# Calculate confidence interval for log determinant of covariance matrices

### Description

This function uses asymptotic results described by Cai et. al (2016), Theorem 1, to calculate approximate, normal theory confidence intervals (CIs) for the log determinant of one or more sample covariance matrices.

Their results are translated into a CI via the approximation

* \log det( \widehat{Σ} ) - bias \pm z_{1 - α/2} \times SE *

where * \widehat{Σ}* is the sample estimate of a population
covariance matrix,
*bias* is a bias correction constant and *SE* is a width factor for
the confidence interval. Both *bias* and *SE* are functions of
the sample size, *n* and number of variables, *p*.

This function is included here only to provide an approximation to
*graphical accuracy* for use with Box's M test for equality of covariance
matrices, `boxM`

and its associated `plot.boxM`

method.

### Usage

1 |

### Arguments

`cov` |
a covariance matrix or a (named) list of covariance matrices, all the same size |

`n` |
sample size, or vector of sample sizes, one for each covariance matrix |

`conf` |
confidence level |

`method` |
Three methods are provided, based on Cai et. al Theorem 1 ( |

`bias.adj` |
logical; set |

### Details

Cai et. al (2016) claim that their Theorem 1 holds with either *p* fixed or *p(n)* growing with *n*, as long as
*p(n) ≤ n*. Their Corollary 1 (`method=2`

) is the special case when *p* is fixed.
Their Corollary 2 (`method=3`

) is the special case when *0 ≤ p/n < 1* is fixed.

The properties of this CI estimator are unknown in small to moderate sample sizes, but it seems to be the only one available. It is therefore experimental in this version of the package and is subject to change in the future.

The *bias* term offsets the confidence interval from the sample
estimate of * \log det( \widehat{Σ} ) *.
When *p* is large relative to *n*, the confidence interval may not
overlap the sample estimate.

Strictly speaking, this estimator applies to the MLE of the covariance matrix * \widehat{Σ}*, i.e.,
using *n* rather than *n-1* in as the divisor. The factor *(n-1 / n)*
has not yet been taken into account here.

### Value

A data frame with one row for each covariance matrix.
`lower`

and `upper`

are the boundaries of the confidence intervals.
Other columns are `logdet, bias, se`

.

### Author(s)

Michael Friendly

### References

Cai, T. T., Liang, T. & Zhou, H. H. (2016). Law of log determinant of sample
covariance matrix and optimal estimation of differential enteropy
for high-dimensional Gaussian distributions. *JMA*, to appear.
Preprint: www.stat.yale.edu/~hz68/Covariance-Determinant.pdf

### See Also

`boxM`

,
`plot.boxM`

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 | ```
data(iris)
iris.mod <- lm(as.matrix(iris[,1:4]) ~ iris$Species)
iris.boxm <- boxM(iris.mod)
cov <- c(iris.boxm$cov, list(pooled=iris.boxm$pooled))
n <- c(rep(50, 3), 150)
CI <- logdetCI( cov, n=n, conf=.95, method=1)
CI
plot(iris.boxm, xlim=c(-14, -8), main="Iris data, Box's M test", gplabel="Species")
arrows(CI$lower, 1:4, CI$upper, 1:4, lwd=3, angle=90, len=.1, code=3)
CI <- logdetCI( cov, n=n, conf=.95, method=1, bias.adj=FALSE)
CI
plot(iris.boxm, xlim=c(-14, -8), main="Iris data, Box's M test", gplabel="Species")
arrows(CI$lower, 1:4, CI$upper, 1:4, lwd=3, angle=90, len=.1, code=3)
``` |