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Statistics / Probability and Statistics 4 / Chapter 8.8 / Problem 6

Probability and Statistics | 4th Edition | ISBN: 9780321500465 | Authors: Morris H. DeGroot, Mark J. Schervish

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Suppose that X is a random variable for which the p.d.f.

Chapter 8, Problem 6

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QUESTION:

Suppose that X is a random variable for which the p.d.f. or the p.f. is f (x| ), where the value of the parameter is unknown but must lie in an open interval . Let I0( ) denote the Fisher information in X. Suppose now that the parameter is replaced by a new parameter , where = (), and is a differentiable function. Let I1() denote the Fisher information in X when the parameter is regarded as . Show that I1() = [ ()] 2I0[()].

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QUESTION:

Suppose that X is a random variable for which the p.d.f. or the p.f. is f (x| ), where the value of the parameter is unknown but must lie in an open interval . Let I0( ) denote the Fisher information in X. Suppose now that the parameter is replaced by a new parameter , where = (), and is a differentiable function. Let I1() denote the Fisher information in X when the parameter is regarded as . Show that I1() = [ ()] 2I0[()].

ANSWER:

Step 1 of 2

Let be a random variable from a distribution depending on , with pdf for the same set and all . Assume that the function is twice differentiable as a function on . The Fisher information in the random variable is

Denote with the probability density (mass) function of where is the parameter. Connection with and is

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