High‐order Corrected Estimator of Asymptotic Variance with Optimal Bandwidth

Estimation of time‐average variance constant (TAVC), which is the asymptotic variance of the sample mean of a dependent process, is of fundamental importance in various fields of statistics. For frequentists, it is crucial for constructing confidence interval of mean and serving as a normalizing constant in various test statistics and so forth. For Bayesians, it is widely used for evaluating effective sample size and conducting convergence diagnosis in Markov chain Monte Carlo method. In this paper, by considering high‐order corrections to the asymptotic biases, we develop a new class of TAVC estimators that enjoys optimal ‐convergence rates under different degrees of the serial dependence of stochastic processes. The high‐order correction procedure is applicable to estimation of the so‐called smoothness parameter, which is essential in determining the optimal bandwidth. Comparisons with existing TAVC estimators are comprehensively investigated. In particular, the proposed optimal high‐order corrected estimator has the best performance in terms of mean squared error.     

Bayesian Analysis of the Proportional Hazards Model with Time‐Varying Coefficients

We study a Bayesian analysis of the proportional hazards model with time‐varying coefficients. We consider two priors for time‐varying coefficients – one based on B‐spline basis functions and the other based on Gamma processes – and we use a beta process prior for the baseline hazard functions. We show that the two priors provide optimal posterior convergence rates (up to the term) and that the Bayes factor is consistent for testing the assumption of the proportional hazards when the two priors are used for an alternative hypothesis. In addition, adaptive priors are considered for theoretical investigation, in which the smoothness of the true function is assumed to be unknown, and prior distributions are assigned based on B‐splines.     

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.     

A Class of Pseudolikelihood Ratio Tests for Homogeneity in Exponential Tilt Mixture Models

Mixture models are commonly used in biomedical research to account for possible heterogeneity in population. In this paper, we consider tests for homogeneity between two groups in the exponential tilt mixture models. A novel pairwise pseudolikelihood approach is proposed to eliminate the unknown nuisance function. We show that the corresponding pseudolikelihood ratio test has an asymptotic distribution as a supremum of two squared Gaussian processes under the null hypothesis. To maintain the appeal of simplicity for conventional likelihood ratio tests, we propose two alternative tests, both shown to have a simple asymptotic distribution of under the null. Simulation studies show that the proposed class of pseudolikelihood ratio tests performs well in controlling type I errors and having competitive powers compared with the current tests. The proposed tests are illustrated by an example of partial differential expression detection using microarray data from prostate cancer patients.     

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.     

Adaptive Deconvolution on the Non‐negative Real Line

In this paper, we consider the problem of adaptive density or survival function estimation in an additive model defined by Z=X+Y with X independent of Y, when both random variables are non‐negative. This model is relevant, for instance, in reliability fields where we are interested in the failure time of a certain material that cannot be isolated from the system it belongs. Our goal is to recover the distribution of X (density or survival function) through n observations of Z, assuming that the distribution of Y is known. This issue can be seen as the classical statistical problem of deconvolution that has been tackled in many cases using Fourier‐type approaches. Nonetheless, in the present case, the random variables have the particularity to be supported. Knowing that, we propose a new angle of attack by building a projection estimator with an appropriate Laguerre basis. We present upper bounds on the mean squared integrated risk of our density and survival function estimators. We then describe a non‐parametric data‐driven strategy for selecting a relevant projection space. The procedures are illustrated with simulated data and compared with the performances of a more classical deconvolution setting using a Fourier approach. Our procedure achieves faster convergence rates than Fourier methods for estimating these functions.     

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.     

Nonparametric Benchmark Dose Estimation with Continuous Dose‐Response Data

We propose a new method for risk‐analytic benchmark dose (BMD) estimation in a dose‐response setting when the responses are measured on a continuous scale. For each dose level d, the observation X(d) is assumed to follow a normal distribution: . No specific parametric form is imposed upon the mean μ(d), however. Instead, nonparametric maximum likelihood estimates of μ(d) and σ are obtained under a monotonicity constraint on μ(d). For purposes of quantitative risk assessment, a ‘hybrid’ form of risk function is defined for any dose d as R(d) = P[X(d) < c], where c > 0 is a constant independent of d. The BMD is then determined by inverting the additional risk functionR_A(d) = R(d) ? R(0) at some specified value of benchmark response. Asymptotic theory for the point estimators is derived, and a finite‐sample study is conducted, using both real and simulated data. When a large number of doses are available, we propose an adaptive grouping method for estimating the BMD, which is shown to have optimal mean integrated squared error under appropriate designs.     

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.     

On Projection‐type Estimators of Multivariate Isotonic Functions

Abstract. Let M be an isotonic real‐valued function on a compact subset of and let be an unconstrained estimator of M. A feasible monotonizing technique is to take the largest (smallest) monotone function that lies below (above) the estimator or any convex combination of these two envelope estimators. When the process is asymptotically equicontinuous for some sequence r_n→∞, we show that these projection‐type estimators are r_n‐equivalent in probability to the original unrestricted estimator. Our first motivating application involves a monotone estimator of the conditional distribution function that has the distributional properties of the local linear regression estimator. Applications also include the estimation of econometric (probability‐weighted moment, quantile) and biometric (mean remaining lifetime) functions.     

Estimation of a Copula when a Covariate Affects only Marginal Distributions

This paper is concerned with studying the dependence structure between two random variables Y₁ and Y₂ in the presence of a covariate X, which affects both marginal distributions but not the dependence structure. This is reflected in the property that the conditional copula of Y₁ and Y₂ given X, does not depend on the value of X. This latter independence often appears as a simplifying assumption in pair‐copula constructions. We introduce a general estimator for the copula in this specific setting and establish its consistency. Moreover, we consider some special cases, such as parametric or nonparametric location‐scale models for the effect of the covariate X on the marginals of Y₁ and Y₂ and show that in these cases, weak convergence of the estimator, at ‐rate, holds. The theoretical results are illustrated by simulations and a real data example.     

On the robustness to outliers of the Student-t process

The theory of Bayesian robustness modeling uses heavy-tailed distributions to resolve conflicts of information by rejecting automatically the outlying information in favor of the other sources of information. In particular, the Student's-t process is a natural alternative to the Gaussian process when the data might carry atypical information. Several works attest to the robustness of the Student $t$ process, however, the studies are mostly guided by intuition and focused mostly on the computational aspects rather than the mathematical properties of the involved distributions. This work uses the theory of regular variation to address the robustness of the Student $t$ process in the context of nonlinear regression, that is, the behavior of the posterior distribution in the presence of outliers in the inputs, in the outputs, or in both sources of information. In all these cases, under certain conditions, it is shown that the posterior distribution tends to a quantity that does not depend on the atypical information, then, for every case, the limiting posterior distribution as the outliers tend to infinity is provided. The impact of outliers on the predictive posterior distribution is also addressed. The theory is illustrated with a few simulated examples.     

Rate efficient estimation of realized Laplace transform of volatility with microstructure noise

In this paper, we consider the problem of estimating the Laplace transform of volatility within a fixed time interval [0,T] using high‐frequency sampling, where we assume that the discretized observations of the latent process are contaminated by microstructure noise. We use the pre‐averaging approach to deal with the effect of microstructure noise. Under the high‐frequency scenario, we obtain a consistent estimator whose convergence rate is , which is known as the optimal convergence rate of the estimation of integrated volatility functionals under the presence of microstructure noise. The related central limit theorem is established. The simulation studies justify the finite‐sample performance of the proposed estimator.     

Sparse concordance-based ordinal classification

Ordinal classification is an important area in statistical machine learning, where labels exhibit a natural order. One of the major goals in ordinal classification is to correctly predict the relative order of instances. We develop a novel concordance-based approach to ordinal classification, where a concordance function is introduced and a penalized smoothed method for optimization is designed. Variable selection using the $L_1$ penalty is incorporated for sparsity considerations. Within the set of classification rules that maximize the concordance function, we find optimal thresholds to predict labels by minimizing a loss function. After building the classifier, we derive nonparametric estimation of class conditional probabilities. The asymptotic properties of the estimators as well as the variable selection consistency are established. Extensive simulations and real data applications show the robustness and advantage of the proposed method in terms of classification accuracy, compared with other existing methods.     

Exponential Family Techniques for the Lognormal Left Tail

Let X be lognormal(μ,σ²) with density f(x); let θ > 0 and define . We study properties of the exponentially tilted density (Esscher transform) f_θ(x) = e^?θxf(x)/L(θ), in particular its moments, its asymptotic form as θ→∞ and asymptotics for the saddlepoint θ(x) determined by . The asymptotic formulas involve the Lambert W function. The established relations are used to provide two different numerical methods for evaluating the left tail probability of the sum of lognormals S_n=X₁+?+X_n: a saddlepoint approximation and an exponential tilting importance sampling estimator. For the latter, we demonstrate logarithmic efficiency. Numerical examples for the cdf F_n(x) and the pdf f_n(x) of S_n are given in a range of values of σ²,n and x motivated by portfolio value‐at‐risk calculations.     

Selection of linear mixed-effects models for clustered data

We consider model selection for linear mixed-effects models with clustered structure, where conditional Kullback–Leibler (CKL) loss is applied to measure the efficiency of the selection. We estimate the CKL loss by substituting the empirical best linear unbiased predictors (EBLUPs) into random effects with model parameters estimated by maximum likelihood. Although the BLUP approach is commonly used in predicting random effects and future observations, selecting random effects to achieve asymptotic loss efficiency concerning CKL loss is challenging and has not been well studied. In this paper, we propose addressing this difficulty using a conditional generalized information criterion (CGIC) with two tuning parameters. We further consider a challenging but practically relevant situation where the number, $m$, of clusters does not go to infinity with the sample size. Hence the random-effects variances are not consistently estimable. We show that via a novel decomposition of the CKL risk, the CGIC achieves consistency and asymptotic loss efficiency, whether $m$ is fixed or increases to infinity with the sample size. We also conduct numerical experiments to illustrate the theoretical findings.     

Asymptotic properties of the maximum smoothed partial likelihood estimator in the change-plane Cox model

The change-plane Cox model is a popular tool for the subgroup analysis of survival data. Despite the rich literature on this model, there has been limited investigation into the asymptotic properties of the estimators of the finite-dimensional parameter. Particularly, the convergence rate, not to mention the asymptotic distribution, has not been fully characterized for the general model where classification is based on multiple covariates. To bridge this theoretical gap, this study proposes a maximum smoothed partial likelihood estimator and establishes the following asymptotic properties. First, it shows that the convergence rate for the classification parameter can be arbitrarily close to $n^{-1}$ up to a logarithmic factor under a certain condition on covariates and the choice of tuning parameter. Given this convergence rate result, it also establishes the asymptotic normality for the regression parameter.     

Empirical Likelihood for Nonparametric Models Under Linear Process Errors

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

Joint Central Limit Theorem for Eigenvalue Statistics from Several Dependent Large Dimensional Sample Covariance Matrices with Application

Let X _n = (x _{i j} ) be a k ×n data matrix with complex‐valued, independent and standardized entries satisfying a Lindeberg‐type moment condition. We consider simultaneously R sample covariance matrices , where the Q _r 's are non‐random real matrices with common dimensions p ×k (k ≥p ). Assuming that both the dimension p and the sample size n grow to infinity, the limiting distributions of the eigenvalues of the matrices { B _{n r} } are identified, and as the main result of the paper, we establish a joint central limit theorem (CLT) for linear spectral statistics of the R matrices { B _{n r} }. Next, this new CLT is applied to the problem of testing a high‐dimensional white noise in time series modelling. In experiments, the derived test has a controlled size and is significantly faster than the classical permutation test, although it does have lower power. This application highlights the necessity of such joint CLT in the presence of several dependent sample covariance matrices. In contrast, all the existing works on CLT for linear spectral statistics of large sample covariance matrices deal with a single sample covariance matrix (R = 1).