section introduces such a measure for information, and we can also see that this informationmeasure can be used to find bounds on the variance of estimators, and it can be used toapproximate the sampling distribution of an estimator obtained from a large sample, andfurther be used to obtain an approximate confidence interval in case of large sample.In this section, we consider a random variableXfor which the pdf or pmf isf(x|θ), whereθis an unknown parameter andθ∈Θ, with Θ is the parameter space.1Fisher InformationMotivation:Intuitively, if an event has small probability, then the occurrence of this eventbrings us much information. For a random variableX∼f(x|θ), ifθwere the true value ofthe parameter, the likelihood function should take a big value, or equivalently, the derivativelog-likelihood function should be close to zero, and this is the basic principle of maximumlikelihood estimation. We definel(x|θ) = logf(x|θ) as the log-likelihood function, andl0(x|θ) =∂∂θlogf(x|θ) =f0(x|θ)f(x|θ)wheref0(x|θ) is the derivative off(x|θ) with respect toθ. Similarly, we denote the secondorder derivative off(x|θ) with respect toθas
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