f 00 x θ According to the above analysis if l X θ is close to zero then it is

# F 00 x θ according to the above analysis if l x θ

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f 00 ( x | θ ). According to the above analysis, if l 0 ( X | θ ) is close to zero, then it is expected, thus the random variable does not provide much information about θ ; on the other hand, if | l 0 ( X | θ ) | or [ l 0 ( X | θ )] 2 is large, the random variable provides much information about θ . Thus, we can use [ l 0 ( X | θ )] 2 to measure the amount of information provided by X . However, since X is a random variable, we should consider the average case. Thus, we introduce the following definition: Fisher information (for θ ) contained in the random variable X is defined as: I ( θ ) = E θ ' [ l 0 ( X | θ ))] 2 = Z [ l 0 ( x | θ ))] 2 f ( x | θ ) dx. (1) 1

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The maximum likelihood method We seek a method which produces good estimators. R. A. Fisher (1912), “On an absolute criterion for fitting frequency curves,” Messenger of Math. 41 , 155–160. Fisher’s first mathematical paper, written while a final-year undergraduate in mathematics and mathematical physics at Gonville and Caius College, Cambridge University. It’s not clear what motivated Fisher to study this subject; perhaps it was the influence of his tutor, the astronomer F. J. M. Stratton. Fisher’s paper started with a criticism of two methods of curve fitting, the least-squares and the method of moments.
∂μ ln L = 1 σ 2 n X i =1 ( x i - μ )

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• Fall '13
• Maximum likelihood, Fisher Information

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