ON HYPOTHESIS TESTS FOR COVARIANCE MATRICES UNDER MULTIVARIATE
               NORMALITY

In this paper we proposed a new statistical test for testing the covariance matrix in one population under multivariate normal assumption. In general, the proposed and the likelihood-ratio tests resulted in larger values of estimated powers than VMAX for bivariate and trivariate cases. VMAX was not sensitive to general changes in the covariance (correlation) structure. The advantage of the new test is that it is based on the comparison of all elements of the postulated covariance matrix under the null hypothesis with their respective maximum likelihood sample estimates and therefore, it does not restrict the information of the covariance matrix into a scalar number such as the determinant or trace, for example. Due to the fact that it is based on the maximum likelihood estimates and the Fisher information matrix, it can be used for data coming from distribution other than the multivariate normal.

covariance matrix; hypothesis testing; Fisher information matrix

Authorship

Letícia Pereira Pinto

Departament of Computer Science, UFMG, Belo Horizonte,Minas Gerais, Brazil. E-mail: leticiap@dcc.ufmg.br UFMGBrazilBelo Horizonte, Minas Gerais, BrazilDepartament of Computer Science, UFMG, Belo Horizonte,Minas Gerais, Brazil. E-mail: leticiap@dcc.ufmg.br

Sueli Aparecida Mingoti

Departament of Statistics, UFMG, Belo Horizonte, Minas Gerais, Brazil. E-mail: sueliam@est.ufmg.br UFMGBrazilBelo Horizonte, Minas Gerais, BrazilDepartament of Statistics, UFMG, Belo Horizonte, Minas Gerais, Brazil. E-mail: sueliam@est.ufmg.br

* Corresponding author.

*
Corresponding author.

SCIMAGO INSTITUTIONS RANKINGS

Figures | Tables | Formulas

Figures (1)
Tables (9)
Formulas (8)

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Figure 1
Part of the example 2. Source: See [Aparisi et. al, 1999].

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Table 1
Simulated models − p = 2 variables.

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Table 2
Simulated models − p = 3 variables.

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Table 3
Type I error and power estimates of the tests − p = 2, n = 5, 10 and 25.

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Table 4
Type I error and power estimates of the tests − p = 2, n = 50 and 100.

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Table 5
Type I error and power estimates of the tests −p = 3, n = 5, 10 and 25.

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Table 6
Type I error and power estimates of the tests −p = 3, n = 50 and 100.

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Table 7
Test statistic values and critical limits - Example 1.

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Table 8
Results of the Example 2 - samples sizes n = 10 and n = 25 - α = 0.005.

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Table 9
Results of the Example 2 - sample size n = 5.

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Figure 1 Part of the example 2. Source: See [Aparisi et. al, 1999].

Table 1 Simulated models − p = 2 variables.

^*
() Det: determinant; T R and DR: ratios between the traces and the determinants of the covariance matrices under H ₁ and H ₀, respectively; HVR: ratio between the maximum of the variances under H ₁ and H ₀. ρ ₁₂ is the correlation between both variables. σ _i is the standard deviation of variable X ₁, i = 1, 2.

Table 2 Simulated models − p = 3 variables.

^*
() Det: determinant; T R and DR: ratios between the traces and the determinants of the covariancematrices under H ₁ and H ₀, respectively; HVR: ratio between the maximum of the variances under H ₁ and H ₀. ρ _{i j} is the correlation between variables. X _i and X _j , i ≠ j .

Table 3 Type I error and power estimates of the tests − p = 2, n = 5, 10 and 25.

					Average proportion of rejection of the null hypothesis
		n = 5			n = 10				n = 25
Case	LRE	SE	VMAX	LRE	SE	VMAX	LR	LRE	SC	SE	VMAX
null	0.05	0.05	0.05	0.05	0.05	0.05	0.07	0.05	0.06	0.05	0.05
1	0.10	0.25	0.05	0.23	0.43	0.05	0.67	0.63	0.79	0.77	0.05
2	0.29	0.58	0.05	0.72	0.89	0.05	1.00	1.00	1.00	1.00	0.05
3	0.11	0.27	0.07	0.26	0.46	0.08	0.71	0.67	0.81	0.79	0.10
4	0.11	0.29	0.09	0.28	0.49	0.12	0.75	0.72	0.84	0.82	0.17
5	0.20	0.41	0.29	0.50	0.68	0.44	0.94	0.93	0.96	0.96	0.74
6	0.26	0.45	0.36	0.62	0.75	0.55	0.98	0.98	0.99	0.98	0.87
7	0.05	0.06	0.08	0.05	0.07	0.08	0.08	0.06	0.09	0.07	0.10
8	0.05	0.07	0.10	0.06	0.08	0.12	0.11	0.09	0.14	0.12	0.17
9	0.09	0.18	0.29	0.21	0.33	0.46	0.61	0.57	0.68	0.65	0.74
10	0.16	0.26	0.37	0.43	0.52	0.56	0.91	0.89	0.92	0.92	0.87

^*
() LR and LRE: are the likelihood-ratio using chi-square approximation and the exact distributions; SC and SE are the adapted Sullivan et al. using chi-square approximation and the exact distributions; VMAX test using the exact distribution.

Table 4 Type I error and power estimates of the tests − p = 2, n = 50 and 100.

				Average proportion of rejection of the null hypothesis
			n = 50					n = 100
Case	LR	LRE	SC	SE	VMAX	LR	LRE	SC	SE	VMAX
null	0.06	0.05	0.06	0.05	0.05	0.05	0.05	0.05	0.05	0.05
1	0.93	0.93	0.96	0.96	0.05	1.00	0.93	1.00	1.00	0.05
2	1.00	1.00	1.00	1.00	0.05	1.00	1.00	1.00	1.00	0.05
3	0.95	0.95	0.97	0.97	0.13	1.00	0.95	1.00	1.00	0.18
4	0.96	0.96	0.98	0.98	0.25	1.00	0.96	1.00	1.00	0.41
5	1.00	1.00	1.00	1.00	0.94	1.00	1.00	1.00	1.00	1.00
6	1.00	1.00	1.00	1.00	0.99	1.00	1.00	1.00	1.00	1.00
7	0.08	0.07	0.10	0.09	0.13	0.11	0.11	0.13	0.13	0.18
8	0.16	0.14	0.19	0.18	0.26	0.28	0.27	0.31	0.31	0.40
9	0.89	0.88	0.92	0.91	0.94	0.99	0.99	0.99	0.99	0.99
10	0.99	0.99	0.99	0.99	0.99	1.00	1.00	1.00	1.00	1.00

^*
() LR and LRE are the likelihood-ratio tests using chi-square and the exact distributions. SC and SE are the adapted Sullivan et al. test using chi-square and the exact distributions. VMAX is the test using the exact distribution.

Table 5 Type I error and power estimates of the tests −p = 3, n = 5, 10 and 25.

				Average proportion of rejection of the null hypothesis
		n = 5			n = 10			n = 25
Case	LRE	SE	VMAX	LRE	SE	VMAX	LRE	SE	VMAX
null	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05
1	0.31	0.73	0.24	0.88	0.98	0.40	1.00	1.00	0.70
2	0.07	0.12	0.05	0.16	0.30	0.05	0.49	0.65	0.05
3	0.11	0.22	0.05	0.34	0.57	0.05	0.84	0.93	0.05
4	0.05	0.06	0.11	0.05	0.07	0.15	0.08	0.11	0.21
5	0.12	0.38	0.05	0.41	0.70	0.05	0.91	0.97	0.05
6	0.17	0.43	0.14	0.57	0.78	0.18	0.97	0.99	0.26
7	0.20	0.46	0.17	0.63	0.82	0.25	0.99	1.00	0.40
8	0.59	0.69	0.66	0.95	0.97	0.88	1.00	1.00	1.00

^*
() LR and LRE are the likelihood-ratio tests using chi-squared and the exact distributions. SC and SE are the adapted Sullivan et al. test using chi-squared and the exact distributions. VMAX is the test using the exact distribution.

Table 6 Type I error and power estimates of the tests −p = 3, n = 50 and 100.

					Average proportion of rejection of the null hypothesis
			n = 50					n = 100
Case	LR	LRE	SC	SE	VMAX	LR	LRE	SC	SE	VMAX
null	0.06	0.05	0.07	0.05	0.05	0.06	0.05	0.05	0.05	0.05
1	1.00	1.00	1.00	1.00	0.93	1.00	1.00	1.00	1.00	1.00
2	0.86	0.84	0.93	0.92	0.05	1.00	1.00	1.00	1.00	0.05
3	0.99	0.99	1.00	1.00	0.05	1.00	1.00	1.00	1.00	0.05
4	0.17	0.14	0.24	0.20	0.30	0.33	0.31	0.40	0.37	0.46
5	1.00	1.00	1.00	1.00	0.06	1.00	1.00	1.00	1.00	0.06
6	1.00	1.00	1.00	1.00	0.37	1.00	1.00	1.00	1.00	0.55
7	1.00	1.00	1.00	1.00	0.61	1.00	1.00	1.00	1.00	0.86
8	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00

^*
() LR and LRE are the likelihood-ratio tests using chi-squared and the exact distributions. SC and SE are the adapted Sullivan et al. test using chi-squared and the exact distributions. VMAX is the test using the exact distribution.

Table 7 Test statistic values and critical limits - Example 1.

Test	Test statistic	Critical limits - α = 0.05
LR	38.69	12.59
LRE	38.69	14.23
SC	85.88	12.59
SE	85.86	14.16
VMAX	0.95	1.77

Table 8 Results of the Example 2 - samples sizes n = 10 and n = 25 - α = 0.005.

			Test Statistics - observed values
		n = 10			n = 25
Sample	LRE	SE	VMAX	LRE	SE	VMAX
1	10.4610	2.9871	1.3467	6.6965	4.5624	1.0794
2	5.4263	4.6919	1.3168	11.4868	13.9057	1.8518
3	5.7642	4.9829	1.4768	2.5405	2.5593	1.2564
4	4.8761	3.8433	0.8694	2.4840	2.8476	1.1432
5	3.4707	4.1756	1.7074	4.7600	5.8345	1.7261
6	37.6804	355.8093	0.7003	62.1800	504.5504	0.4022
7	36.0395	185.7810	0.4293	77.2557	367.9003	0.5291
8	21.1632	84.7545	1.0847	89.3189	257.7654	1.4477
9	57.8377	186.1254	2.5899	219.0934	480.5102	4.0263
10	46.2343	60.5017	4.2255	160.5388	207.9995	3.4506
		LLR = LSC = 18.5476; LRE = 24.1789; LSE = 52.0888; LVMAX = 2.7559.			LLR = LSC = 18.5476; LRE = 20.5275; LSE = 29.5167; LVMAX = 2.0007.

^*
() LLR and LSC: likelihood-ratio and the adapted Sullivan et. al tests critical limits using chi-square distribution. LRE and LSE: likelihood-ratio and the adapted Sullivan et. al tests critical limits using exact distribution. LVMAX: critical limit for VMAX test.

Table 9 Results of the Example 2 - sample size n = 5.

^*
() LLR and LSC: likelihood-ratio and the adapted Sullivan et al. tests critical limits using chi-square distribution. LRE and LSE: likelihood-ratio and the adapted Sullivan et. al tests critical limits using exact distribution. LVMAX: critical limit for VMAX test.

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ON HYPOTHESIS TESTS FOR COVARIANCE MATRICES UNDER MULTIVARIATE NORMALITY

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[1] *
Corresponding author.