Further Properties of Frequentist Confidence Intervals in Regression that Utilize Uncertain Prior Information

Consider a linear regression model with n‐dimensional response vector, regression parameter and independent and identically distributed errors. Suppose that the parameter of interest is where a is a specified vector. Define the parameter where c and t are specified. Also suppose that we have uncertain prior information that . Part of our evaluation of a frequentist confidence interval for is the ratio (expected length of this confidence interval)/(expected length of standard confidence interval), which we call the scaled expected length of this interval. We say that a confidence interval for utilizes this uncertain prior information if: (i) the scaled expected length of this interval is substantially less than 1 when ; (ii) the maximum value of the scaled expected length is not too much larger than 1; and (iii) this confidence interval reverts to the standard confidence interval when the data happen to strongly contradict the prior information. Kabaila and Giri (2009) present a new method for finding such a confidence interval. Let denote the least squares estimator of . Also let and . Using computations and new theoretical results, we show that the performance of this confidence interval improves as increases and decreases.     

Empirical Likelihood for Compound Poisson Processes

Let {N(t), t > 0} be a Poisson process with rate λ > 0, independent of the independent and identically distributed random variables with mean μ and variance . The stochastic process is then called a compound Poisson process and has a wide range of applications in, for example, physics, mining, finance and risk management. Among these applications, the average number of objects, which is defined to be λμ, is an important quantity. Although many papers have been devoted to the estimation of λμ in the literature, in this paper, we use the well‐known empirical likelihood method to construct confidence intervals. The simulation results show that the empirical likelihood method often outperforms the normal approximation and Edgeworth expansion approaches in terms of coverage probabilities. A real data set concerning coal‐mining disasters is analyzed using these methods.     

Central Limit Theorems for Reduced U‐Statistics Under Dependence and Their Usefulness

A reduced ‐statistic is a ‐statistic with its summands drawn from a restricted but balanced set of pairs. In this article, central limit theorems are derived for reduced ‐statistics under ‐mixing, which significantly extends the work of Brown & Kildea in various aspects. It will be shown and illustrated that reduced ‐statistics are quite useful in deriving test statistics in various nonparametric testing problems.     

Random Weighting Estimation of Confidence Intervals for Quantiles

This paper presents a new random weighting method for confidence interval estimation for the sample ‐quantile. A theory is established to extend ordinary random weighting estimation from a non‐smoothed function to a smoothed function, such as a kernel function. Based on this theory, a confidence interval is derived using the concept of backward critical points. The resultant confidence interval has the same length as that derived by ordinary random weighting estimation, but is distribution‐free, and thus it is much more suitable for practical applications. Simulation results demonstrate that the proposed random weighting method has higher accuracy than the Bootstrap method for confidence interval estimation.     

Variable Selection in Linear Mixed Models Using an Extended Class of Penalties

There is an emerging need to advance linear mixed model technology to include variable selection methods that can simultaneously choose and estimate important effects from a potentially large number of covariates. However, the complex nature of variable selection has made it difficult for it to be incorporated into mixed models. In this paper we extend the well known class of penalties and show that they can be integrated succinctly into a linear mixed model setting. Under mild conditions, the estimator obtained from this mixed model penalised likelihood is shown to be consistent and asymptotically normally distributed. A simulation study reveals that the extended family of penalties achieves varying degrees of estimator shrinkage depending on the value of one of its parameters. The simulation study also shows there is a link between the number of false positives detected and the number of true coefficients when using the same penalty. This new mixed model variable selection (MMVS) technology was applied to a complex wheat quality data set to determine significant quantitative trait loci (QTL).     

Autocovariance Estimation in Regression with a Discontinuous Signal and m‐Dependent Errors: A Difference‐Based Approach

We discuss a class of difference‐based estimators for the autocovariance in nonparametric regression when the signal is discontinuous and the errors form a stationary m‐dependent process. These estimators circumvent the particularly challenging task of pre‐estimating such an unknown regression function. We provide finite‐sample expressions of their mean squared errors for piecewise constant signals and Gaussian errors. Based on this, we derive biased‐optimized estimates that do not depend on the unknown autocovariance structure. Notably, for positively correlated errors, that part of the variance of our estimators that depend on the signal is minimal as well. Further, we provide sufficient conditions for ‐consistency; this result is extended to piecewise Hölder regression with non‐Gaussian errors. We combine our biased‐optimized autocovariance estimates with a projection‐based approach and derive covariance matrix estimates, a method that is of independent interest. An R package, several simulations and an application to biophysical measurements complement this paper.     

Empirical Likelihood for Linear Models Under Linear Process Errors

In this article, we study the construction of confidence intervals for regression parameters in a linear model under linear process errors by using the blockwise technique. It is shown that the blockwise empirical likelihood (EL) ratio statistic is asymptotically χ²-type distributed. The result is used to obtain EL based confidence regions for regression parameters. The finite-sample performance of the method is evaluated through a simulation study.     

Confidence intervals for probability density functions under associated samples

In this paper, we study the construction of confidence intervals for a probability density function under a (positively) associated sample by using the blockwise technique. It is shown that the blockwise empirical likelihood (EL) ratio statistic is asymptotically χ²-type${\chi }^2\mathrm{-}\mathrm{type}$ distributed. The result is used to obtain EL-based confidence interval for the probability density function.     

Empirical likelihood for probability density functions under negatively associated samples

In this paper, we study the construction of confidence intervals for a probability density function under a negatively associated sample by using the blockwise technique. It is shown that the blockwise empirical likelihood (EL) ratio statistic is asymptotically χ²‐type distributed. The result is used to obtain EL based confidence interval on the probability density function.     

Wavelet Thresholding Estimation in a Poissonian Interactions Model with Application to Genomic Data

This paper deals with the study of dependencies between two given events modelled by point processes. In particular, we focus on the context of DNA to detect favoured or avoided distances between two given motifs along a genome suggesting possible interactions at a molecular level. For this, we naturally introduce a so‐called reproduction function h that allows to quantify the favoured positions of the motifs and that is considered as the intensity of a Poisson process. Our first interest is the estimation of this function h assumed to be well localized. The estimator based on random thresholds achieves an oracle inequality. Then, minimax properties of on Besov balls are established. Some simulations are provided, proving the good practical behaviour of our procedure. Finally, our method is applied to the analysis of the dependence between promoter sites and genes along the genome of the Escherichia coli bacterium.     

Joint estimation of multiple high‐dimensional Gaussian copula graphical models

A joint estimation approach for multiple high‐dimensional Gaussian copula graphical models is proposed, which achieves estimation robustness by exploiting non‐parametric rank‐based correlation coefficient estimators. Although we focus on continuous data in this paper, the proposed method can be extended to deal with binary or mixed data. Based on a weighted minimisation problem, the estimators can be obtained by implementing second‐order cone programming. Theoretical properties of the procedure are investigated. We show that the proposed joint estimation procedure leads to a faster convergence rate than estimating the graphs individually. It is also shown that the proposed procedure achieves an exact graph structure recovery with probability tending to 1 under certain regularity conditions. Besides theoretical analysis, we conduct numerical simulations to compare the estimation performance and graph recovery performance of some state‐of‐the‐art methods including both joint estimation methods and estimation methods for individuals. The proposed method is then applied to a gene expression data set, which illustrates its practical usefulness.     

Modelling columnarity of pyramidal cells in the human cerebral cortex

For modelling the location of pyramidal cells in the human cerebral cortex, we suggest a hierarchical point process in that exhibits anisotropy in the form of cylinders extending along the z-axis. The model consists first of a generalised shot noise Cox process for the xy-coordinates, providing cylindrical clusters, and next of a Markov random field model for the z-coordinates conditioned on the xy-coordinates, providing either repulsion, aggregation or both within specified areas of interaction. Several cases of these hierarchical point processes are fitted to two pyramidal cell data sets, and of these a final model allowing for both repulsion and attraction between the points seem adequate. We discuss how the final model relates to the so-called minicolumn hypothesis in neuroscience.     

Semiparametric Mixtures of Symmetric Distributions

We consider in this paper the semiparametric mixture of two unknown distributions equal up to a location parameter. The model is said to be semiparametric in the sense that the mixed distribution is not supposed to belong to a parametric family. To insure the identifiability of the model, it is assumed that the mixed distribution is zero symmetric, the model being then defined by the mixing proportion, two location parameters and the probability density function of the mixed distribution. We propose a new class of M‐estimators of these parameters based on a Fourier approach and prove that they are ‐consistent under mild regularity conditions. Their finite sample properties are illustrated by a Monte Carlo study, and a benchmark real dataset is also studied with our method.     

Confidence intervals following Box-Cox transformation

What is the interpretation of a confidence interval following estimation of a Box-Cox transformation parameter λ? Several authors have argued that confidence intervals for linear model parameters ψ can be constructed as if λ. were known in advance, rather than estimated, provided the estimand is interpreted conditionally given $\hat \lambda$. If the estimand is defined as $\psi \left( {\hat \lambda } \right)$, a function of the estimated transformation, can the nominal confidence level be regarded as a conditional coverage probability given $\hat \lambda$, where the interval is random and the estimand is fixed? Or should it be regarded as an unconditional probability, where both the interval and the estimand are random? This article investigates these questions via large-n approximations, small- σ approximations, and simulations. It is shown that, when model assumptions are satisfied and n is large, the nominal confidence level closely approximates the conditional coverage probability. When n is small, this conditional approximation is still good for regression models with small error variance. The conditional approximation can be poor for regression models with moderate error variance and single-factor ANOVA models with small to moderate error variance. In these situations the nominal confidence level still provides a good approximation for the unconditional coverage probability. This suggests that, while the estimand may be interpreted conditionally, the confidence level should sometimes be interpreted unconditionally.     

Empirical Likelihood Confidence Intervals for Distribution Functions under Negatively Associated Samples

In this article, we discuss the construction of the confidence intervals for distribution functions under negatively associated samples. It is shown that the blockwise empirical likelihood (EL) ratio statistic for a distribution function is asymptotically χ²-type distributed. The result is used to obtain an EL-based confidence interval for the distribution function.     

Uncertainty Quantification in Case of Imperfect Models: A Non‐Bayesian Approach

The starting point in uncertainty quantification is a stochastic model, which is fitted to a technical system in a suitable way, and prediction of uncertainty is carried out within this stochastic model. In any application, such a model will not be perfect, so any uncertainty quantification from such a model has to take into account the inadequacy of the model. In this paper, we rigorously show how the observed data of the technical system can be used to build a conservative non‐asymptotic confidence interval on quantiles related to experiments with the technical system. The construction of this confidence interval is based on concentration inequalities and order statistics. An asymptotic bound on the length of this confidence interval is presented. Here we assume that engineers use more and more of their knowledge to build models with order of errors bounded by . The results are illustrated by applying the newly proposed approach to real and simulated data.     

Optimal adaptive sequential designs for crossover bioequivalence studies

In prior works, this group demonstrated the feasibility of valid adaptive sequential designs for crossover bioequivalence studies. In this paper, we extend the prior work to optimize adaptive sequential designs over a range of geometric mean test/reference ratios (GMRs) of 70–143% within each of two ranges of intra‐subject coefficient of variation (10–30% and 30–55%). These designs also introduce a futility decision for stopping the study after the first stage if there is sufficiently low likelihood of meeting bioequivalence criteria if the second stage were completed, as well as an upper limit on total study size. The optimized designs exhibited substantially improved performance characteristics over our previous adaptive sequential designs. Even though the optimized designs avoided undue inflation of type I error and maintained power at 80%, their average sample sizes were similar to or less than those of conventional single stage designs. Copyright © 2015 John Wiley & Sons, Ltd.     

Constrained nonparametric maximum likelihood estimation of stochastically ordered survivor functions

This paper considers estimators of survivor functions subject to a stochastic ordering constraint based on right censored data. We present the constrained nonparametric maximum likelihood estimator (C‐NPMLE) of the survivor functions in one‐and two‐sample settings where the survivor distributions could be discrete or continuous and discuss the non‐uniqueness of the estimators. We also present a computationally efficient algorithm to obtain the C‐NPMLE. To address the possibility of non‐uniqueness of the C‐NPMLE of $S_1(t)$ when $S_1(t)\le S_2(t)$ , we consider the maximum C‐NPMLE (MC‐NPMLE) of $S_1(t)$ . In the one‐sample case with arbitrary upper bound survivor function $S_2(t)$ , we present a novel and efficient algorithm for finding the MC‐NPMLE of $S_1(t)$ . Dykstra ( 1982 ) also considered constrained nonparametric maximum likelihood estimation for such problems, however, as we show, Dykstra's method has an error and does not always give the C‐NPMLE. We corrected this error and simulation shows improvement in efficiency compared to Dykstra's estimator. Confidence intervals based on bootstrap methods are proposed and consistency of the estimators is proved. Data from a study on larynx cancer are analysed to illustrate the method. The Canadian Journal of Statistics 40: 22–39; 2012 © 2012 Statistical Society of Canada