# Maximum likelihood estimation of the parameters of a matrix normal distribution

### Description

Maximum likelihood estimation for the parameters of a matrix normal
distribution **X**, which is characterized by a simply separable
variance-covariance structure. In the general case, which is the case
considered here, two unstructured factor
variance-covariance matrices determine the covariability of random
matrix entries, depending on the row (one factor matrix)
and the column (the other factor matrix) where two
**X**-entries are. In the required function, the Id1 and Id2 variables correspond to
the row and column subscripts, and are the second and third columns in the matrix
(2d) data file, respectively; “value”
indicates the observed variable, and is the fourth column in the matrix data file.

### Usage

1 |

### Arguments

`formula` |
value2d~Id1+Id2 |

`rep` |
the replicate, also called subject or individual, the first column in the matrix (2d) data file |

`data` |
the name of the matrix data |

`eps` |
the threshold in the stopping criterion for the iterative mle algorithm |

`maxiter` |
the maximum number of iterations for the iterative mle algorithm |

`startmat` |
the value of the second factor variance-covariance matrix used for initialization, i.e., to start the algorithm and obtain the initial estimate of the first factor variance-covariance matrix |

### Output

“Convergence”, TRUE or FALSE

“Iter”, will indicate the number of iterations needed for the mle algorithm to converge

“Xmeanhat”, the estimated mean matrix (i.e., the sample mean)

“First”, the row subscript, or the second column in the data file

“U1hat”, the estimated variance-covariance matrix for the rows

“Standardized.U1hat”, the standardized estimated variance-covariance matrix for the rows; the standardization is performed by dividing each entry of U1hat by entry(1, 1) of U1hat

“Second”, the column subscript, or the third column in the data file

“U2hat”, the estimated variance-covariance matrix for the columns

“Standardized.U2hat”, the standardized estimated variance-covariance matrix for the columns; the standardization is performed by multiplying each entry of U2hat by entry(1, 1) of U1hat

“Shat”, is the sample variance-covariance matrix computed from of the vectorized data matrices

### References

Dutilleul P. 1990. Apport en analyse spectrale d'un periodogramme modifie et modelisation des series chronologiques avec repetitions en vue de leur comparaison en frequence. D.Sc. Dissertation, Universite catholique de Louvain, Departement de mathematique.

Dutilleul P. 1999. The mle algorithm for the matrix normal distribution. Journal of Statistical Computation and Simulation 64: 105-123.

### Examples

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