Kernel and pseudokernel estimators for the a priori density of a multivariate parameter (Q1269938)

Consider a sequence of independent identically distributed pairs of random vectors \((X_1, \theta_1)\), \((X_2, \theta_2), \dots, (X_N, \theta_N)\), \(X_j \in\chi \subset R^s\), \(\theta_j \in\Omega \subset R^s\), \(j=1, \dots,N\). We assume that \(X_j\) are observable and for a given \(\theta_j\) have unknown regular conditional density \(q(x; \theta)\) with respect to the Lebesgue measure. Here \(\theta_j\) are independent nonobservable parameters having unknown density \(g(\theta)\), called the a priori density. The problem is to estimate the density \(g(\theta)\) in the case of the multivariate parameter \(\theta\). Introduce the following restriction: A. There are one-to-one continuously differentiable transformations \[ U:\mathbb{R}^s \to\chi \subset R^s,\qquad V:R^2 \to \Omega \subset R^s, \] and a function \(Q(\beta)\) satisfying the condition \[ q\bigl(U(y); V(\beta) \bigr)\bigl( {\mathcal T} U(y)\bigr) =Q(y-\beta), \quad y=(y_1, \dots, y_s),\;\beta= (t_1, \dots, t_s). \] Here and in what follows, \({\mathcal T} U(y)\) stands for the Jacobian of the transformation \(U(y)\). Bold print is used to indicate \(s\)-dimensional row vectors. Transposition will be denoted by the prime. Under restriction \(A\) two problems are solved. Namely, the case where \(\theta\) is a multivariate location parameter: \(q(x, \theta)= q(x-\theta)\), \(x,\theta \in(- \infty; \infty)\); a kernel estimator for \(g (\theta)\) will be obtained. If \(\theta\) is not a shift parameter, but the restriction \(A\) holds, a pseudokernel estimate for \(g(\theta)\) will also be presented. The asymptotic properties of the constructed estimates are studied, and some examples are discussed.