Prior specification for nonparametric Bayesian inference involves the
difficult task of quantifying prior knowledge about a parameter of high, often
infinite, dimension. Realistically, a statistician is unlikely to have informed
opinions about all aspects of such a parameter, but may have real information
about functionals of the parameter, such the population mean or variance.
Novel dose-finding designs, using estimation to assign the best estimated
maximum- tolerated-dose (MTD) at each point in the experiment, most commonly
via Bayesian techniques, have recently entered large-scale implementation in
Phase I cancer clinical trials. We examine the small-sample behavior of these
"Bayesian Phase I" (BP1) designs, and also of non-Bayesian designs sharing the
same main "long-memory" traits (hereafter: LMP1s).
Often of primary interest in the analysis of multivariate data are the copula
parameters describing the dependence among the variables, rather than the
univariate marginal distributions. Since the ranks of a multivariate dataset
are invariant to changes in the univariate marginal distributions, rank-based
procedures are natural candidates as semiparametric estimators of copula
parameters. Asymptotic information bounds for such estimators can be obtained
from an asymptotic analysis of the rank likelihood, i.e. the probability of the
multivariate ranks.
The focus of this paper is an approach to the modeling of longitudinal social
relational or network data. Such data arise from measurements on pairs of
objects or actors made at regular temporal intervals, resulting in a social
network for each point in time. In this article we represent the network and
temporal dependencies with a random effects model, resulting in a stochastic
process defined by a set of stationary covariance matrices.
Modern datasets are often in the form of matrices or arrays,potentially
having correlations along each set of data indices. For example, data involving
repeated measurements of several variables over time may exhibit temporal
correlation as well as correlation among the variables. A possible model for
matrix-valued data is the class of matrix normal distributions, which is
parametrized by two covariance matrices, one for each index set of the data. In
this article we describe an extension of the matrix normal model to accommodate
multidimensional data arrays, or tensors.
In this article we evaluate the statistical evidence that a population of
students learn about the sub-game perfect Nash equilibrium of the centipede
game via repeated play of the game. This is done by formulating a model in
which a player's error in assessing the utility of decisions changes as they
gain experience with the game. We first estimate parameters in a statistical
model where the probabilities of choices of the players are given by a Quantal
Response Equilibrium (QRE) (McKelvey and Palfrey, 1995, 1996, 1998), but are
allowed to change with repeated play.
We examine Phase I cancer clinical trial designs that use toxicity estimates
based on all available data at each dose-allocation decision, but refrain from
employing parametric models or Bayesian decision rules. We show that one such
design family, called here "interval designs", converges almost surely to the
maximum tolerated dose under fairly general conditions. Another family called
"point designs" does not converge.