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Let X~Geometric(p), with parameter p unknown, 0<p<1. a) Find I(p), the Fisher Information in X about...

Let X~Geometric(p), with parameter p unknown, 0<p<1.

a) Find I(p), the Fisher Information in X about p.

b) Suppose that pˆ is some unbiased estimator of p. Determine the Cramér-Rao Lower Bound for Var p[ ]ˆ based on one observation from this distribution.

c) Show that p= I _{1}(X) is an unbiased estimator of p. Does its variance achieve the Cramer-Rao Lower Bound?

Solutions

Expert Solution

Answer:-

Given That:-

Let X~Geometric(p), with parameter p unknown, 0<p<1.

a) Find I(p), the Fisher Information in X about p.

Fisher information in X about p is

Now,

ln f(x; p) = ln p + (x - 1) ln (1 - p) fro x = 1, 2, -------

b) Suppose that is some unbiased estimator of p. Determine the Cramér-Rao Lower Bound for Var p[ ]ˆ based on one observation from this distribution.

Based on one observation X from this distribution CRLB fro var (); being an Unbiased estimator of p; is

c) Show that p= I _{1}(X) is an unbiased estimator of p. Does its variance achieve the Cramer-Rao Lower Bound?

Hence,

is an unbiased estimator of p.

= p[X = 1] - p² [X = 1] = p - p² = p(1 - p)

Which is greater than p²(1 - p) = CRLB

Thus variance of does not achieve CRLB.

orchestra answered 1 year ago

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