Description Usage Arguments Details Value Note References See Also Examples

Calculates p-value of the test for testing equality of two-sample high-dimensional mean vectors proposed by Srivastava and Du (2008) based on the asymptotic distribution of the test statistic.

1 | ```
apval_Sri2008(sam1, sam2)
``` |

`sam1` |
an n1 by p matrix from sample population 1. Each row represents a |

`sam2` |
an n2 by p matrix from sample population 2. Each row represents a |

Suppose that the two groups of *p*-dimensional independent and identically distributed samples *\{X_{1i}\}_{i=1}^{n_1}* and *\{X_{2j}\}_{j=1}^{n_2}* are observed; we consider high-dimensional data with *p \gg n := n_1 + n_2 - 2*. Assume that the two groups share a common covariance matrix. The primary object is to test *H_{0}: μ_1 = μ_2* versus *H_{A}: μ_1 \neq μ_2*. Let *\bar{X}_{k}* be the sample mean for group *k = 1, 2*. Also, let *S = n^{-1} ∑_{k = 1}^{2} ∑_{i = 1}^{n_{k}} (X_{ki} - \bar{X}_k) (X_{ki} - \bar{X}_k)^T* be the pooled sample covariance matrix from the two groups.

Srivastava and Du (2008) proposed the following test statistic:

*T_{SD} = \frac{(n_1^{-1} + n_2^{-1})^{-1} (\bar{X}_1 - \bar{X}_2)^T D_S^{-1} (\bar{X}_1 - \bar{X}_2) - (n - 2)^{-1} n p}{√{2 (tr R^2 - p^2 n^{-1}) c_{p, n}}},*

where *D_S = diag (s_{11}, s_{22}, ..., s_{pp})*, *s_{ii}*'s are the diagonal elements of *S*, *R = D_S^{-1/2} S D_S^{-1/2}* is the sample correlation matrix and *c_{p, n} = 1 + tr R^2 p^{-3/2}*. This test statistic follows normal distribution under the null hypothesis.

A list including the following elements:

`sam.info` |
the basic information about the two groups of samples, including the samples sizes and dimension. |

`cov.assumption` |
this output reminds users that the two sample populations have a common covariance matrix. |

`method` |
this output reminds users that the p-values are obtained using the asymptotic distributions of test statistics. |

`pval` |
the p-value of the test proposed by Srivastava and Du (2008). |

The asymptotic distribution of the test statistic was derived under normality assumption in Bai and Saranadasa (1996). Also, this function assumes that the two sample populations have a common covariance matrix.

Srivastava MS and Du M (2008). "A test for the mean vector with fewer observations than the dimension." *Journal of Multivariate Analysis*, **99**(3), 386–402.

1 2 3 4 5 6 7 8 9 10 11 |

```
Loading required package: mvtnorm
$sam.info
n1 n2 p
50 50 200
$cov.assumption
[1] "the two groups have same covariance"
$method
[1] "asymptotic distribution"
$pval
Sri2008
0.61321
```

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