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Suppose we have some data sample $x_1, x_2, ... x_n$ that came from some true probability distribution function $f(x; \theta)$.

Based on this sample, i am interested in estimating $\theta$ using Maximum Likelihood Estimation, but I incorrectly assume that it came from some distribution $g(x; \theta)$.

I have the following question about misspecification and robustness:

As my choice of $g(x; \theta)$ deviates further away from $f(x; \theta)$ : How strongly do the properties of the MLE's of $\hat{\theta}$ get affected relative to their true values $\theta$ ?

Here are my guesses:

• Bias: Even under the correct distribution MLE estimates can be unbiased. Therefore, under the incorrect distribution, the MLE estimates should be even more biased?
• Consistency: The MLE estimates for $g(x, \theta)$ will converge in probability (for large samples) to the true values of $g(x, \theta)$ .... but obviously will not converge to the correct values of $f(x, \theta)$
• Asymptotic Normal: For large sample sizes, under the incorrect distribution, the estimates of $g(x, \theta)$ will still be asymptotically normal - but this will be meaningless as they are not cantered around the correct location of $f(x, \theta)$
• Minimum Variance: Again, I think this property will also be respected under the incorrect choice of distribution, but unfortunately this will be meaningless as the distribution choice was incorrect.

Is this the right intuition?

Suppose we have some data sample $x_1, x_2, ... x_n$ that came from some true probability distribution function $f(x; \theta)$.

Based on this sample, i am interested in estimating $\theta$ using Maximum Likelihood Estimation, but I incorrectly assume that it came from some distribution $g(x; \theta)$.

I have the following question about misspecification and robustness:

As my choice of $g(x; \theta)$ deviates further away from $f(x; \theta)$ : How strongly do the properties of the MLE's of $\hat{\theta}$ get affected relative to their true values $\theta$ ?

Here are my guesses:

• Bias: Even under the correct distribution MLE estimates can be biased. Therefore, under the incorrect distribution, the MLE estimates have the possibility of being even more biased?
• Consistency: The MLE estimates for $g(x, \theta)$ will converge in probability (for large samples) to the true values of $g(x, \theta)$ .... but obviously will not converge to the correct values of $f(x, \theta)$
• Asymptotic Normal: For large sample sizes, under the incorrect distribution, the estimates of $g(x, \theta)$ will still be asymptotically normal - but this will be meaningless as they are not cantered around the correct location of $f(x, \theta)$
• Minimum Variance: Again, I think this property will also be respected under the incorrect choice of distribution, but unfortunately this will be meaningless as the distribution choice was incorrect.

Is this the right intuition?