1-γ percentile of the sample (Wj)j=1m.
The obtained value ofcγ may be validated by generating a new sample (Wj′)j=1m′ in which the proportion 1-γ′ of (Wj′) less than cγ may be estimated. Table 1 shows the results obtained by means of amatlab program, using the data k=10, σ12=2, σ22=4, λi=1/i for i=1,…,10, t=25, s=50, and m=m′=1000.Table 1
Sample percentiles and its validation.
1
-
γ
c
γ
1
-
γ
′
0.95
2.7182
0.97
0.90
2.3249
0.935
0.85
1.8182
0.848The next algorithm is designed to obtain the expected volume of the generalized confidence region and the coverage probability of a given vectorδ=(δ1,…,δk).Algorithm 6.
Given a large number of iterationsm, for each j=1,…,m,(1)
generate ak-dimensional vector T~j with independent components, such that T~j,i has a distribution N(δi,σ12/t+σ22λi/s) for each i=1,…,k;(2)
generateη~1,j and η~2,j as in Algorithm 5;(3)
compute(40)Wj=∑i=1kδi-T~j,i2tsη~1,js+η~2,jλit,where the coefficients are the terms of the diagonal matrix (η~1t-1I+η~2s-1Λ)-1;(4)
use the valuecγ given in Algorithm 5 to compute the estimated coverage probability of δ as the proportion pc(δ) of Wj less than cγ;(5)
according to (27) compute the volume(41)vj=πk/2cγk/2Γ1+k/2∏i=1kη~1,jt-1+η~2,js-1λi1/2;and the estimated expected volume v-=(1/m)∑j=1mvj.
Taking for instanceδ=(1,-1,2,0,3,4,-2.,5,-3,1) and with the remaining data as before, the obtained coverage probability and estimated expected volume are shown in Table 2.
Finally an algorithm may be designed in order to simulate the simultaneous confidence intervals for the components ofδ and to estimate the joint coverage probability.Table 2
Coverage probability and expected volume of the confidence region.
1
-
γ
p
c
(
δ
)
v
-
0.95
0.982
271.85
0.90
0.947
130.45
0.85
0.891
37.30Algorithm 7.
Given a large number of iterationsm, for each j=1,…,m the following hold.(1)
For eachi=1,…,k, compute, according to the results of the previous algorithms, the extremes of the intervals Ij,i=(lj,i,Lj,i), where (42)lj,i=T~j,i-cγ1/2η~1,jt+η~2,jλis1/2,Lj,i=T~j,i+cγ1/2η~1,jt+η~2,jλis1/2.(2)
Compute the estimated joint coverage probability ofδ as the proportion pc(δ) of iterations such that δi∈Ij,i for all i=1,…,k.(3)
Compute(43)li=1m∑j=1mlj,i,Li=1m∑j=1mLj,ithat gives the expected length Li-li of the interval I-i=(li,Li).
With the same data as before and the same vectorδ, the obtained results are shown in Table 3. Let us observe that the coverage probability of the generalized confidence region pc(δ) always exceeds the confidence level and that the coverage probability of the confidence intervals is always very close to 1. Taking alternative parameter values, some other simulations have been made with similar results.Table 3
Confidence intervals and joint coverage probabilities.
1
-
γ
=
0.95
1
-
γ
=
0.90
1
-
γ
=
0.85
I
-
i
L
i
-
l
i
I
-
i
L
i
-
l
i
I
-
i
L
i
-
l
i
(
-
1.29,3.32
)
4.61
(
-
1.09,3.14
)
4.23
(
-
0.88,2.89
)
3.73
(
-
2.79,0.79
)
3.58
(
-
2.64,0.65
)
3.29
(
-
2.46,0.44
)
2.90
(
0.41,3.57
)
3.16
(
0.56,3.46
)
2.90
(
0.72,3.28
)
2.56
(
-
1.47,1.45
)
2.92
(
-
1.34,1.35
)
2.69
(
-
1.19,1.18
)
2.37
(
1.60,4.36
)
2.76
(
1.73,4.28
)
2.55
(
1.87,4.13
)
2.25
(
2.67,5.33
)
2.66
(
2.77,5.22
)
2.45
(
2.93,5.09
)
2.16
(
-
3.29
,
-
0.71
)
2.58
(
-
3.19
,
-
0.81
)
2.38
(
-
3.05
,
-
0.95
)
2.10
(
3.73,6.25
)
2.52
(
3.83,6.16
)
2.33
(
3.99,6.04
)
2.05
(
-
4.24
,
-
1.77
)
2.47
(
-
4.14
,
-
1.86
)
2.28
(
-
4.01
,
-
1.99
)
2.01
(
-
0.21,2.22
)
2.43
(
-
0.12,2.12
)
2.24
(
0.01,1.99
)
1.98
p
c
(
δ
)
=
1
p
c
(
δ
)
=
0.9997
p
c
(
δ
)
=
0.9997


## 5. Inferences about the Vector Means of Several Independent Gaussian Processes

As a generalization we now consider the case ofN independent multivariate Gaussian processes Xt1(1),Xt2(2),…,XtN(N) with dimensions p1,p2,…,pN and time parameters t1,t2,…,tN varying in some interval T. We assume, for each i=1,…,N, the following: (44)EθiXtii=Aitiθi,CovXtii,Xti′i=σi2Kti,ti′,where θ1,…,θN∈Rk and σ12,…,σN2∈R are unknown parameters.


### 5.1. Testing the Equality of Means

We first consider a test of the null hypothesisH0:θ1=θ2=⋯=θN, or H0:δ=0 with δi=θi-θN for i=1,2,…,N-1.According to the method of Section3, we may consider the estimators θti(i) of the parameters θi, whose distribution is Nk(θi,σi2σti(i)), and also the differences Tti,tN(i)=θti(i)-θtN(N) for i=1,2,…,N-1 with distributions Nk(δi,σi2σti(i)+σN2σtN(N)).The(N-1)k-dimensional vector (45)Tt′=Tt1,…,tN′=Tt1,tN1,Tt2,tN2,…,TtN-1,tNN-1will have a multivariate normal distribution with vector mean δ and covariance matrix (46)
U
t
=
σ
1
2
σ
t
1
1
σ
2
2
σ
t
2
2
⋱
σ
N
-
1
2
σ
t
N
-
1
N
-
1
(47)
+
σ
N
2
σ
t
N
N
σ
N
2
σ
t
N
N
…
σ
N
2
σ
t
N
N
σ
N
2
σ
t
N
N
σ
N
2
σ
t
N
N
…
σ
N
2
σ
t
N
N
⋮
⋮
⋱
⋮
σ
N
2
σ
t
N
N
σ
N
2
σ
t
N
N
…
σ
N
2
σ
t
N
N
(48)
=
diag
⁡
σ
1
2
σ
t
1
1
,
…
,
σ
N
-
1
2
σ
t
N
-
1
N
-
1
+
J
σ
N
2
σ
t
N
N
J
′
,where J is the Kronecker product 1N-1⊗Ik (composed by a column of N-1 copies of the identity matrix Ik).For eachi=1,…,N, given ti,1<ti,2<ti such that σti,1(i)-σti,2(i) is nonsingular, let (49)ηi=θti,2i-θti,1i′σti,1i-σti,2i-1θti,2i-θti,1iwhich are independent random variables, independent also of Tt and such that ui=ηi/σi2 has χk2 distribution.The covariance matrixUt may be estimated by means of (50)Vt=diag⁡η1σt11,…,ηN-1σtN-1N-1+JηNσtNNJ′with observed value (51)V~t=diag⁡η~1σt11,…,η~N-1σtN-11+Jη~NσtNNJ′.As in (17), the matrix (52)Rt=V~t-1/2diag⁡σ12σt11η~1η1,…,σN-12σtN-1N-1η~N-1ηN-1+JσN2σtNNη~NηNJ′V~t-1/2=V~t-1/2diag⁡σt11η~1u1,…,σtN-1N-1η~N-1uN-1+JσtNNη~NuNJ′V~t-1/2does not depend on any parameter and is independent of Tt; moreover its observed value is (53)R~t=V~t-1/2UtV~t-1/2.Let us define (54)dt=V~t-1/2Tt,Zt=R~t-1/2dtthat have, under H0, distributions N(N-1)k(0,R~t) and N(N-1)k(0,I), respectively. So that, finally, (55)Wt=Zt′RtZtand  its  observed  valueW~t=T~t′V~t-1T~tare scalars not depending on the parameters and the distribution of Wt neither depends on (σ12,…,σN2). Moreover, Wt is a positive definite quadratic form in Zt whose distribution under H1 is N(N-1)k(δ-,I) with δ-=(V~t-1/2UtV~t-1/2)-1/2V~t-1/2δ. Hence Wt is stochastically larger under H1 than under H0 and we get the following.Proposition 8.
W
t is a generalized test process for testing H0:δ=0 against H1:δ≠0.An alternative expression will be more suitable for the calculation of thep values. For each i=1,…,N-1, let Gi be the (N-1)k×(N-1)k square matrix obtained by replacing in the matrix (46) the i block with η~iσti(i) while the remaining are replaced with 0 (of order k×k); that means (56)Gi=Giti=diag⁡0,…,0,η~iσtii,0,…,0.Now (57)Mi=V~t-1/2GiV~t-1/2,MN=V~t-1/2JηN~σtNNJ′V~t-1/2satisfy ∑i=1N-1Mi+MN=I and allow expressing (58)Rt=∑i=1N-1Miui+MNuN=∑i=1N-1Miui+I-∑i=1N-1MiuN.Consequently (59)Wt=Zt′∑i=1N-1Miui+I-∑i=1N-1MiuNZt=1NkZ-t′∑i=1N-1Miβi+I-∑i=1N-1Mi1-∑i=1N-1βiZ-t,where (60)Z-t=Nk∑i=1Nui1/2Zthas a (N-1)k-multivariate Student t-distribution, with Nk degrees of freedom, while βi=ui/(∑i=1N-1ui) is such that (β1,…,βN-1) has a Dirichlet distribution with parameters (k/2,…,k/2) and is independent of Z-t.After the observation ofM1,…,MN-1, the generalized p value of the given test is (61)ptM1,…,MN-1=Pδ=01NkZ-t′∑i=1N-1Miβi+I-∑i=1N-1Mi1-∑i=1N-1βiZ-t≥Wt~.As in Section4.2, an analogous property of invariance of the test statistic Wt may be stated with respect to the transformations of each process (62)Xtii⋆=QiXtiifor nonsingular matrices Qi.


### 5.2. Confidence Region forδ

For any value of the unknown parameterδ, the difference Ttδ=Tt-δ has distribution N(N-1)k(0,Ut) and Ztδ=(V~t-1/2UtV~t-1/2)-1/2V~t-1/2Ttδ is N(N-1)k(0,I). Hence (63)Wtδ=Ztδ′RtZtδhas the same distribution as Wt under H0:δ=0, which is independent of all the parameters, while its observed value W~tδ does not depend on the nuisance parameters. The following proposition is thus established.Proposition 9.
W
t
δ is a generalized pivotal quantity and a generalized confidence region for δ is given by {δ∈R(N-1)k∣W~tδ≤cγ}, if P{Wtδ≤cγ}=1-γ.As observed in Section4.3, the just obtained confidence region may be written as (64)R=δ∈RN-1k∣a′T~t-δ≤cγ1/2a′V~ta1/2∀a∈RN-1k,a≠0.For j≤N-1 and i≤k, let aj,i represent the vector in R(N-1)k whose only nonzero term is 1 in the (j-1)k+i position, whereas aj,m,i=aj,i-am,i for j<m≤N-1. Then (65)aj,i′δ=θij-θiN,aj,i′T~t=θ~tjj-θ~tNNi,aj,m,i′δ=θij-θim,aj,m,i′T~t=θ~tjj-θ~tmmi. And from (51) we get (66)aj,i′V~taj,i=η~jσtjji,i+η~NσtNNi,i,aj,m,i′V~taj,m,i=η~jσtjji,i+η~mσtmmi,i.Therefore, as in Section 4.3, R is contained in the intersection of all the intervals: (67)θj-θmi-θ~tjj-θ~tmmi≤cγ1/2η~jσtjj+η~mσtmmi,ifor 1≤j<m≤N and 1≤i≤k. This gives a set of simultaneous confidence intervals for all the components of the differences between the mean parameters of the N processes.


### 5.3. General Linear Hypotheses

When the null hypothesisH0:θ1=θ2=⋯=θN is rejected, one can focus on testing a linear relation between the mean parameters. That is, if θ′=(θ1′,…,θN′), L is an r×Nk matrix, and l is a vector in Rr, the objective is to test the null hypothesis H0′:Lθ=l against the alternative H1′:Lθ≠l.We denoteθt′=(θt11′,…,θtNN′) which has distribution NNk(θ,Ut), where Ut=diag⁡(σ12σt1(1),…,σN2σtN(N)). Now, Tt=Lθt-l has Nr(Lθ-l,LUtL′) distribution, whose mean vector is 0 under H0′. Moreover Ut may be estimated by (68)Vt=Ldiag⁡η1σt11,…,ηNσtNNL′which has observed value V~t=Ldiag⁡(η~1σt1(1),…,η~NσtN(N))L′. Observing that, under H0′, (69)Dt=V~t-1/2Tthas distribution Nr(0,V~t-1/2LUtL′V~t-1/2), we get that (70)Zt=V~t-1/2LUtL′V~t-1/2-1/2Dthas Nr(0,I) distribution. Therefore, if (71)Rt=V~t-1/2Ldiag⁡σ12σt11η~1η1,…,σN2σtNNη~NηNL′V~t-1/2,then (72)Wt=Zt′RtZtis a one-dimensional random variable with distribution, under H0′, independent of the parameters and whose observed value, given by W~t=Tt′V~tTt, does not depend on the parameters. Since Wt is a positive quadratic form in the normal vector Zt whose mean under H1 is different from 0, Wt is stochastically larger under H1 than under H0, so that the following proposition is proved.Proposition 10.
W
t is a generalized test process for the contrast of H0′:Lθ=l against H1′:Lθ≠l.With the same matrixGi defined in (56), one can put (73)Mi=Miti=V~t-1/2LGiL′V~t-1/2that satisfy ∑i=1NMi=I. Since (74)Rt=∑i=1NMiui,where u1,…,uN are mutually independent, χk2 distributed and independent of Zt, we get (75)Wt=1NkZ-t′∑i=1N-1Miβi+I-∑i=1N-1Mi1-∑i=1N-1βiZ-twith Z-t, given by (60), distributed according to a (N-1)k multivariate Student t distribution and βi=ui/(∑i=1N-1ui) such that (β1,…,βN-1) has a Dirichlet distribution with parameters (k/2,…,k/2) and is independent of Z-t. Thus, the generalized p value of the given test can be calculated as in (61), although the expressions of Mi are now different.


## 5.1. Testing the Equality of Means

We first consider a test of the null hypothesisH0:θ1=θ2=⋯=θN, or H0:δ=0 with δi=θi-θN for i=1,2,…,N-1.According to the method of Section3, we may consider the estimators θti(i) of the parameters θi, whose distribution is Nk(θi,σi2σti(i)), and also the differences Tti,tN(i)=θti(i)-θtN(N) for i=1,2,…,N-1 with distributions Nk(δi,σi2σti(i)+σN2σtN(N)).The(N-1)k-dimensional vector (45)Tt′=Tt1,…,tN′=Tt1,tN1,Tt2,tN2,…,TtN-1,tNN-1will have a multivariate normal distribution with vector mean δ and covariance matrix (46)
U
t
=
σ
1
2
σ
t
1
1
σ
2
2
σ
t
2
2
⋱
σ
N
-
1
2
σ
t
N
-
1
N
-
1
(47)
+
σ
N
2
σ
t
N
N
σ
N
2
σ
t
N
N
…
σ
N
2
σ
t
N
N
σ
N
2
σ
t
N
N
σ
N
2
σ
t
N
N
…
σ
N
2
σ
t
N
N
⋮
⋮
⋱
⋮
σ
N
2
σ
t
N
N
σ
N
2
σ
t
N
N
…
σ
N
2
σ
t
N
N
(48)
=
diag
⁡
σ
1
2
σ
t
1
1
,
…
,
σ
N
-
1
2
σ
t
N
-
1
N
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1
+
J
σ
N
2
σ
t
N
N
J
′
,where J is the Kronecker product 1N-1⊗Ik (composed by a column of N-1 copies of the identity matrix Ik).For eachi=1,…,N, given ti,1<ti,2<ti such that σti,1(i)-σti,2(i) is nonsingular, let (49)ηi=θti,2i-θti,1i′σti,1i-σti,2i-1θti,2i-θti,1iwhich are independent random variables, independent also of Tt and such that ui=ηi/σi2 has χk2 distribution.The covariance matrixUt may be estimated by means of (50)Vt=diag⁡η1σt11,…,ηN-1σtN-1N-1+JηNσtNNJ′with observed value (51)V~t=diag⁡η~1σt11,…,η~N-1σtN-11+Jη~NσtNNJ′.As in (17), the matrix (52)Rt=V~t-1/2diag⁡σ12σt11η~1η1,…,σN-12σtN-1N-1η~N-1ηN-1+JσN2σtNNη~NηNJ′V~t-1/2=V~t-1/2diag⁡σt11η~1u1,…,σtN-1N-1η~N-1uN-1+JσtNNη~NuNJ′V~t-1/2does not depend on any parameter and is independent of Tt; moreover its observed value is (53)R~t=V~t-1/2UtV~t-1/2.Let us define (54)dt=V~t-1/2Tt,Zt=R~t-1/2dtthat have, under H0, distributions N(N-1)k(0,R~t) and N(N-1)k(0,I), respectively. So that, finally, (55)Wt=Zt′RtZtand  its  observed  valueW~t=T~t′V~t-1T~tare scalars not depending on the parameters and the distribution of Wt neither depends on (σ12,…,σN2). Moreover, Wt is a positive definite quadratic form in Zt whose distribution under H1 is N(N-1)k(δ-,I) with δ-=(V~t-1/2UtV~t-1/2)-1/2V~t-1/2δ. Hence Wt is stochastically larger under H1 than under H0 and we get the following.Proposition 8.
W
t is a generalized test process for testing H0:δ=0 against H1:δ≠0.An alternative expression will be more suitable for the calculation of thep values. For each i=1,…,N-1, let Gi be the (N-1)k×(N-1)k square matrix obtained by replacing in the matrix (46) the i block with η~iσti(i) while the remaining are replaced with 0 (of order k×k); that means (56)Gi=Giti=diag⁡0,…,0,η~iσtii,0,…,0.Now (57)Mi=V~t-1/2GiV~t-1/2,MN=V~t-1/2JηN~σtNNJ′V~t-1/2satisfy ∑i=1N-1Mi+MN=I and allow expressing (58)Rt=∑i=1N-1Miui+MNuN=∑i=1N-1Miui+I-∑i=1N-1MiuN.Consequently (59)Wt=Zt′∑i=1N-1Miui+I-∑i=1N-1MiuNZt=1NkZ-t′∑i=1N-1Miβi+I-∑i=1N-1Mi1-∑i=1N-1βiZ-t,where (60)Z-t=Nk∑i=1Nui1/2Zthas a (N-1)k-multivariate Student t-distribution, with Nk degrees of freedom, while βi=ui/(∑i=1N-1ui) is such that (β1,…,βN-1) has a Dirichlet distribution with parameters (k/2,…,k/2) and is independent of Z-t.After the observation ofM1,…,MN-1, the generalized p value of the given test is (61)ptM1,…,MN-1=Pδ=01NkZ-t′∑i=1N-1Miβi+I-∑i=1N-1Mi1-∑i=1N-1βiZ-t≥Wt~.As in Section4.2, an analogous property of invariance of the test statistic Wt may be stated with respect to the transformations of each process (62)Xtii⋆=QiXtiifor nonsingular matrices Qi.


## 5.2. Confidence Region forδ

For any value of the unknown parameterδ, the difference Ttδ=Tt-δ has distribution N(N-1)k(0,Ut) and Ztδ=(V~t-1/2UtV~t-1/2)-1/2V~t-1/2Ttδ is N(N-1)k(0,I). Hence (63)Wtδ=Ztδ′RtZtδhas the same distribution as Wt under H0:δ=0, which is independent of all the parameters, while its observed value W~tδ does not depend on the nuisance parameters. The following proposition is thus established.Proposition 9.
W
t
δ is a generalized pivotal quantity and a generalized confidence region for δ is given by {δ∈R(N-1)k∣W~tδ≤cγ}, if P{Wtδ≤cγ}=1-γ.As observed in Section4.3, the just obtained confidence region may be written as (64)R=δ∈RN-1k∣a′T~t-δ≤cγ1/2a′V~ta1/2∀a∈RN-1k,a≠0.For j≤N-1 and i≤k, let aj,i represent the vector in R(N-1)k whose only nonzero term is 1 in the (j-1)k+i position, whereas aj,m,i=aj,i-am,i for j<m≤N-1. Then (65)aj,i′δ=θij-θiN,aj,i′T~t=θ~tjj-θ~tNNi,aj,m,i′δ=θij-θim,aj,m,i′T~t=θ~tjj-θ~tmmi. And from (51) we get (66)aj,i′V~taj,i=η~jσtjji,i+η~NσtNNi,i,aj,m,i′V~taj,m,i=η~jσtjji,i+η~mσtmmi,i.Therefore, as in Section 4.3, R is contained in the intersection of all the intervals: (67)θj-θmi-θ~tjj-θ~tmmi≤cγ1/2η~jσtjj+η~mσtmmi,ifor 1≤j<m≤N and 1≤i≤k. This gives a set of simultaneous confidence intervals for all the components of the differences between the mean parameters of the N processes.


## 5.3. General Linear Hypotheses

When the null hypothesisH0:θ1=θ2=⋯=θN is rejected, one can focus on testing a linear relation between the mean parameters. That is, if θ′=(θ1′,…,θN′), L is an r×Nk matrix, and l is a vector in Rr, the objective is to test the null hypothesis H0′:Lθ=l against the alternative H1′:Lθ≠l.We denoteθt′=(θt11′,…,θtNN′) which has distribution NNk(θ,Ut), where Ut=diag⁡(σ12σt1(1),…,σN2σtN(N)). Now, Tt=Lθt-l has Nr(Lθ-l,LUtL′) distribution, whose mean vector is 0 under H0′. Moreover Ut may be estimated by (68)Vt=Ldiag⁡η1σt11,…,ηNσtNNL′which has observed value V~t=Ldiag⁡(η~1σt1(1),…,η~NσtN(N))L′. Observing that, under H0′, (69)Dt=V~t-1/2Tthas distribution Nr(0,V~t-1/2LUtL′V~t-1/2), we get that (70)Zt=V~t-1/2LUtL′V~t-1/2-1/2Dthas Nr(0,I) distribution. Therefore, if (71)Rt=V~t-1/2Ldiag⁡σ12σt11η~1η1,…,σN2σtNNη~NηNL′V~t-1/2,then (72)Wt=Zt′RtZtis a one-dimensional random variable with distribution, under H0′, independent of the parameters and whose observed value, given by W~t=Tt′V~tTt, does not depend on the parameters. Since Wt is a positive quadratic form in the normal vector Zt whose mean under H1 is different from 0, Wt is stochastically larger under H1 than under H0, so that the following proposition is proved.Proposition 10.
W
t is a generalized test process for the contrast of H0′:Lθ=l against H1′:Lθ≠l.With the same matrixGi defined in (56), one can put (73)Mi=Miti=V~t-1/2LGiL′V~t-1/2that satisfy ∑i=1NMi=I. Since (74)Rt=∑i=1NMiui,where u1,…,uN are mutually independent, χk2 distributed and independent of Zt, we get (75)Wt=1NkZ-t′∑i=1N-1Miβi+I-∑i=1N-1Mi1-∑i=1N-1βiZ-twith Z-t, given by (60), distributed according to a (N-1)k multivariate Student t distribution and βi=ui/(∑i=1N-1ui) such that (β1,…,βN-1) has a Dirichlet distribution with parameters (k/2,…,k/2) and is independent of Z-t. Thus, the generalized p value of the given test can be calculated as in (61), although the expressions of Mi are now different.



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*Source: 479762-2015-10-29.xml*