Home page for accesible maths 14 Distribution of the MLE

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14.2 Summary

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    Under certain regularity conditions, the maximum likelihood estimator has, asymptotically, a normal distribution with mean equal to the true parameter value, and variance equal to the inverse of the Fisher information.

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    The Fisher information is minus the expectation of the second derivative of the log-likelihood evaluated at the true parameter value.

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    Based on this, we can construct approximate 95% confidence intervals for the true parameter value based on the MLE and the observed information.

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    Importantly, this is an asymptotic result so is only approximate. In particular, it is a bad approximation to a 95% confidence interval when the sample size, n, is small.