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Mean and covariance of incomplete multivariate normal data

```
[Mean,Covariance] = ecmnmle(Data,InitMethod,MaxIterations,Tolerance,Mean0,Covar0)
```

`[Mean,Covariance] = ecmnmle(Data,InitMethod,MaxIterations,Tolerance,Mean0,Covar0)`

estimates
the mean and covariance of a data set. If the data set has missing
values, this routine implements the ECM algorithm of Meng and Rubin
[2] with enhancements by Sexton and Swensen [3]. ECM stands for *expectation conditional maximization*, a conditional
maximization form of the EM algorithm of Dempster, Laird, and Rubin
[4].

This routine has two operational modes.

With no output arguments, this mode displays the convergence of the ECM algorithm. It estimates and plots objective function values for each iteration of the ECM algorithm until termination, as shown in the following plot.

Display mode can determine `MaxIter`

and `Tolerance`

values
or serve as a diagnostic tool. The objective function is the negative
log-likelihood function of the observed data and convergence to a
maximum likelihood estimate corresponds with minimization of the objective.

With output arguments, this mode estimates the mean and covariance via the ECM algorithm.

To see an example of how to use `ecmnmle`

,
run the program `ecmguidemo`

.

[1] Little, Roderick J. A. and Donald B. Rubin. *Statistical
Analysis with Missing Data.* 2nd Edition. John Wiley &
Sons, Inc., 2002.

[2] Meng, Xiao-Li and Donald B. Rubin. “Maximum Likelihood
Estimation via the ECM Algorithm.” *Biometrika.* Vol.
80, No. 2, 1993, pp. 267–278.

[3] Sexton, Joe and Anders Rygh Swensen. “ECM Algorithms
that Converge at the Rate of EM.” *Biometrika.* Vol.
87, No. 3, 2000, pp. 651–662.

[4] Dempster, A. P., N. M. Laird, and Donald B. Rubin. “Maximum
Likelihood from Incomplete Data via the EM Algorithm.” *Journal
of the Royal Statistical Society.* Series B, Vol. 39, No.
1, 1977, pp. 1–37.

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