Mathematical Statistics I · Loss, risk, and introductory decision theory
A hidden state θ is either zero or one
Problem
A hidden state \(\theta\) is either zero or one. A binary observation has \[ \Pr_0(X=1)=\frac15,\qquad \Pr_1(X=1)=\frac7{10}. \] The action \(a\in\{0,1\}\) declares a state. Correct declarations have loss zero. Declaring one when the state is zero costs \(2\); declaring zero when the state is one costs \(3\). For a decision rule \(\delta\), its risk is \[ R(\theta,\delta)=E_\theta L(\theta,\delta(X)). \] Compute the risk pair \((R(0,\delta),R(1,\delta))\) for each deterministic rule: always declare zero, always declare one, report \(X\), and report \(1-X\). Identify a rule that is dominated by another, meaning the other rule has no larger risk at either state and strictly smaller risk at at least one. Explain the difference between a realized loss and a risk.
Hint
Hold the state fixed while averaging over observations.
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