Question: Suppose that we want to estimate the parameter of the geometric distribution on the basis of a single observation. If the loss function is

Suppose that we want to estimate the parameter θ of the geometric distribution on the basis of a single observation. If the loss function is given by

x(x + 1)6(1-)"-1 for 0

And Θ is looked upon as a random variable having the uniform density h(θ) = 1 for 0

(a) The conditional density of given X = x is

(b) The Bayes risk is minimized by the decision function d(x) = 2 / x + 2

x(x + 1)6(1-)"-1 for 0

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