Abstract: This paper deals with the Intersection Property, or Basu's First Theorem, which is valid under a condition of no common information, also known as measurable separability. After formalizing this notion, the paper reviews general properties and give operational characterizations in two topical cases the finite one and the multivariate normal one. The paper concludes discussing the relevance of these characterizations for different fields as graphical models, zero entries in contingency tables, causal analysis and estimability in Markov processes.

Abstract: The SiZer Map, proposed by Chaudhuri and Marron (1999), is a statistical tool for finding which features in noisy data are strong enough to be distinguished from background noise.In this paper, we propose the local likelihood SiZer map.Some simulation examples illustrate that the newly proposed SiZer map is more efficient in distinguishing features than the original one, because of the inferential advantage of the local likelihood approach.Some computational problems are addressed, with the result that the computational cost in constructing the local likelihood SiZer map is close to that of the original one.

Abstract: Since its introduction by Bak, Tang and Wiessenfeld, the abelian sandpile dynamics has been studied extensively in finite volume. There are many problems posed by the existence of a sandpile dynamics in an infinite volume S: its invariant distribution should be the thermodynamic limit (does the latter exist?) of the invariant measure for the finite volume dynamics; the extension of the sand grains addition operator to infinite volume is related to the boundary effects of the dynamics in finite volume; finally, the crucial difficulty of the definition of a Markov process in infinite volume is that, due to sand avalanches, the interaction is long range, so that no use of the Hille-Yosida theorem is possible. In this review paper, we recall the needed results in finite volume, then explain how to deal with infinite volume when $S={\Bbb Z},S={\Bbb T}$ is an infinite tree, $S={\Bbb Z}^{d}$ with d large, and when the dynamics is dissipative (i.e. sand grains may disappear at each toppling).

Abstract: This paper considers a nonparametric model in which (i) the conditional quantile function is assumed to be nondecreasing and (ii) the distribution of the error disturbance may depend upon the covariate. For this general model, the (pointwise) limiting distribution of the isotonic quantile regression estimator (Casady and Cryer (1976)) is derived. Since the bootstrap is not consistent for this estimator, an adjusted version of the bootstrap is discussed as an alternative method for inference. An empirical application to birthweights is considered.

Abstract: We consider estimation of quantiles when data are generated from ranked set sampling. A new estimator is proposed and is shown to have a smaller asymptotic variance for all distributions. It is also shown that the optimal sampling strategy is to select observations with one fixed rank from different ranked sets. Both the optimal rank and the relative efficiency gain with respect to simple random sampling are distribution-free and depend on the set size and the given probability only. In the case of median estimation, it is analytically shown that the optimal design is to select the median from each ranked set.

Abstract: The estimation of a mean of a normal distribution with an unknown variance is addressed under the restriction that the coefficient of variation is within a bounded interval. The paper constructs a class of estimators improving upon the best location-scale equivariant estimator of the mean. It is demonstrated that the class includes three typical estimators: the generalized Bayes esti mator based on the uniform prior over the restricted region, the generalized Bayes estimator based on the prior putting mass on the boundary, and a truncated estimator. The non-minimaxity of the best location-scale equiv ariant estimator is shown in the general location-scale family. When another type of restriction is considered, however, we have a different story that the best location-scale equivariant estimator remains minimax. AMS (2000) subject classification. Primary 62C10; secondary 62C20, 62F10.

Abstract: c-optimal design problems for weighted polynomial models are discussed. Vectors c, where c-optimal designs are supported by some extreme points of a certain equioscillating function, are characterized, and this equioscillating function is a linear combination of the regression functions. These results are then applied to the no-intercept model in which the optimal designs for estimating certain individual parameters can be found. Examples of applications of the above results in finding locally c-optimal designs for some nonlinear models are discussed. Finally the results are extended to a more general linear model. AMS (2000) subject classification. 62K05.