9Large-Sample Inference Based on Maximum Likelihood Estimates

9.1 Standard Errors Based on The Information Matrix

We noted in Chapter 6 that large-sample maximum likelihood (ML) inferences can be based on Approximation 6.1, namely that

(9.1)numbered Display Equation

where C is an estimate of the d × d covariance matrix of , for example,

(9.2)numbered Display Equation

the inverse of the observed information evaluated at , or

(9.3)numbered Display Equation

the inverse of the expected information evaluated at , or

(9.4)numbered Display Equation

the sandwich estimator. The estimate (9.2) is computed as part of the Newton–Raphson algorithm for ML estimation, and (9.3) is computed as part of the scoring algorithm. When the expectation–maximization (EM) algorithm or one of the variants described in Chapter 8 is used for ML estimation, additional steps are needed to compute standard errors ...

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