## Asymptotic Normality

theorem 7.4.3 Let the likelihood function be L(XbX2,. . . ,Xn 0). Then, under general conditions, the maximum likelihood estimator 0 is asymptotically distributed as

(Here we interpret the maximum likelihood estimator as a solution to the likelihood equation obtained by equating the derivative to zero, rather than the global maximum likelihood estimator. Since the asymptotic normality can be proved only for this local maximum likelihood estimator, henceforth this is always what we mean by the maximum likelihood estimator.)

Sketch of Proof. By definition, 31ogL/30 evaluated at 0 is zero. We expand it in a Taylor series around 0O to obtain

3 log L |
_ 3 log L |
! 32 log L |

30 |
9 30 |
e0 302 |

(7.4.17) 0 |

(0 – 0o), |

where 0* lies between 0 and 0O...

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