The MVE subroutine computes the robust estimation of multivariate location and scatter, defined by minimizing the volume of an ellipsoid that contains points.

The MVE subroutine computes the minimum volume ellipsoid estimator. These robust locations and covariance matrices can be used to detect multivariate outliers and leverage points. For this purpose, the MVE subroutine provides a table of robust distances.

In the following discussion, is the number of observations and is the number of regressors. The input arguments to the MVE subroutine are as follows:

opt

refers to an options vector with the following components (missing values are treated as default values):

opt[1]

specifies the amount of printed output. Higher option values request additional output and include the output of lower values.

0

prints no output except error messages.

1

prints most of the output.

2

additionally prints case numbers of the observations in the best subset and some basic history of the optimization process.

3

additionally prints how many subsets result in singular linear systems.

The default is opt[1]=0.

opt[2]

specifies whether the classical, initial, and final robust covariance matrices are printed. The default is opt[2]=0. The final robust covariance matrix is always returned in coef.

opt[3]

specifies whether the classical, initial, and final robust correlation matrices are printed or returned. The default is opt[3]=0.

0

does not return or print.

1

prints the robust correlation matrix.

2

returns the final robust correlation matrix in coef.

3

prints and returns the final robust correlation matrix.

opt[4]

specifies the quantile used in the objective function. The default is opt[5]= . If the value of is specified outside the range , it is reset to the closest boundary of this region.

opt[5]

specifies the number of subset generations. This option is the same as described previously for the LMS and LTS subroutines. Due to computer time restrictions, not all subset combinations can be inspected for larger values of and . If opt[5] is zero or missing, the default number of subsets is taken from the following table.

n	1	2	3	4	5	6	7	8	9	10
	500	50	22	17	15	14	0	0	0	0
		1414	182	71	43	32	27	24	23	22
	500	1000	1500	2000	2500	3000	3000	3000	3000	3000

n	11	12	13	14	15
	0	0	0	0	0
	22	22	22	23	23
	3000	3000	3000	3000	3000

If the number of cases (observations) is smaller than , as given in the table, then all possible subsets are used; otherwise, subsets are chosen randomly. This means that an exhaustive search is performed for opt[5]. If is larger than , a note is printed in the log file that indicates how many subsets exist.

x

refers to an matrix of regressors. Missing values are not permitted in .

s

refers to an vector that contains observation numbers of a subset for which the objective function should be evaluated, where is the number of parameters. In other words, the MVE algorithm computes the minimum volume of the ellipsoid that contains the observation numbers contained in .

The MVE subroutine returns the following values:

sc

is a column vector that contains the following scalar information:

sc[1]

the quantile used in the objective function

sc[2]

number of subsets generated

sc[3]

number of subsets with singular linear systems

sc[4]

number of nonzero weights

sc[5]

lowest value of the objective function attained (volume of smallest ellipsoid found)

sc[6]

Mahalanobis-like distance used in the computation of the lowest value of the objective function

sc[7]

the cutoff value used for the outlier decision

coef

is a matrix with columns that contains the following results in its rows:

coef[1,]

location of ellipsoid center

coef[2,]

eigenvalues of final robust scatter matrix

coef[3:2+n,]

the final robust scatter matrix for opt[2]=1 or opt[2]=3

coef[2+n+1:2+2n,]

the final robust correlation matrix for opt[3]=1 or opt[3]=3

dist

is a matrix with columns that contains the following results in its rows:

dist[1,]

Mahalanobis distances

dist[2,]

robust distances based on the final estimates

dist[3,]

weights (1 for small robust distances; 0 for large robust distances)

        /* X1  X2  X3   Y    Stackloss data */
 aa = { 1  80  27  89  42,
        1  80  27  88  37,
        1  75  25  90  37,
        1  62  24  87  28,
        1  62  22  87  18,
        1  62  23  87  18,
        1  62  24  93  19,
        1  62  24  93  20,
        1  58  23  87  15,
        1  58  18  80  14,
        1  58  18  89  14,
        1  58  17  88  13,
        1  58  18  82  11,
        1  58  19  93  12,
        1  50  18  89   8,
        1  50  18  86   7,
        1  50  19  72   8,
        1  50  19  79   8,
        1  50  20  80   9,
        1  56  20  82  15,
        1  70  20  91  15 };

a = aa[, 2:4];
opt = j(8, 1, .);
opt[1] = 2;              /* ipri */
opt[2] = 1;              /* pcov: print COV */
opt[3] = 1;              /* pcor: print CORR */
opt[5] = -1;             /* nrep: use all subsets */

call mve(sc, xmve, dist, opt, a);