section introduces such a measure for information and we can also see that this

# Section introduces such a measure for information and

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section introduces such a measure for information, and we can also see that this information measure can be used to find bounds on the variance of estimators, and it can be used to approximate the sampling distribution of an estimator obtained from a large sample, and further be used to obtain an approximate confidence interval in case of large sample. In this section, we consider a random variable X for which the pdf or pmf is f ( x | θ ), where θ is an unknown parameter and θ Θ, with Θ is the parameter space. 1 Fisher Information Motivation: Intuitively, if an event has small probability, then the occurrence of this event brings us much information. For a random variable X f ( x | θ ), if θ were the true value of the parameter, the likelihood function should take a big value, or equivalently, the derivative log-likelihood function should be close to zero, and this is the basic principle of maximum likelihood estimation. We define l ( x | θ ) = log f ( x | θ ) as the log-likelihood function, and l 0 ( x | θ ) = ∂θ log f ( x | θ ) = f 0 ( x | θ ) f ( x | θ ) where f 0 ( x | θ ) is the derivative of f ( x | θ ) with respect to θ . Similarly, we denote the second order derivative of f ( x | θ ) with respect to θ as

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• Fall '13
• Maximum likelihood, Fisher Information

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