Hi Ítalo,

Generally, maximum likelihood parameter estimates exhibit asymptotic normality. Thus, it is essential that the estimator does not estimating on boundary. In addition, the estimator does not suffer from situation like data boundary parameter-dependent. Moreover, avoid having a number of nuisance parameters and increasing information as well.

HTH.

Samer

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some examples have been given in a previous discussion :

https://www.researchgate.net/post/What_is_the_basic_difference_between_the_maximum_likelihood_estimator_and_the_least_square_estimator

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sufficient conditions for the MLE to be consistent are given in section 7.3.2 of Theory of Statistics, Mark J. Schervish, Springer

when such conditions are not fullfilled, strange things can happen ...

but these are rather broad conditions

see also

https://projecteuclid.org/euclid.aoms/1177729952

Note on the Consistency of the Maximum Likelihood Estimate

Abraham Wald

Ann. Math. Statist.

Volume 20, Number 4 (1949), 595-601.

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What kind of models are you considering? To use MLE you need to know the probability density function and you have to assume random sampling. The classical example is when estimating the parameters in a regression model (the coefficients and the variance of the error term). There is an old theorem that shows that the MLE estimator is the same as the least squares estimator(s) when the response variable is multivariate normal. This result is related to but not the same as the discussion of whether the MLE is consistent.

Your question is posed in the "geostatistics" forum but the assumptions pertaining to kriging are quite different and in general MLE would not be applicable.

I would like to thank everybody for your answers so far.

I am currently researching the application of machine learning methods in the geosciences, so my question is related to the applicability of MLE in general, as opposed to a specific method.

If we must choose a specific model in order to better formulate the question, I would choose the Gaussian Process model. In the reference below the authors mention the possibility of multiple maxima of the likelihood function. I wonder if there are other possible complications one must be aware of when using MLE, or if its use will in general always provide a good, if not best, set of parameters for a given model.

http://www.gaussianprocess.org/gpml/