Is MLE consistent if my data is independent but not identically distributed? Specifically I have n samples where each sample is from a Gaussian distribution with equal variance but different mean. The mean for each Gaussian distribution is a different function of the true parameter.
MLE are used for regression (whose data are knowingly non iid) with no problems with regard to consistency.
In the case you described, through an ANOVA, you will make a comparison among the sample means to evaluate whether or not the means are equal to each other.
ANOVA have been long studied and its properties are well stablished. Among several others, you have the consistency of the parameters' estimation.
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