Statistics | Part IB, 2004

It is required to estimate the unknown parameter θ\theta after observing XX, a single random variable with probability density function f(xθ)f(x \mid \theta); the parameter θ\theta has the prior distribution with density π(θ)\pi(\theta) and the loss function is L(θ,a)L(\theta, a). Show that the optimal Bayesian point estimate minimizes the posterior expected loss.

Suppose now that f(xθ)=θeθx,x>0f(x \mid \theta)=\theta e^{-\theta x}, x>0 and π(θ)=μeμθ,θ>0\pi(\theta)=\mu e^{-\mu \theta}, \theta>0, where μ>0\mu>0 is known. Determine the posterior distribution of θ\theta given XX.

Determine the optimal Bayesian point estimate of θ\theta in the cases when

(i) L(θ,a)=(θa)2L(\theta, a)=(\theta-a)^{2}, and

(ii) L(θ,a)=(θa)/θL(\theta, a)=|(\theta-a) / \theta|.

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