Confidence curves and improved exact confidence intervals for discrete distributions

The author describes a method for improving standard “exact” confidence intervals in discrete distributions with respect to size while retaining correct level. The binomial, negative binomial, hypergeometric, and Poisson distributions are considered explicitly. Contrary to other existing methods, the author's solution possesses a natural nesting condition: if α < α', the 1 ‐ α' confidence interval is included in the 1 ‐ α interval. Nonparametric confidence intervals for a quantile are also considered.     

Tolerance‐based error reduction in multi‐agent bioassays

Some degree of error is inevitable in multi‐agent bioassays regardless of design or measurement technology. Estimation error can be reduced post facto by exploiting the matrix partial ordering of the bioassay survival estimates. The standard method for this is order‐restricted regression (ORR). If the joint action of the bioassay agents admits a tolerance‐based interpretation, additional structure beyond matrix partial ordering is available, leading to a new method of error reduction. This tolerance‐based error reduction (TBER) procedure almost always outperforms ORR. Like ORR, TBER applies to complete factorial bioassay designs and, using weighting, to incomplete designs.     

Differential equation models for statistical functions

Differential equations have been used in statistics to define functions such as probability densities. But the idea of using differential equation formulations of stochastic models has a much wider scope. The author gives several examples, including simultaneous estimation of a regression model and residual density, monotone smoothing, specification of a link function, differential equation models of data, and smoothing over complicated multidimensional domains. This paper aims to stimulate interest in this approach to functional estimation problems, rather than provide carefully worked out methods.     

Directional mixture models and optimal estimation of the mixing density

The authors develop consistent nonparametric estimation techniques for the directional mixing density. Classical spherical harmonics are used to adapt Euclidean techniques to this directional environment. Minimax rates of convergence are obtained for rotation ally invariant densities verifying various smoothness conditions. It is found that the differences in smoothness between the Laplace, the Gaussian and the von Mises‐Fisher distributions lead to contrasting inferential conclusions.     

Posterior Analysis for Normalized Random Measures with Independent Increments

Abstract.  One of the main research areas in Bayesian Nonparametrics is the proposal and study of priors which generalize the Dirichlet process. In this paper, we provide a comprehensive Bayesian non-parametric analysis of random probabilities which are obtained by normalizing random measures with independent increments (NRMI). Special cases of these priors have already shown to be useful for statistical applications such as mixture models and species sampling problems. However, in order to fully exploit these priors, the derivation of the posterior distribution of NRMIs is crucial: here we achieve this goal and, indeed, provide explicit and tractable expressions suitable for practical implementation. The posterior distribution of an NRMI turns out to be a mixture with respect to the distribution of a specific latent variable. The analysis is completed by the derivation of the corresponding predictive distributions and by a thorough investigation of the marginal structure. These results allow to derive a generalized Blackwell–MacQueen sampling scheme, which is then adapted to cover also mixture models driven by general NRMIs.     

A simple two-stage design for quantitative responses with application to a study in diabetic neuropathic pain

Two-stage designs offer substantial advantages for early phase II studies. The interim analysis following the first stage allows the study to be stopped for futility, or more positively, it might lead to early progression to the trials needed for late phase II and phase III. If the study is to continue to its second stage, then there is an opportunity for a revision of the total sample size. Two-stage designs have been implemented widely in oncology studies in which there is a single treatment arm and patient responses are binary. In this paper the case of two-arm comparative studies in which responses are quantitative is considered. This setting is common in therapeutic areas other than oncology. It will be assumed that observations are normally distributed, but that there is some doubt concerning their standard deviation, motivating the need for sample size review. The work reported has been motivated by a study in diabetic neuropathic pain, and the development of the design for that trial is described in detail.     

Models for the analysis of C-reactive protein in statin trials

Elevation in C-reactive protein (CRP) is an independent risk factor for cardiovascular disease progression and levels are reduced by treatment with statins. However, on-treatment CRP, given baseline CRP and treatment, is not normally distributed and outliers exist even when transformations are applied. Although classical non-parametric tests address some of these issues, they do not enable straightforward inclusion of covariate information. The aims of this study were to produce a model that improved efficiency and accuracy of analysis of CRP data. Estimation of treatment effects and identification of outliers were addressed using controlled trials of rosuvastatin. The robust statistical technique of MM-estimation was used to fit models to data in the presence of outliers and was compared with least-squares estimation. To develop the model, appropriate transformations of the response and baseline variables were selected. The model was used to investigate how on-treatment CRP related to baseline CRP and estimated treatment effects with rosuvastatin. On comparing least-squares and MM-estimation, MM-estimation was superior to least-squares estimation in that parameter estimates were more efficient and outliers were clearly identified. Relative reductions in CRP were higher at higher baseline CRP levels. There was also evidence of a dose-response relationship between CRP reductions from baseline and rosuvastatin. Several large outliers were identified, although there did not appear to be any relationships between the incidence of outliers and treatments. In conclusion, using robust estimation to model CRP data is superior to least-squares estimation and non-parametric tests in terms of efficiency, outlier identification and the ability to include covariate information.     

Partitioning k multivariate normal populations according to equivalence with respect to a standard vector

We propose optimal procedures to achieve the goal of partitioning k multivariate normal populations into two disjoint subsets with respect to a given standard vector. Definition of good or bad multivariate normal populations is given according to their Mahalanobis distances to a known standard vector as being small or large. Partitioning k multivariate normal populations is reduced to partitioning k non-central Chi-square or non-central F distributions with respect to the corresponding non-centrality parameters depending on whether the covariance matrices are known or unknown. The minimum required sample size for each population is determined to ensure that the probability of correct decision attains a certain level. An example is given to illustrate our procedures.     

Inference of non-centrality parameter of a truncated non-central chi-squared distribution

Non-central chi-squared distribution plays a vital role in statistical testing procedures. Estimation of the non-centrality parameter provides valuable information for the power calculation of the associated test. We are interested in the statistical inference property of the non-centrality parameter estimate based on one observation (usually a summary statistic) from a truncated chi-squared distribution. This work is motivated by the application of the flexible two-stage design in case–control studies, where the sample size needed for the second stage of a two-stage study can be determined adaptively by the results of the first stage. We first study the moment estimate for the truncated distribution and prove its existence, uniqueness, and inadmissibility and convergence properties. We then define a new class of estimates that includes the moment estimate as a special case. Among this class of estimates, we recommend to use one member that outperforms the moment estimate in a wide range of scenarios. We also present two methods for constructing confidence intervals. Simulation studies are conducted to evaluate the performance of the proposed point and interval estimates.     

Characteristic function estimation of non-Gaussian Ornstein–Uhlenbeck processes

Continuous non-Gaussian stationary processes of the OU-type are becoming increasingly popular given their flexibility in modelling stylized features of financial series such as asymmetry, heavy tails and jumps. The use of non-Gaussian marginal distributions makes likelihood analysis of these processes unfeasible for virtually all cases of interest. This paper exploits the self-decomposability of the marginal laws of OU processes to provide explicit expressions of the characteristic function which can be applied to several models as well as to develop efficient estimation techniques based on the empirical characteristic function. Extensions to OU-based stochastic volatility models are provided.