On lower bounds for errors of prior density estimators (Q1355318)

Consider a sequence of independent identically distributed random pairs \((X_1, \theta_1), \dots, (X_N,\theta_N)\) defined on the product of measurable spaces \((\mathcal{X} \times \Omega, A \times B)\), where \(\mathcal{X}, \Omega\) are some sets of Euclidean spaces, \(A,B\) are \(\sigma\)-algebras, and \(\nu,\mu\) are some measures on \(A\) and \(B\), respectively. It is assumed that there exists a known regular conditional density \(q(x \mid\theta)\) of the random variable \(X\) given \(\theta\). Let the values \(X_i\in \mathcal{X}\) be observable and \(\theta_i\in \Omega\) be not observable and have unknown density \(g(\theta)\) called prior. It is required to estimate the density \(g(\theta)\) given the observations \(X_1, \dots, X_N\). We characterize the performance of the prior density estimator by the lower bounds of its error. For a wide class of sets \(\mathcal{X}, \Omega\) and densities \(q(x\mid \theta)\), the lower bounds of errors are presented. In particular, no restrictions will be imposed on the dimensions of \(X\) and \(\theta\).