Search

We consider the estimation of two-sample integral functionals, of the type that occur naturally, for example, when the object of interest is a divergence between unknown probability densities. Our first main result is that, in wide generality, a weighted nearest neighbour estimator is efficient, in the sense of achieving the local asymptotic minimax lower bound. Moreover, we also prove a corresponding central limit theorem, which facilitates the construction of asymptotically valid confidence intervals for the functional, having asymptotically minimal width. One interesting consequence of our results is the discovery that, for certain functionals, the worst-case performance of our estimator may improve on that of the natural `oracle' estimator, which is given access to the values of the unknown densities at the observations., Comment: 76 pages

There are a variety of settings where vague prior information may be available on the importance of predictors in high-dimensional regression settings. Examples include ordering on the variables offered by their empirical variances (which is typically discarded through standardisation), the lag of predictors when fitting autoregressive models in time series settings, or the level of missingness of the variables. Whilst such orderings may not match the true importance of variables, we argue that there is little to be lost, and potentially much to be gained, by using them. We propose a simple scheme involving fitting a sequence of models indicated by the ordering. We show that the computational cost for fitting all models when ridge regression is used is no more than for a single fit of ridge regression, and describe a strategy for Lasso regression that makes use of previous fits to greatly speed up fitting the entire sequence of models. We propose to select a final estimator by cross-validation and provide a general result on the quality of the best performing estimator on a test set selected from among a number $M$ of competing estimators in a high-dimensional linear regression setting. Our result requires no sparsity assumptions and shows that only a $\log M$ price is incurred compared to the unknown best estimator. We demonstrate the effectiveness of our approach when applied to missing or corrupted data, and time series settings. An R package is available on github., Comment: To appear in Statistics and Computing

Mixed outcome endpoints that combine multiple continuous and discrete components to form co-primary, multiple primary or composite endpoints are often employed as primary outcome measures in clinical trials. There are many advantages to joint modelling the individual outcomes using a latent variable framework, however in order to make use of the model in practice we require techniques for sample size estimation. In this paper we show how the latent variable model can be applied to the three types of joint endpoints and propose appropriate hypotheses, power and sample size estimation methods for each. We illustrate the techniques using a numerical example based on the four dimensional endpoint in the MUSE trial and find that the sample size required for the co-primary endpoint is larger than that required for the individual endpoint with the smallest effect size. Conversely, the sample size required for the multiple primary endpoint is reduced from that required for the individual outcome with the largest effect size. We show that the analytical technique agrees with the empirical power from simulation studies. We further illustrate the reduction in required sample size that may be achieved in trials of mixed outcome composite endpoints through a simulation study and find that the sample size primarily depends on the components driving response and the correlation structure and much less so on the treatment effect structure in the individual endpoints., 36 pages, 8 figures, 7 tables

When simultaneously testing multiple hypotheses, the usual approach in the context of confirmatory clinical trials is to control the familywise error rate (FWER), which bounds the probability of making at least one false rejection. In many trial settings, these hypotheses will additionally have a hierarchical structure that reflects the relative importance and links between different clinical objectives. The graphical approach of Bretz et al. (2009) is a flexible and easily communicable way of controlling the FWER while respecting complex trial objectives and multiple structured hypotheses. However, the FWER can be a very stringent criterion that leads to procedures with low power, and may not be appropriate in exploratory trial settings. This motivates controlling generalised error rates, particularly when the number of hypotheses tested is no longer small. We consider the generalised familywise error rate (k-FWER), which is the probability of making k or more false rejections, as well as the tail probability of the false discovery proportion (FDP), which is the probability that the proportion of false rejections is greater than some threshold. We also consider asymptotic control of the false discovery rate (FDR), which is the expectation of the FDP. In this paper, we show how to control these generalised error rates when using the graphical approach and its extensions. We demonstrate the utility of the resulting graphical procedures on three clinical trial case studies., Biometrika Trust; Medical Research Council, Grant/Award Numbers: MC∖UU∖00002/6, MR/N028171/1