Inferences for hierarchical models with partial prior information

Suppose items can be purchased from one of k-suppliers and it is required to purchase from the one with the smaller failure rate or equivalently from the one with the larger mean-time-to-failure. It is assumed that data $\mathit{d}$, in the form of the times-to-failure for $n_1,\dots ,n_k$ items from suppliers $1,\dots ,k$, respectively is available. There are two suggested selection criteria studied in this paper and when comparing only two suppliers they reduce to$P({\theta }_1<b{\theta }_2|d)\text{and}P(Y_1>{\mathit{cY}}_2|d)\text{,}$where b and c are prespecified practical constants; ${\theta }_1$ and ${\theta }_2$ are the respective mean failure rates; $Y_1$ and $Y_2$ are the predicted times to failure for individual items purchased from each supplier.In addition partial prior information about the $k$-suppliers collectively is assumed to have been elicited. This situation is modelled using the hierarchical Bayesian approach, which easily facilitates interpreting the elicited partial prior information as constraints on the hyperpriors, i.e. hyperpriors that are known only to be contained in families with specified properties. In this paper these properties are assumed to be in the form of specifying certain quantiles arising from the elicited information. Minimum and maximum values of the above selection criteria are obtained and are used to indicate whether or not the elicited prior information is useful. Specific examples are given for comparing two suppliers but generalisation to k-suppliers follows easily.     

Asymptotic normality of the deconvolution kernel density estimator under the vanishing error variance

Let $X_1,\dots ,X_n$ be i.i.d. observations, where $X_i=Y_i+{\sigma }_n{Z}_i$ and the $Y$’s and $Z$’s are independent. Assume that the $Y$’s are unobservable and that they have the density $f$ and also that the $Z$’s have a known density $k$. Furthermore, let ${\sigma }_n$ depend on $n$ and let ${\sigma }_n\to 0$ as $n\to \infty$. We consider the deconvolution problem, i.e. the problem of estimation of the density $f$ based on the sample $X_1,\dots ,X_n$. A popular estimator of $f$ in this setting is the deconvolution kernel density estimator. We derive its asymptotic normality under two different assumptions on the relation between the sequence ${\sigma }_n$ and the sequence of bandwidths $h_n$. We also consider several simulation examples which illustrate different types of asymptotics corresponding to the derived theoretical results and which show that there exist situations where models with ${\sigma }_n\to 0$ have to be preferred to the models with fixed $\sigma$.     

On the rate of convergence of uniform approximations for sequences of distribution functions

In this paper, we develop uniform bounds for the sequence of distribution functions of $g(V_n+{\mu }_n)$, where $g$ is some smooth function, $\{V_n,n\ge 1\}$ is a sequence of identically distributed random variables with common distribution having a bounded derivative and $\left\{{\mu }_n\right\}$ are constants such that ${\mu }_n\to \infty$. These bounds allow us to identify a suitable sequence of random variables which is asymptotically of the same type of $g(V_n+{\mu }_n)$ showing that the rate of convergence for these uniform approximations depends on the ratio of the second derivative to the first derivative of $g$. The corresponding generalization to the multivariate case is also analyzed. An application of our results to the STATIS-ACT method is provided in the final section.     

Laguerre and Hermite bases for inverse problems

We present inverse problems of nonparametric statistics which have a smart solution using projection estimators on bases of functions with non compact support, namely, a Laguerre basis or a Hermite basis. The models are $Y_i=X_i{U}_i,Z_i=X_i+{\Sigma }_i,$ where the $X_i$’s are i.i.d. with unknown density $f$, the ${\Sigma }_i$’s are i.i.d. with known density $f_{\Sigma }$, the $U_i$’s are i.i.d. with uniform density on $[0,1]$. The sequences $\left(X_i\right),\left(U_i\right),\left({\Sigma }_i\right)$ are independent. We define projection estimators of $f$ in the two cases of indirect observations of $(X_1,\dots ,X_n)$, and we give upper bounds for their ${\mathbb{L}}^2$-risks on specific Sobolev–Laguerre or Sobolev–Hermite spaces. Data-driven procedures are described and proved to perform automatically the bias–variance compromise.