For given samples X1...Xn, which follow the distribution F with parameters p1,...pm, one may use the maximum likelihood method to estimate p1,...pm. In the technical process, the method is only an optimization problem in m-dimensional field.
In some cases, it is irritating to acquire different estimated values of p1,...pm than the one acquired if one only estimate p1 (assuming this parameter does not depend on p2,...pm). E.g. the joint estimators (\mu, \sigma) may be different than \mu, when estimated alone. Since \mu is reserved for expected value, which one is "correct"? the joint-\mu or the single-\mu?
1) if the joint one is correct, then any single estimator should be doubted.
2) if the single one is correct, then joint-model does not have any values. So one should always choose univariate model.
In some cases, the joint-\mu does not make any sense. The interpretation that \mu is an expected value is somehow no longer valid in a joint-model.
Sure- In additional to small samples, poor saturation and correlated residuals can detriment estimators.
Hope this helps,
Matt
Maximum likelihood estimation works very well for multi-dimensional data. And you are right that singe point estimators are quite useless if the multidimensional space is not orthogonal. In that case the correlations among the parameters must be accommodated,
In other words the parameters that make up the multidimensional space are inaccurate unless the model accommodates any correlations among them.
Certainly if the model is biased by failure to accommodate interactions in the parameter then the estimators will be wrong.
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