Mathematical Problems in Engineering
Volume 2016 (2016), Article ID 8975902, 25 pages
http://dx.doi.org/10.1155/2016/8975902

Research Article

Near-Exact Distributions for Likelihood Ratio Statistics Used in the Simultaneous Test of Conditions on Mean Vectors and Patterns of Covariance Matrices

Carlos A. Coelho,1,2 Filipe J. Marques,1,2 and Sandra Oliveira1,3

¹Centro de Matemática e Aplicações (CMA), FCT/UNL, 2829-516 Caparica, Portugal
²Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal
³Departamento de Economia e Gestão, Instituto Politécnico de Setúbal, 2910-761 Setúbal, Portugal

Received 31 May 2015; Accepted 23 November 2015

Academic Editor: Andrzej Swierniak

Copyright © 2016 Carlos A. Coelho et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The authors address likelihood ratio statistics used to test simultaneously conditions on mean vectors and patterns on covariance matrices. Tests for conditions on mean vectors, assuming or not a given structure for the covariance matrix, are quite common, since they may be easily implemented. But, on the other hand, the practical use of simultaneous tests for conditions on the mean vectors and a given pattern for the covariance matrix is usually hindered by the nonmanageability of the expressions for their exact distribution functions. The authors show the importance of being able to adequately factorize the c.f. of the logarithm of likelihood ratio statistics in order to obtain sharp and highly manageable near-exact distributions, or even the exact distribution in a highly manageable form. The tests considered are the simultaneous tests of equality or nullity of means and circularity, compound symmetry, or sphericity of the covariance matrix. Numerical studies show the high accuracy of the near-exact distributions and their adequacy for cases with very small samples and/or large number of variables. The exact and near-exact quantiles computed show how the common chi-square asymptotic approximation is highly inadequate for situations with small samples or large number of variables.

1. Introduction

Testing conditions on mean vectors is a common procedure in multivariate statistics. Often a given structure is assumed for the covariance matrix, without testing it, or otherwise this test to the covariance structure is carried out apart. This is often due to the fact that the exact distribution of the test statistics used to test simultaneously conditions on mean vectors and patterns on covariance matrices is too elaborate to be used in practice. The authors show how this problem may be overcome with the development of very sharp and manageable near-exact distributions for the test statistics. These distributions may be obtained from adequate factorizations of the characteristic function (c.f.) of the logarithm of the likelihood ratio (l.r.) statistics used for these tests.

The conditions tested on mean vectors are

(i)the equality of all the means in the mean vector,(ii)the nullity of all the means in the mean vector

and the patterns tested on covariance matrices are

(i)circularity,(ii)compound symmetry,(iii)sphericity.

Let be a random vector with . The covariance matrix is said to be circular, or circulant, if ,  , with

where , for ; .

For example, for and , we have

Besides the almost obvious area of times series analysis, there is a wealth of other areas and research fields where circular or circulant matrices arise, such as statistical signal processing, information theory and cryptography, biological sciences, psychometry, quality control, and signal detection, as well as spatial statistics and engineering, when observations are made on the vertices of a regular polygon.

We say that a positive-definite covariance matrix is compound-symmetric if we can write

For example, for , we have

If, in (3), , we say that the matrix is spheric.

The l.r. tests for equality and nullity of means, assuming circularity, and the l.r. tests for the simultaneous test of equality or nullity of means and circularity of the covariance matrix were developed by [1], while the test for equality of means, assuming compound symmetry, and the test for equality of means and compound symmetry were formulated by [2] and the test for nullity of the means, assuming compound symmetry, and the simultaneous test for nullity of the means and compound symmetry of the covariance matrix were worked out by [3]. The exact distribution for the l.r. test statistic for the simultaneous test of equality of means and circularity of the covariance matrix was obtained in [4] and is briefly referred to in Section 2, for the sake of completeness, while near-exact distributions for the l.r. test statistic for the simultaneous test of nullity of the means and circularity of the covariance matrix are developed in Section 3. Near-exact distributions for the l.r. test statistics for the simultaneous test of equality and nullity of the means and compound symmetry of the covariance matrix are developed in Sections 4 and 5, using a different approach from the one used in Section 3. The l.r. statistics for the tests of equality and nullity of all means, assuming sphericity of the covariance matrix, may be analyzed in Appendix C and the l.r. statistics for the simultaneous tests of equality and nullity of all means and sphericity, together with the development of near-exact distributions for these statistics, may be examined in Sections 6 and 7.

Since, as referred above, the exact distributions for the statistics for the simultaneous tests of conditions on means vectors and patterns of covariance matrices are too elaborate to be used in practice, the authors propose in this paper the use of near-exact distributions for these statistics. These are asymptotic distributions which are built using a different concept in approximating distributions which combines an adequately developed decomposition of the c.f. of the statistic or of its logarithm, most often a factorization, with the action of keeping then most of this c.f. unchanged and replacing the remaining smaller part by an adequate asymptotic approximation [5, 6]. All this is done in order to obtain a manageable and very well-fitting approximation, which may be used to compute near-exact quantiles or values. These distributions are much useful in situations where it is not possible to obtain the exact distribution in a manageable form and the common asymptotic distributions do not display the necessary precision. Near-exact distributions show very good performances for very small samples, and when correctly developed for statistics used in Multivariate Analysis, near-exact distributions display a sharp asymptotic behavior both for increasing sample sizes and for increasing number of variables.

In Sections 3–7, near-exact distributions are obtained using different techniques and results, according to the structure of the exact distribution of the statistic.

In order to study, in each case, the proximity between the near-exact distributions developed and the exact distribution, we will use the measure

with

where represents the l.r. statistic, is the exact c.f. of , is the near-exact c.f., and , , , and are the exact and near-exact c.d.f.’s of and .

This measure is particularly useful, since in our cases we do not have the exact c.d.f. of or in a manageable form, but we have both the exact and near-exact c.f.’s for .

2. The Likelihood Ratio Test for the Simultaneous Test of Equality of Means and the Circularity of the Covariance Matrix

Let , where . Then, for a sample of size , the th power of the l.r. statistic to test the null hypothesis

is

where , is the maximum likelihood estimator (m.l.e.) of , , where is the matrix with running element

with , and where is the th diagonal element of , and

with , where is the vector of sample means.

This test statistic was derived by [1, sec. 5.2], where the expression for the l.r. test statistic has to be slightly corrected.

According to [1],

where

are a set of independent r.v.’s.

From this fact we may write the c.f. of as

By adequately handling this c.f., the exact distribution of is obtained in [4] as a Generalized Integer Gamma (GIG) distribution (see [7] for the GIG distribution), since we may write

for

A popular asymptotic approximation for the distribution of is the chi-square asymptotic distribution with a number of degrees of freedom equal to the difference of the number of unknown parameters under the alternative hypothesis and the number of parameters under the null hypothesis, which gives for , for in (8), a chi-square asymptotic distribution with degrees of freedom. Although this is a valid approximation for large sample sizes, in practical terms, this approximation is somewhat useless given the fact that it gives quantiles that are much lower than the exact ones, as it may be seen from the quantiles in Table 1, namely, for small samples or when the number of variables involved is somewhat large.

Table 1: Exact and asymptotic 0.95 and 0.99 quantiles for where for the statistic in (8), for different values of and samples of size .

exact
Asymptotic-

10

76.77780315606147980433710659

11
184.84579506364826855487849906

60
82.86779631112725385496956047

460
77.50088072977322094345813820

15

153.19790274395621072198817490

16
356.83946609433702153375390686

65
169.23132191434840041430238602

465
155.17420633635277455721974156

25

379.74587752919253597245376194

26
853.62442647392551959929457598

75
437.12290346767321994020024210

475
387.31925318201483716457949700

50

1382.92839770564012472044120417

51
2983.52950629554250120199516974

100
1719.07640203276757900109720368

500
1434.09183007253302711800352147

10

85.95017624510346845181671517

11
221.13637373719535956938670312

60
92.78317859393323169599466291

460
86.75984117402977424037787646

15

165.84100085082047675645088502

16
409.92566639020778120425384446

65
183.23718212829228159346647123

465
167.98095654076846741881112565

25

399.22970790268112734530953113

26
940.55141434060365229501805667

75
459.68274728064743270409254333

475
407.19370031104569525049581690

50

1419.46244733465596475819616876

51
3156.01716925813527651187643029

100
1765.17588807596249988258749774

500
1471.99072215013613268320536543

From the values in Table 1 we may see that even for quite large sample sizes and rather small number of variables as in the case of and , the asymptotic chi-square quantile does not even match the units digit of the exact quantile, a difference that gets even larger as the number of variables increases. The chi-square asymptotic quantiles are always smaller than the exact ones, their use leading to an excessive number of rejections of the null hypotheses, a problem that becomes a grievous one when we use smaller samples or larger numbers of variables.

3. The Likelihood Ratio Test for the Simultaneous Test of Nullity of Means and the Circularity of the Covariance Matrix

For a sample of size , the th power of the l.r. test statistic to test the null hypothesis

is

where , , and , as well as the matrices and , are defined as in the previous section.

According to [1],

where

are a set of independent r.v.’s.

Taking and following similar steps to the ones used in [4] to handle the c.f. of , we may write the c.f. of as

for given by (15).

This shows that the exact distribution of is the same as that of the sum of GIG distributions of depth with an independent distributed r.v.

But then, using the result in expression (3) of [8], we know that we can replace asymptotically a distribution by an infinite mixture of distributions , for large values of . This means that we can replace asymptotically

As such, in order to obtain a very sharp and manageable near-exact distribution for , we will use, as near-exact c.f. for ,

where the weights , , will be determined in such a way that

with .

is the c.f. of a mixture of Generalized Near-Integer Gamma (GNIG) distributions of depth (see [5] for the GNIG distribution).

As such, using the notation for the p.d.f. and c.d.f. of the GNIG distribution used in Section 3 of [6], the near-exact p.d.f.’s and c.d.f.’s for and are

with given by (15).

In Table 2 we may analyze values of the measure in (5) for the near-exact distributions developed in this section, for different values of and different sample sizes. We may see how these near-exact distributions display very low values of the measure , indicating an extremely good proximity to the exact distribution, even for very small sample sizes, and how they display a sharp asymptotic behavior for increasing values of and .

Table 2: Values of the measure in (5), for the near-exact distributions of the l.r. test statistic in (17), which match exact moments, for different values of and samples of size .

2
4
6
10
20

10
11

60

460

15
16

65

465

25
26

75

475

50
51

100

500

In Table 3 we may analyze the asymptotic quantiles for for the common chi-square asymptotic approximation for l.r. statistics, here with degrees of freedom and the quantiles for the near-exact distributions that equate 2, 6, and 10 exact moments. These quantiles are shown with 26 decimal places in order to make it possible to identify the number of correct decimal places for the quantiles of the near-exact distributions that match 2 and 6 exact moments. We should note that the quantiles of the near-exact distributions that match 10 exact moments always have much more than 26 decimal places that are correct. Also for the statistic in this section, we may see the lack of precision of the asymptotic chi-square quantiles.

Table 3: Quantiles of orders and for the chi-square approximation and for the near-exact distributions that match , 6, or 10 exact moments, of for the l.r. statistic in (17), for different values of and samples of size .

Near-exact distributions

2

6
10

10

77.93052380523042221626519134

11
186.05876112572581565724084671
186.05432193047513242686020674
186.05432193047314525997232991

60
84.04125105999276667186524643
84.04108458609231348943338211
84.04108458609231109537970540

460
78.65634165506500024084199822
78.65633892721702104079228658
78.65633892721702104079204160

15

154.30151616535021445238246295

16
357.97669643528506074060759290
357.97507556215083974984357415
357.97507556215080939585842368

65
170.35237275637365647369739224
170.35226008885071534567845906
170.35226008885071540240672638

465
156.28032082820735784475589844
156.28031868060615422818671849
156.28031868060615422818672130

25

380.80932798326309965351661308

26
854.70444721767252577468152620
854.70394734650287271287479556
854.70394734650287269133917757

75
438.20003415718340026159076324
438.19996662815447062882655526
438.19996662815447063138401124

475
388.38496822730515375937449496
388.38496656458350909591459832
388.38496656458350909591459983

50

1383.96068056800040468077204943

51
2984.56855943704772921090729785
2984.56844789958359734870454884
2984.56844789958359734870371056

100
1720.11786940941582246751263410
1720.11783827238527706678307676
1720.11783827238527706678545071

500
1435.12611970183181982922209152
1435.12611845907109035050829355
1435.12611845907109035050829357

10

87.16571139978756939605535555

11
222.35460048933098793717199428
222.34999712883572718401028732
222.34999712881867010107412453

60
94.01616090177012981866436862
94.01591912908288010423242638
94.01591912908288738648975948

460
87.97774784195631422822549370
87.97774379040648875818772244
87.97774379040648875818827763

15

166.98739013667388353875528529

16
411.06647044542264851519342470
411.06480936217954062715053117
411.06480936217950312171917698

65
184.39761554112671734787489011
184.39747019675240094311740291
184.39747019675240137066064190

465
169.12943676298651871191909376
169.12943392987951859377231013
169.12943392987951859377239666

25

400.31941006633016981735907054

26
941.63369479396272075223880115
941.63318710455464318301861791
941.63318710455464316544796384

75
460.78287302204290413806922579
460.78279435481890211302276154
460.78279435481890211439671412

475
408.28522573430588726946677528
408.28522375722866010999960584
408.28522375722866010999960757

50

1420.50810052822545308190548728

51
3157.05746081267985238368475051
3157.05734831670317027719825824
3157.05734831670317027719755606

100
1766.22802929701017442904996266
1766.22799581952383557719092088
1766.22799581952383557718851667

500
1473.03793287996522918580780119
1473.03793152230567784068549373
1473.03793152230567784068549371

4. The Likelihood Ratio Test for the Simultaneous Test of Equality of Means and Compound Symmetry of the Covariance Matrix

Let us assume that , with . We are interested in testing the hypothesis

where represents a compound symmetric matrix, as defined in (3).

For a sample of size , the th power of the l.r. test statistic is (see [2])

where

with being the sample matrix and a matrix of 1’s of dimension ,

with

Wilks [2] has also shown that

where

form a set of independent r.v.’s.

As such, the th moment of may be written as

Since the expression in (33) remains valid for any complex , we may write the c.f. of as

which may be rewritten as

Then, we may apply on the relation

to obtain

with

Expression (37) shows that the exact distribution of is the same as that of the sum of GIG distributed r.v.’s of depth with an independent sum of independent r.v.’s.

Our aim in building the near-exact distribution will be to keep unchanged and approximate asymptotically .

In order to obtain this asymptotic approximation, we will need to use a different approach from the one used in the previous section. We will use the result in sec. 5 of [9], which implies that a distribution may be asymptotically replaced by an infinite mixture of distributions.

Using a somewhat heuristic approach, we will thus approximate by a mixture of distributions where

is the sum of the second parameters of the Logbeta r.v.’s in and is the common rate parameter in the mixture of two Gamma distributions that matches the first 4 moments of , that is, in

As such, in order to build the near-exact distributions for , we will use, as near exact c.f. for ,

where the weights , , will be determined in such a way that

with .

The c.f. in (41) is, for integer , the c.f. of a mixture of GIG distributions of depth or, for noninteger , the c.f. of a mixture of GNIG distributions of depth , with shape parameters and rate parameters .

This will yield, for noninteger , near-exact distributions whose p.d.f.’s and c.d.f.’s for and are

with given by (38). For integer , we will only have to replace in the above expressions the GNIG p.d.f. and c.d.f. by the GIG p.d.f. and c.d.f., respectively.

In Table 4, we may analyze values of the measure in (5) for the near-exact distributions developed in this section, for different values of and different sample sizes. We may see how these near-exact distributions display, once again, very low values of the measure even for very small sample sizes, indicating an extremely good proximity to the exact distribution and how, once again, they display a sharp asymptotic behavior for increasing values of and , although for large values of , namely, for in Table 4, one may have to consider larger values of in order to be able to observe the asymptotic behavior in terms of sample size.

Table 4: Values of the measure in (5), for the near-exact distributions of the l.r. test statistic in (26), which match exact moments, for different values of and samples of size .

2
4
6
10
20

10
11

60

460

15
16

65

465

25
26

75

475

50
51

100

500

The asymptotic quantiles for in Table 5, for the common chi-square asymptotic approximation for l.r. statistics, now with degrees of freedom, display again, as in the previous sections, an almost shocking lack of precision, mainly for small sample sizes and/or larger numbers of variables. On the other hand, the near-exact quantiles show a steady evolution towards the exact quantiles for increasing number of exact moments matched, with the quantiles for the near-exact distributions that match 6 exact moments displaying more than 20 correct decimal places, for the larger sample sizes.

Table 5: Quantiles of orders and for the chi-square approximation and for the near-exact distributions that match , 6, or 10 exact moments, of for the l.r. statistic in (26), for different values of and samples of size .

Near-exact distributions

2
6
10

10

81.381015188899104508431120785

11
189.114696480423388015515754393
189.114697238826289320495944292
189.114697238826277646711500921

60
87.443617040369847972033082381
87.443617050273318041561932389
87.443617050273318056851954690

460
82.100788323800277056582786653
82.100788323820173111875365635
82.100788323820173111875372688

15

159.813546850997802977506766879

16
363.087896604766629226939377100
363.087896734501151262354131399
363.087896734501151251811277829

65
175.805353263604236148015901820
175.805353269595285156952328009
175.805353269595285157155780468

465
161.784540428765594956506807022
161.784540428780828916102538276
161.784540428780828916102538558

25

391.438719112192616721837671209

26
864.891282269279116189851387185
864.891282284150965012514477481
864.891282284150965012477034037

75
448.748369857001308907641564890
448.748369859173975960465295344
448.748369859173975960473758210

475
399.002746812117152321633172649
399.002746812124288653675424722
399.002746812124288653675424753

50

1407.69978493252055471492271999

51
3007.81592652688874585708833686
3007.81592652788317452613238449
3007.81592652788317452613235100

100
1743.73166206895650376506106613
1743.73166206957708782548197403
1743.73166206957708782548209645

500
1458.84419233676150570472235786
1458.84419233676635867029771753
1458.84419233676635867029771754

10

90.801532030838687874709640406

11
225.421470930453454966690527975
225.421472009727318561373265594
225.421472009727311663857900086

60
97.582829539160859251491143413
97.582829585977762108505255066
97.582829585977762137425094394

460
91.604850363213970123907600250
91.604850363315475545131468587
91.604850363315475545131487705

15

172.710824396692046362684067121

16
416.192057135707772720106372243
416.192057320949648013699720809
416.192057320949648008342780372

65
190.035075104220296563348087039
190.035075126350713655861714506
190.035075126350713655903270996

465
174.841471634002327233344093337
174.841471634061877548052071914
174.841471634061877548052072072

25

411.209208332240728524234613091

26
951.840639147683569759583191462
951.840639169340274602907166879
951.840639169340274602887597489

75
471.550355225483649333455494949
471.550355232304350604344289953
471.550355232304350604337719505

475
419.157414390428896468907403837
419.157414390452257215547282031
419.157414390452257215547282016

50

1444.55331478957048828683472593

51
3180.33179707887525737674116194
3180.33179708038051828516553103
3180.33179708038051828516550812

100
1790.07969304544686923930101362
1790.07969304718352521730352610
1790.07969304718352521730331125

500
1497.05022951059504341326946354
1497.05022951060908489974722503
1497.05022951060908489974722503

5. The Likelihood Ratio Test for the Simultaneous Test of Nullity of Means and Compound Symmetry of the Covariance Matrix

Let us assume now that . We are interested in testing the hypothesis

We may write

where

While, for a sample of size , the th power of the l.r. statistic to test may be shown to be (see Appendix A for details)

where and are given by (28) and

with

where is the sample matrix, the l.r. test statistic to test is shown by [2] to be

again with and given by (28) and given by (27).

The l.r. test to test in (44) is thus

For a sample of size , , the th power of the l.r. test statistic to test , may be shown to be distributed as (see [3] and Appendix A for details) , where and are independent, with

while [2] shows that the th power of the l.r. statistic to test is distributed as , where are independent, with

Based on Theorem  5 in [10], it is then possible to show that the l.r. statistics to test and are independent, since is independent of and is built only on this statistic, since is the same statistic in a constrained subspace.

From this fact, we may show that the th power of the l.r. statistic to test in (44), , is distributed as (see Appendix B for details)

where all r.v.’s are independent, with

We note that the r.v.’s are the same as the r.v.’s in (31) and (32).

As such, the c.f. of may be written as

with given by (38).

Then, following a similar approach to the one used for and , in the previous section, we obtain near-exact distributions with a similar structure to those in that section, now with

and with determined as the solution of a system of equations similar to the one in (40), with replaced by .

This will yield for and near-exact distributions with p.d.f.’s and c.d.f.’s given by (43), now with given by (57).

We should note that as it happens with and , also for and , may be either an integer or a half-integer, so that, in those cases where is an integer, the near-exact distributions are mixtures of GIG distributions, while when is noninteger, they are mixtures of GNIG distributions.

In Table 6 we may analyze values of the measure in (5) for the near-exact distributions developed in this section, for different values of and different sample sizes. We may see how these near-exact distributions display similar properties to those of the near-exact distributions developed for in the previous section.

Table 6: Values of the measure in (5), for the near-exact distributions of the l.r. test statistic in (51), which match exact moments, for different values of and samples of size .

2
4
6
10
20

10
11

60

460

15
16

65

465

25
26

75

475

50
51

100

500

In Table 7, the asymptotic chi-square quantiles are made available for the common chi-square asymptotic approximation for l.r. statistics, now with degrees of freedom, as well as the near-exact quantiles for . Similar conclusions to those drawn in the previous sections apply here.

Table 7: Quantiles of orders and for the chi-square approximation and for the near-exact distributions that match , 6, or 10 exact moments, of for the l.r. statistic in (51), for different values of and samples of size .

Near-exact distributions

2
6
10

10

82.52872654147179311367711677

11
190.32756331545176062637260063
190.32756312341501096656402680
190.32756312341502158305183289

60
88.61292917028511318678026436
88.61292916079636593558509765
88.61292916079636591605025196

460
83.25135114394460784736871943
83.25135114392165525365869519
83.25135114392165525365868449

15

160.91477802694323358920183897

16
364.22508588466772907687245064
364.22508590299516153552837462
364.22508590299516166009461668

65
176.92458777551078581129813667
176.92458777274377296466188482
176.92458777274377296219866202

465
162.88835115579315719900816060
162.88835115578197146107641778
162.88835115578197146107641501

25

392.50115579019226559339157016

26
865.97128758699096427158883014
865.97128759463038534181481168
865.97128759463038534192831533

75
449.82482829542519016999435848
449.82482829549700283679630302
449.82482829549700283676131146

475
400.06750221064414330250460763
400.06750221064079210854726486
400.06750221064079210854726472

50

1408.73177341938097671969938629

51
3008.85497566956079322838948966
3008.85497567039266902344201858
3008.85497567039266902344199893

100
1744.77297987136924686391338212
1744.77297987182100521614403517
1744.77297987182100521614409508

500
1459.87821714249309834106087509
1459.87821714249577662643142818
1459.87821714249577662643142818

10

92.01002361413199132182815244

11
226.63967070294372053364533245
226.63967043727843805017814101
226.63967043727844439241178794

60
98.81015393413627584334298342
98.81015389070350242742117425
98.81015389070350239170134559

460
92.81589286301973694456386019
92.81589286290397669965973982
92.81589286290397669965971186

15

173.85385431432844593057930059

16
417.33284870330949699905167521
417.33284873028538888270377699
417.33284873028538894546001788

65
191.19301774559926635976479112
191.19301773577866477925423064
191.19301773577866477873083849

465
175.98671397271816998582919704
175.98671397267474771413147408
175.98671397267474771413147254

25

412.29747839373905678663071280

26
952.92291432362881946300916855
952.92291433479892788245985114
952.92291433479892788251876064

75
472.64955572272794762005618804
472.64955572304282388887164444
472.64955572304282388889904345

475
420.24758865175907740683966303
420.24758865174822800295256496
420.24758865174822800295256503

50

1445.59855177795512096240506173

51
3181.37208704154942688955747637
3181.37208704280924834914384041
3181.37208704280924834914382694

100
1791.13162751307828413931876980
1791.13162751434597504689343387
1791.13162751434597504689332917

500
1498.09706702524552623910040811
1498.09706702525328984531816943
1498.09706702525328984531816942

6. The Likelihood Ratio Test for the Simultaneous Test of Equality of Means and Sphericity of the Covariance Matrix

If , where , and we are interested in testing the null hypothesis

we may write

where

where, for a sample of size , the th power of the l.r. statistic to test , versus an alternative hypothesis that assumes sphericity for the covariance matrix and no structure for the mean vector, may be shown to be (see Appendix C for details)

where is the matrix in (27) and

with

We have

where is the sample matrix and

In (65), from standard theory on normal r.v.’s, since , independent for ,

and, since, under , we have , i.i.d. for , under this null hypothesis,

Thus, since the r.v.’s in (67) are independent for ,

while, from (68),

Since and are independent, given that is independent of all and is defined only from the , then

From [6, 11] and [12, sec. 10.7], the th power of the l.r. statistic to test in (60) is given by

with (see [6])

where, for ,

are a set of independent r.v.’s.

The th power of the l.r. statistic to test in (60) is thus

where, from Theorem  5 in [10], we may assure the independence between and and as such say that

where all r.v.’s are independent, are the r.v.’s in (75), and is a r.v. with the same distribution as in (72).

Let us take , , and . Then we will have

where, using (36), we may write, for odd ,

and, for even , following similar steps,

Taking for the expression for in (A.6) in [6], we may write

where ,

Then, following a similar approach to the one used for and , in Section 4, we obtain near-exact distributions with a somewhat similar structure to those in that section, now with

and with determined as the solution of a system of equations similar to the one in (40), with replaced by .

This will yield for near-exact distributions which are mixtures of GIG or GNIG distributions, according to the fact that is an integer or a noninteger, of depth , with shape parameters

with by (82) and given by (83), and corresponding rate parameters

with p.d.f.’s and c.d.f.’s, respectively, given by

and for with p.d.f.’s and c.d.f.’s, respectively, given by

We should note that as it happens with and , also for and , may be either an integer or a half-integer, so that, in those cases where is an integer, the near-exact distributions are mixtures of GIG distributions, while when is noninteger, they are mixtures of GNIG distributions.

In Table 8 are displayed the values of the measure in (5) for the near-exact distributions developed for and and in Table 9 we may find the chi-square asymptotic quantiles for , based on a chi-square distribution with degrees of freedom and the quantiles for the near-exact distributions with , 6, and 10. Similar conclusions to those drawn for the asymptotic and near-exact distributions for the l.r. statistics in the previous sections also apply here.

Table 8: Values of the measure in (5), for the near-exact distributions of the l.r. test statistic in (76), which match exact moments, for different values of and samples of size .

2
4
6
10
20

10
11

60

460

15
16

65

465

25
26

75

475

50
51

100

500

Table 9: Quantiles of orders and for the chi-square approximation and for the near-exact distributions that match , 6, or 10 exact moments, of for the l.r. statistic in (76), for different values of and samples of size .

Near-exact distributions

2
6
10

10

82.52872654147179311367711677

11
190.34577352567912574418958472
190.34577224119045860164933939
190.34577224119046305269448848

60
88.61692747841779863573642579
88.61692745278611013951753747
88.61692745278611013324695622

460
83.25188239392602534312672397
83.25188239386900029725019512
83.25188239386900029725019084

15

160.91477802694323358920183897

16
364.24055367674081921965459933
364.24055364729180144370579916
364.24055364729180141324533927

65
176.92888004902098965390468116
176.92888004652298399068321382
176.92888004652298398935665716

465
162.88896091620593405370503267
162.88896091619839192532218731
162.88896091619839192532218500

25

392.50115579019226559339157016

26
865.98217181891738253073510565
865.98217181564421769408452503
865.98217181564421769411599634

75
449.82888066877263241039729095
449.82888066790844106600627651
449.82888066790844106598248210

475
400.06815103634197188114099440
400.06815103633783183795559146
400.06815103633783183795559133

50

1408.73177341938097671969938629

51
3008.86102439239945267715486761
3008.86102439143486337077345854
3008.86102439143486337077347761

100
1744.77616968248446695304159926
1744.77616968166143981696831893
1744.77616968166143981696823169

500
1459.87886196492823624335068318
1459.87886196491897609569196780
1459.87886196491897609569196779

10

92.01002361413199132182815244

11
226.65802233268160965656246440
226.65802050594440639840352032
226.65802050594440942100275312

60
98.81456601487869161798863928
98.81456589548535463063927369
98.81456589548535461913454163

460
92.81648438468645753157179639
92.81648438439817704231467008
92.81648438439817704231465890

15

173.85385431432844593057930059

16
417.34840709478673000029559780
417.34840705292241842439350332
417.34840705292241840622917724

65
191.19761414641994048857514633
191.19761413728546916412651829
191.19761413728546916385955487

465
175.98737199442095758355328501
175.98737199439157553407869592
175.98737199439157553407869464

25

412.29747839373905678663071280

26
952.93384227143576783559100319
952.93384226668216247367925905
952.93384226668216247369526304

75
472.65378089035668898963824328
472.65378088767304149154578277
472.65378088767304149156442088

475
420.24826946954476567253237555
420.24826946953125669939828606
420.24826946953125669939828612

50

1445.59855177795512096240506173

51
3181.37814988860070147625187269
3181.37814988714219606939294787
3181.37814988714219606939296092

100
1791.13488232226638659185402987
1791.13488231997376953941120636
1791.13488231997376953941135956

500
1498.09772807257343724270969706
1498.09772807254669066052003540
1498.09772807254669066052003541

7. The Likelihood Ratio Test for the Simultaneous Test of Nullity of Means and Sphericity of the Covariance Matrix

We now assume , and we now want to test the null hypothesis

which may be written as

where

For a sample of size , the th power of the l.r. statistic to test , versus an alternative hypothesis that assumes sphericity for the covariance matrix and no structure for the mean vector, may be shown to be (see Appendix D for details)

where is the matrix in (27) and now

which was already used in Section 5.

We now have

where is the r.v. defined in (65), is the sample matrix, and

In (93), from standard theory on normal r.v.’s, since under we have , independent for , under this null hypothesis,

Since and are independent, given that is independent of all and is defined only from , then, given the distribution of in (69),

Since is the same hypothesis as in Section 6, the th power of the l.r. statistic to test in (90) is thus

where is given by (27) and by (92) and where, from Theorem  5 in [10], we may assure the independence between and , so that

where all r.v.’s are independent, are the r.v.’s in (75), and is a r.v. with the same distribution as in (97).

Then, if we take , , and , we may write

where, using (36), we may write, for even ,

while, for odd , we may write

Following then a similar procedure to the one used in the previous section, in order to build near-exact distributions for and , we take for the expression for in (A.6) in [6] and write

where , and the are defined in (82).

Then, we may obtain near-exact distributions for and , with a similar structure to those in the previous section, now with

with determined as the solution of a system of equations similar to the one in (40), with replaced by .

This yields for near-exact distributions which are mixtures of GIG or GNIG distributions, according to the fact that is an integer or a noninteger, of depth , with shape parameters

with by (82) and given by (104), and corresponding rate parameters

with p.d.f.’s and c.d.f.’s, respectively, given by

and for with p.d.f.’s and c.d.f.’s, respectively, given by

We should note that as it happens with the statistics in Sections 3–6, also for and , may be either an integer or a half-integer, so that, in those cases where is an integer, the near-exact distributions are mixtures of GIG distributions, while when is noninteger, they are mixtures of GNIG distributions.

In Tables 10 and 11 are displayed the values of the measure in (5) for the near-exact distributions developed for and and the chi-square asymptotic quantiles for , based on a chi-square distribution with degrees of freedom, together with the quantiles for the near-exact distributions with , 6, and 10. In these tables we may observe the same developments discovered in previous sections. Although in Table 10 we may observe a slight increase in the values of the measure when we go from to as well as when we compare the near-exact distributions that match only 2 exact moments for and , the near-exact distributions developed end up having a sharp asymptotic behavior for increasing , which is clearly visible when we compare the values of for with those for any other .

Table 10: Values of the measure in (5), for the near-exact distributions of the l.r. test statistic in (98), which match exact moments, for different values of and samples of size .

2
4
6
10
20

10
11

60

460

15
16

65

465

25
26

75

475

50
51

100

500

Table 11: Quantiles of orders and for the chi-square approximation and for the near-exact distributions that match , 6, or 10 exact moments, of for the l.r. statistic in (98), for different values of and samples of size .

Near-exact distributions

2
6
10

10

83.675260742720985209859769504

11
191.40417508216086413376362096
191.40417507456156657285101953
191.40417507456156582171789369

60
89.75551474727646979869969732
89.75551474640967644843717358
89.75551474640967644824901262

460
84.39744286268007803380889520
84.39744286267799553786765774
84.39744286267799553786765754

15

162.015627915781020974421513240

16
365.27598959457250439641270781
365.27598924568078011317902861
365.27598924568078023771489768

65
178.02159933390715505406007972
178.02159930927581887701221583
178.02159930927581887281868717

465
163.98875588508246868863047277
163.98875588500965129426910391
163.98875588500965129426909762

25

393.563502697733190559496512362

26
867.00163199459511920686859768
867.00163196313224586608216176
867.00163196313224586631572990

75
450.88354614389745101761538811
450.88354613612423832735039952
450.88354613612423832726057126

475
401.12939936195988958155797530
401.12939936192330144108737902
401.12939936192330144108737858

50

1409.76374981471409967132946025

51
3009.86996103494312689347231192
3009.86996103484414649104682359
3009.86996103484414649104682436

100
1745.80181083720649016504206416
1745.80181083711886310915937759
1745.80181083711886310915937366

500
1460.90974356899046369444463197
1460.90974356898945187656106283
1460.90974356898945187656106283

10

93.216859660238415548016837195

11
227.72045780686361924157771904
227.72045779716228291792684432
227.72045779716228232600823452

60
100.00766587721934070773055100
100.00766587324444064978254945
100.00766587324444064947003150

460
94.02162317660297597321472166
94.02162317659253570112582966
94.02162317659253570112582916

15

174.996346513999368603209796319

16
418.38682050544091438546913353
418.38682000788295279610953591
418.38682000788295286039688584

65
192.32704662492917996667148895
192.32704653485001159062688254
192.32704653485001158976312093

465
177.12815947623512459860873742
177.12815947595193142347546688
177.12815947595193142347546347

25

413.385621639270785606729896736

26
953.95532266934168704083850077
953.95532262361714954158839471
953.95532262361714954171095313

75
473.73022881819913551147338725
473.73022879405047104192607988
473.73022879405047104199617631

475
421.33473289045564827205481429
421.33473289033629484717703330
421.33473289033629484717703352

50

1446.64377167058432609278629314

51
3182.38825614178288689315436637
3182.38825614163324534715229586
3182.38825614163324534715229638

100
1792.17081609574757121169227449
1792.17081609550358487097066245
1792.17081609550358487097066936

500
1499.14133899336021309354248845
1499.14133899335729098476088160
1499.14133899335729098476088160

8. Conclusions

The near-exact approximations developed in this paper allow the practical and precise implementation of simultaneous tests on conditions on mean vectors and of patterns on covariance matrices. These approximations are based on mixtures of Generalized Near-Integer Gamma or Generalized Integer Gamma distributions which are highly manageable and for which there are computational modules available on the Internet: https://sites.google.com/site/nearexactdistributions/home. Numerical studies show the quality and accuracy of near-exact distributions developed, contrary to what happens, for example, with usual chi-square approximation. A natural extension of this work will be to develop approximations that will allow the implementation of simultaneous tests on the equality or nullity of several mean vectors and on the equality of the corresponding covariance matrices to a given matrix which may have a specific structure.

Appendices

A. On the Likelihood Ratio Test Statistic to Test the Nullity of Means Assuming Compound Symmetry of the Covariance Matrix

The l.r. statistic to test the null hypotheses

is the l.r. statistic to test the null hypothesis in (46), which is

where , the likelihood function under in (A.1) above, is, for , defined as in (3), for which we have ,

where is the sample matrix, and , the likelihood function under in (A.1), is

where is an vector of 1’s.

Since under we have and , the m.l.e.’s of and given by (28), and the m.l.e. of is , then

where is the matrix defined in (27), with , the vector sample means, which is the m.l.e. of .

In (A.6),

so that

Under the m.l.e.’s of and are and , given by (48), and the m.l.e. of is , so that

where is the matrix defined in (49).

Given the definition of and in (48), following similar steps to the ones used under , , so that finally

As such, we have

and its th power given by in (47).

Let be a Helmert matrix of order . Then

and if we take , we will have

where are the diagonal elements of .

Since, for in (27), we have

then

so that the diagonal elements of are independent, with

where is the th diagonal element of .

As such, is distributed as times a chi-square r.v. with degrees of freedom, independently distributed from which is distributed as times a chi-square with degrees of freedom.

Concerning the matrix in (49), we know that

so that

which shows that the diagonal elements of , , are independently distributed, with

But then, the m.l.e.’s of and , under in (A.1), are given by (48) or, equivalently,

as such, with distributed as times a chi-square r.v. with degrees of freedom, independently distributed from which is distributed as times a chi-square with degrees of freedom.

We can write

or

where, by application of Cochran’s Theorem (see, e.g., [12, Thm. 7.4.1]), it is easy to show that, under in (A.1), and are independent, since we can write

where, under the null hypothesis in (A.1), the th column of is

and we have

with

which yields for the distribution in (A.14) and

so that

so that, for , each th diagonal element of , has a distribution such that

with

where

so that is independent of and it has a distribution which is that of a chi-square with degrees of freedom, multiplied by , while we also have

where is distributed as a chi-square with 1 degree of freedom, multiplied by , and distributed independently of .

As such

which is independently distributed from

yielding for in (47) the distribution stated in Section 5, as also stated by Geisser in [3], but where the expression for the l.r. statistic should be corrected to be stated as in (A.11).

B. On the Distribution of in (51)

In order to show the distribution of in (51) as mentioned in (54) and (55), all we need to do is to show that

for in (53), in (52), and in (55), where is independent of all .

Let us take as a r.v. whose distribution is the same as that of the product of r.v.’s on the left hand side of (B.1) and as a r.v. whose distribution is the same as that of the product of r.v.’s on the right hand side of (B.1).

Then, using the multiplication formula for the Gamma function

we may write

But then, since both and have a delimited support, more precisely, between zero and 1, their distribution is determined by their moments and as such .

C. Derivation of the l.r. Statistic to Test Equality of Means, Assuming Sphericity of the Covariance Matrix

Let us suppose that , where , and that we want to test the hypotheses

based on a sample of size .

Under , the m.l.e. of is

with

where is the sample matrix and is a matrix of dimensions of 1’s, that is, a column vector of 1’s. Also, under this same hypothesis, the m.l.e. of is

where

where is the element in the th row and th column of and is the matrix in (27).

Since the likelihood function is

its maximum under in (C.1) is thus

Under the null hypothesis in (C.1), the m.l.e. of is

where and the m.l.e. of is

with

where is the matrix in (62), so that the maximum of the likelihood function under in (C.1) is

The l.r. statistic to test the hypotheses in (C.1) is thus

so that its th power is the statistic in (72).

D. Derivation of the l.r. Statistic to Test the Nullity of Means, Assuming Sphericity of the Covariance Matrix

Let us suppose that , where , and that we want to test the hypotheses

based on a sample of size .

Under in (D.1), the m.l.e.’s of and are the same as those in Appendix C, since this hypothesis is the same as the alternative hypothesis in (C.1) in Appendix C, and, as such, the function , the maximum of the likelihood function under in (D.1), is the same as in (C.7) in Appendix C.

Under the null hypothesis in (D.1), the m.l.e. of is

with

where is the matrix in (92), so that the maximum of the likelihood function under in (D.1) is given by a similar function to that in (C.11), now with given by (92) and given by (D.3).

The l.r. statistic to test the hypotheses in (D.1) is thus, for in (27) and in (92),

so that its th power is the statistic in (91).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

Research was partially supported by National Funds through FCT Fundação para a Ciência e a Tecnologia, project UID/MAT/00297/2013 (CMA/UNL).

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