Test for \(\Sigma\) in a \(Np(\mu, \Sigma)\)

This function can be used to test \(H_0: \Sigma = \Sigma_0\) versus \(H_1: \Sigma\) not = \(\Sigma_0\).

one_covar_matrix_test(Sigma0, S, n, method = "lrt")

Arguments

Sigma0

a matrix indicating the hypothesized value of the covariance matrix \(\Sigma\).

S

a matrix with sample variances and covariances.

n

sample size.

method

a character string specifying the method, it must be one of "lrt" (default), "modlrt1" (modified LRT test) or "modlrt2" (modified LRT test for moderate "n"). You can specify just the initial letter. See details.

Value

A list with class "htest" containing the following components:

statistic

the value of the statistic.

parameter

the degrees of freedom for the test.

p.value

the p-value for the test.

estimate

the estimated covariance matrix S.

method

a character string indicating the type of test performed.

Details

When method="lrt" (default) the function performs the LRT test given in Mardia et. al (1979), page 126, expression 5.2.7. For method="modlrt1" or method="modlrt2" the function performs the LRT test given in Rencher and Christensen (2012), page 260, expressions 7.2 and 7.4.

See also

one_mean_vector_test for test \(\mu\) in a \(Np(\mu, \Sigma)\).

Author

Freddy Hernandez

Examples

# Example 5.3.2 from Mardia (1979) page 127
# Test H0: Sigma = diag(100, 100) versus H1: Sigma != diag(100, 100)

Sigma0 <- matrix(c(100, 0, 0, 100), ncol=2)
S <- matrix(c(91.481, 66.875, 66.875, 96.775), ncol=2)

res1 <- one_covar_matrix_test(Sigma0=Sigma0, S=S, n=25, method='lrt')
res1
#> 
#> 	LRT test for Sigma matrix
#> 
#> data:  this test uses summarized data
#> lrt = 17.698, df = 3, p-value = 0.0005077
#> alternative hypothesis: true Sigma matrix is not equal to Sigma0 
#> 
#> sample estimates:
#>        xbar_1 xbar_2
#> xbar_1 91.481 66.875
#> xbar_2 66.875 96.775
#> 
plot(res1, from=12, to=20, shade.col='dodgerblue2')


# Example from Morrison (1990) page 293
# Test H0: Sigma = Sigma0 versus H1: Sigma != Sigma0
# using the modified LRT test versions

n <- 20

Sigma0 <- matrix(c(4, 3, 2,
                   3, 6, 5,
                   2, 5, 10), ncol=3)

S <- matrix(c(3.42, 2.60, 1.89,
              2.60, 8.00, 6.51,
              1.89, 6.51, 9.62), ncol=3)

res2 <- one_covar_matrix_test(Sigma0=Sigma0, S=S, n=n, method='modlrt1')
res2
#> 
#> 	Modified LRT test for Sigma matrix
#> 
#> data:  this test uses summarized data
#> lrt = 3.6374, df = 6, p-value = 0.7256
#> alternative hypothesis: true Sigma matrix is not equal to Sigma0 
#> 
#> sample estimates:
#>        xbar_1 xbar_2 xbar_3
#> xbar_1   3.42   2.60   1.89
#> xbar_2   2.60   8.00   6.51
#> xbar_3   1.89   6.51   9.62
#> 
plot(res2, from=0, to=20, shade.col='indianred1')


res3 <- one_covar_matrix_test(Sigma0=Sigma0, S=S, n=n, method='modlrt2')
res3
#> 
#> 	Modified LRT test for Sigma matrix with moderate n
#> 
#> data:  this test uses summarized data
#> lrt = 3.43, df = 6, p-value = 0.7533
#> alternative hypothesis: true Sigma matrix is not equal to Sigma0 
#> 
#> sample estimates:
#>        xbar_1 xbar_2 xbar_3
#> xbar_1   3.42   2.60   1.89
#> xbar_2   2.60   8.00   6.51
#> xbar_3   1.89   6.51   9.62
#> 
plot(res3, from=0, to=20, shade.col='aquamarine3')