Mathematical Statistics II · Consistency, asymptotic normality, and delta methods
Let X₁,…,Xₙ be iid N(θ²,1), with parameter θ ≥ 0
Problem
Let \(X_1,\ldots,X_n\) be iid \(N(\theta^2,1)\), with parameter \(\theta\ge0\). Find the maximum-likelihood estimator \(\hat\theta_n\). For a fixed \(\theta>0\), derive its usual root-\(n\) limit. At \(\theta=0\), find the scaling giving a nondegenerate limit, identify its CDF including any atom, and explain why the interior formula cannot be continued to the boundary. Finally, along \(\theta_n=c\,n^{-1/4}\), with fixed \(c\ge0\), find the distribution of \[ n^{1/4}(\hat\theta_n-\theta_n) \] and the probability that the estimator equals zero. The parameter is identifiable on \([0,\infty)\); explain how that fact can coexist with this nonregular local behavior.
Hint
Maximize first over the nonnegative mean \(u=\theta^2\), then transform back.
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