Generalized Inferences about the Mean Vector of Several Multivariate Gaussian Processes

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 parameter θ 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.


Introduction
The generalized  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  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  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  populations with heteroscedastic variances.They provided a new generalized  value procedure for testing the equality of means, assuming that the variables are univariate and normally distributed.Numerical results show that their generalized  value test works better than a generalized -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 such  processes.In both cases, for the testing problem, the main step is constructing a generalized test process and analyzing the associated generalized  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.

Journal of Probability and Statistics
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 epochs Δ, different models may be necessary for distinct values of Δ.On the contrary a continuous time model embodies simultaneously all the statistical properties of the time series obtained for each value of Δ.

Continuous Time Generalized Tests and Confidence Regions
Let {  } ∈ be a -dimensional stochastic process with distribution depending on the unknown parameter  = (, ),  being the vector of parameters of interest and  a nuisance parameter vector.For any random vector , Ỹ will denote its observed value.
For the problem of testing a null hypothesis  0 :  ≤  0 against the alternative  1 :  >  0 , where  0 is a given vector (the inequalities like  1 ≤  2 should be understood componentwise.), a generalized test process is defined, following [3], as follows.
Definition 1.A generalized test process   (, X, , ) is, for each  ∈ , a one-dimensional function depending on {  } ≤ and its observed value { X } ≤ , as well as the parameter value  = (, ), satisfying the following: (1) the distribution of   (, X,  0 , ) does not depend on , for any fixed X, (2) the observed value W =   ( X, X,  0 , ) does not depend on , (3)   {  (, X, , ) ≥ } is nondecreasing in every component of , for any  ∈ R and any fixed X and .
Under the above conditions, the generalized  value is defined as When testing  0 :  =  0 versus  1 :  ̸ =  0 , condition (3) must be replaced by (3  )   (, X, , ) is stochastically larger under  1 than under  0 , for any fixed X and .
In this case, the generalized  value is given by Towards the confidential estimation of , we give the following definition.Definition 2. A generalized pivotal quantity   (, X, , ) is, for each  ∈ , a one-dimensional function satisfying the following: (1) the distribution of   (, X, , ) does not depend on  nor , (2) the observed value W =   ( X, X, , ) does not depend on .

Estimation Method
In the previous paper [7] we have considered the confidence estimation of a -dimensional parameter , when observing a continuous time -dimensional Gaussian process   , with covariances function (,   ) =  2 (,   ) and mean function () = (), where (,   ) and () are known matrices, but  2 and  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 of , described in [7], is based on the estimator where f  is the  × -matrix with columns in  2, [0, ] satisfying the equation and Σ  is given by As proved in [7], {  } does not depend on  2 and constitutes a Gaussian process such that Consequently,   and Moreover,   is a mean square consistent estimator of , since all the eigenvalues of the covariance matrix   converge to 0.
In order to estimate  2 , if  1 <  2 are arbitrarily chosen so that   1 −   2 is nonsingular, we can consider the random variable such that / 2 has a  2  distribution.
We will focus on the Behrens-Fisher type problem of comparing the parameters  1 and  2 and, more concretely, our aim is to make inferences about  1 −  2 , based on the progressive observation of both processes   and   .
The covariance matrix of  , is and can be estimated by means of Recall that T, , η , and Ṽ, represent the observed values, obtained when  (1)   and  (2)   are replaced with θ(1)  and θ(2)  , and let us define Under  0 ,  , has distribution does not depend on  and its distribution is independent of the parameters, since is a one-dimensional random variable with distribution not depending on ( In order to simplify the expression of  , , we will put which are positive definite matrices such that  1 +  2 = .Then where, under  0 ,  , = √ 2 , /( 1 +  2 ) where  1 ,  2 are nonsingular square matrices of order  and , respectively.The characteristics of the transformed processes are where  ⋆  (⋅) =     (⋅) and  ⋆  (⋅, ⋅) =     (⋅, ⋅)   .For  ⋆  , the solution f⋆  () of ( 5) with the new characteristics will be related to f () by means of f⋆  () = (  1 ) −1 f ().Therefore, according to (4) and ( 6), we get so that  (1)⋆  =  (1)   , has the same distribution as  , under  0 :  = 0, which is independent of all the parameters, while its observed value W , does not depend on the nuisance parameters ( According to (20), once  1 is observed, the constant   may be determined and the confidence region for  is the ellipsoid in R  : which is centered at T, and with axes in the direction of each eigenvector of Ṽ, of length ±√  , where  is the corresponding eigenvalue.Thus the -dimensional volume of the confidence region results in Simultaneous confidence intervals for the  components of  can be obtained from the following consequence of the Cauchy-Schwarz inequality:  ∈ R  satisfies    ≤  2 if and only if |  | < |  | 1/2 for all  ̸ = 0. Thus, with  = Ṽ−1/2 , (− T, ) and  = Ṽ1/2 , , the above equivalence allows to express This last set provides simultaneous confidence intervals for the components of  with confidence level greater than 1 − .

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,   and   , with The estimators  ()  defined in (4) together with their covariance matrices  ()   have been determined in [8]: Similar results hold for  (2)   and  (2)   .We will take  =  = 1, so that Since (  1  −1 1  1 ) −1 and (  2  −1 2  2 ) −1 are positive definite symmetric matrices, there exists a nonsingular  ×  matrix  that simultaneously diagonalizes them: and therefore   Ṽ,  = η1  −1  + η2  −1 Λ.
The random variable  ⋆ , =    , has distribution   ( ⋆ ,  2  1  −1  +  2 2  −1 Λ) with  ⋆ =    and the confidence region (26) may be written as Thus, the basic case corresponds to  (1)    =  −1  and  (2)    =  −1 Λ, from which the confidence region for more general cases may be constructed.With this simple choice, the matrix  1 in ( 19) is the diagonal matrix and (20) may be written as ) . (37) In the following algorithms we suppose that the values of the dimension  and the variance parameters:  2 1 ,  2 2 , and  1 , . . .,   are given.The values of , , and  are also fixed.
The next algorithm is designed to obtain the expected volume of the generalized confidence region and the coverage probability of a given vector  = ( 1 , . . .,   ).Algorithm 6.Given a large number of iterations , for each  = 1, . . ., , (1) generate a -dimensional vector T with independent components, such that T, has a distribution (  ,  2 1 / +  2 2   /) for each  = 1, . . ., ; (2) generate η1, and η2, as in Algorithm 5; (3) compute where the coefficients are the terms of the diagonal matrix (η (4) use the value   given in Algorithm 5 to compute the estimated coverage probability of  as the proportion   () of   less than   ; (5) according to (27) compute the volume and the estimated expected volume V = (1/) ∑  =1 V  .
Finally an algorithm may be designed in order to simulate the simultaneous confidence intervals for the components of  and to estimate the joint coverage probability.
Algorithm 7. Given a large number of iterations , for each  = 1, . . .,  the following hold.
With the same data as before and the same vector , the obtained results are shown in Table 3.Let us observe that the coverage probability of the generalized confidence region   () 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.

Inferences about the Vector Means of Several Independent Gaussian Processes
As a generalization we now consider the case of  independent multivariate Gaussian processes  (1)   1 , where  1 , . . .,   ∈ R  and  2 1 , . . .,  2  ∈ R are unknown parameters.
1/2 has a multivariate Student's -distribution with 2 degrees of freedom and  =  1 /( 1 +  2 ) is a random variable with distribution Beta(/2, /2), which is independent of  , .Thus the generalized  value of the given test:  , ( 1 ) =  =0 { , ≥ W, } Invariance Properties of the Generalized  Value.Let us assume that the basic processes are transformed by means of =  1 .The same results hold for  ⋆  yielding  ⋆ , =  , , as well as  ⋆ , =  , .In this sense the generalized test process and the corresponding generalized  value are invariant under the proposed transformations.4.3.Generalized Confidence Region for .For any value of the unknown parameter , the difference   , =  , −  has distribution   (0,  , ) and ,   , , is a generalized pivotal quantity and { ∈ R  | W , ≤   } is a generalized confidence region for , whenever {  , ≤   } = 1 − .

Table 2 :
Coverage probability and expected volume of the confidence region.