In a recent paper Birke and Bissantz (2009) considered the problem of nonparametric estimation in inverse regression models with convolution-type operators. For multivariate predictors nonparametric methods suffer from the curse of dimensionality and we consider inverse regression models with the additional qualitative assumption of additivity. In these models several additive estimators are studied. In particular, we propose a new estimation method for observations on regular spaced grid and investigate estimators under the random design assumption which are applicable when observations are not available on a grid. Finally, we compare these estimators with the marginal integration and the non-additive estimator by means of a simulation study. It is demonstrated that the new method yields a substantial improvement of the currently available procedures.

In this paper, we consider a semiparametric model for lifetime data with competing risks and missing causes of death. We assume that an additive hazards model holds for each cause-specific hazard rate function and that a random right censoring occurs. Our goal is to estimate the regression parameters as well as the functional parameters such as the baseline and cause-specific cumulative hazard rate functions/cumulative incidence functions.

We first introduce preliminary estimators of the unknown (Euclidean and functional) parameters when cause of death indicators are missing completely at random (MCAR). These estimators are obtained using the observations with known cause of failure. The advantage of considering the MCAR model is that the information given by the observed lifetimes with unknown failure cause can be used to improve the preliminary estimates in order to attain an asymptotic optimality criterion. This is the main purpose of our work. However, since it is often more realistic to consider a missing at random (MAR) mechanism, we also derive estimators of the regression and functional parameters under the MAR model. We study the large sample properties of our estimators through martingales and empirical process techniques. We also provide a simulation study to compare the behavior of our three types of estimators under the different mechanisms of missingness. It is shown that our improved estimators under MCAR assumption are quite robust if only the MAR assumption holds. Finally, three illustrations on real datasets are also given.

Given $n$ independent, identically distributed random vectors in $\mathbb{R}^{d}$, drawn from a common density $f$, one wishes to find out whether the support of $f$ is convex or not. In this paper we describe a decision rule which decides correctly for sufficiently large $n$, with probability $1$, whenever $f$ is bounded away from zero in its compact support. We also show that the assumption of boundedness is necessary. The rule is based on a statistic that is a second-order $U$-statistic with a random kernel. Moreover, we suggest a way of approximating the distribution of the statistic under the hypothesis of convexity of the support. The performance of the proposed method is illustrated on simulated data sets. As an example of its potential statistical implications, the decision rule is used to automatically choose the tuning parameter of ISOMAP, a nonlinear dimensionality reduction method.

The problem of estimating covariance and precision matrices of multivariate normal distributions is addressed when both the sample size and the dimension of variables are large. The estimation of the precision matrix is important in various statistical inference including the Fisher linear discriminant analysis, confidence region based on the Mahalanobis distance and others. A standard estimator is the inverse of the sample covariance matrix, but it may be instable or can not be defined in the high dimension. Although (adaptive) ridge type estimators are alternative procedures which are useful and stable for large dimension. However, we are faced with questions about how to choose ridge parameters and their estimators and how to set up asymptotic order in ridge functions in high dimensional cases. In this paper, we consider general types of ridge estimators for covariance and precision matrices, and derive asymptotic expansions of their risk functions. Then we suggest the ridge functions so that the second order terms of risks of ridge estimators are smaller than those of risks of the standard estimators.

We consider an adaptive smoothing spline with a piecewise-constant penalty function $\lambda(x)$, in which a univariate smoothing parameter $\lambda$ in the classic smoothing spline is converted into an adaptive multivariate parameter $\boldsymbol{\lambda}$. Choosing the optimal value of $\boldsymbol{\lambda}$ is critical for obtaining desirable estimates. We propose to choose $\boldsymbol{\lambda}$ by minimizing a multivariate version of the generalized cross validation function; the resulting estimator is shown to be consistent and asymptotically optimal under some general conditions—i.e., the counterparts of the nice asymptotic properties of the generalized cross validation in the ordinary smoothing spline are still provable. This provides theoretical justification of adopting the multivariate version of the generalized cross validation principle in adaptive smoothing splines.

It is well-known that if a time series is not sampled at a fast enough rate to capture all the high frequencies then aliasing may occur. Aliasing is a distortion of the spectrum of a series which can cause severe problems for time series modelling and forecasting. The situation is more complex and more interesting for nonstationary series as aliasing can be intermittent. Recent work has shown that it is possible to test for the absence of aliasing in nonstationary time series and this article demonstrates that additional benefits can be obtained by modelling a series using a Shannon locally stationary wavelet (LSW) process. We show that for Shannon LSW processes the effects of dyadic-sampling-induced aliasing can be reversed. We illustrate our method by simulation on Shannon LSW processes and also a time-varying autoregressive process where aliasing is detected. We present an analysis of a wind power time series and show that it can be adequately modelled by a Shannon LSW process, the absence of aliasing can not be inferred and present a dealiased estimate of the series.

The conditional Akaike information criterion, AIC, has been frequently used for model selection in linear mixed models. We develop a general framework for the calculation of the conditional AIC for different exponential family distributions. This unified framework incorporates the conditional AIC for the Gaussian case, gives a new justification for Poisson distributed data and yields a new conditional AIC for exponentially distributed responses but cannot be applied to the binomial and gamma distributions. The proposed conditional Akaike information criteria are unbiased for finite samples, do not rely on a particular estimation method and do not assume that the variance-covariance matrix of the random effects is known. The theoretical results are investigated in a simulation study. The practical use of the method is illustrated by application to a data set on tree growth.

The varying coefficient model is a useful alternative to the classical linear model, since the former model is much richer and more flexible than the latter. We propose estimators of the coefficient functions for the varying coefficient model in the case where different coefficient functions depend on different covariates and the response is subject to random right censoring. Since our model has an additive structure and requires multivariate smoothing we employ a smooth backfitting technique, that is known to be an effective way to avoid “the curse of dimensionality” in structured nonparametric models. The estimators are based on synthetic data obtained by an unbiased transformation. The asymptotic normality of the estimators is established, a simulation study illustrates the reliability of our estimators, and the estimation procedure is applied to data on drug abuse.

Tetrad correlations were obtained historically for Gaussian distributions when tasks are designed to measure an ability or attitude so that a single unobserved variable may generate the observed, linearly increasing dependences among the tasks. We connect such generating processes to a particular type of directed graph, the star graph, and to the notion of traceable regressions. Tetrad correlation conditions for the existence of a single latent variable are derived. These are needed for positive dependences not only in joint Gaussian but also in joint binary distributions. Three applications with binary items are given.

We consider centered compound Poisson processes with finite variance, discretely observed over $[0,T]$ and let the sampling rate $\Delta=\Delta_{T}\rightarrow\infty$ as $T\rightarrow\infty$. From the central limit theorem, the law of each increment converges to a Gaussian variable. Then, it should not be possible to estimate more than one parameter at the limit. First, from the study of a parametric example we identify two regimes for $\Delta_{T}$ and we observe how the Fisher information degenerates. Then, we generalize these results to the class of compound Poisson processes. We establish a lower bound showing that consistent estimation is impossible when $\Delta_{T}$ grows faster than $\sqrt{T}$. We also prove an asymptotic equivalence result, from which we identify, for instance, regimes where the increments cannot be distinguished from Gaussian variables.

We consider the problem of aggregating a general collection of affine estimators for fixed design regression. Relevant examples include some commonly used statistical estimators such as least squares, ridge and robust least squares estimators. Dalalyan and Salmon [DS12] have established that, for this problem, exponentially weighted (EW) model selection aggregation leads to sharp oracle inequalities in expectation, but similar bounds in deviation were not previously known. While results [DRZ12] indicate that the same aggregation scheme may not satisfy sharp oracle inequalities with high probability, we prove that a weaker notion of oracle inequality for EW that holds with high probability. Moreover, using a generalization of the newly introduced $Q$-aggregation scheme we also prove sharp oracle inequalities that hold with high probability. Finally, we apply our results to universal aggregation and show that our proposed estimator leads simultaneously to all the best known bounds for aggregation, including $\ell_{q}$-aggregation, $q\in(0,1)$, with high probability.

In the context of a linear model with a sparse coefficient vector, exponential weights methods have been shown to be achieve oracle inequalities for denoising/prediction. We show that such methods also succeed at variable selection and estimation under the near minimum condition on the design matrix, instead of much stronger assumptions required by other methods such as the Lasso or the Dantzig Selector. The same analysis yields consistency results for Bayesian methods and BIC-type variable selection under similar conditions.

Variational Bayesian inference with a Gaussian posterior approximation provides an alternative to the more commonly employed factorization approach and enlarges the range of tractable distributions. In this paper, we propose an extension to the Gaussian approach which uses Gaussian mixtures as approximations. A general problem for variational inference with mixtures is posed by the calculation of the entropy term in the Kullback-Leibler distance, which becomes analytically intractable. We deal with this problem by using a simple lower bound for the entropy and imposing restrictions on the form of the Gaussian covariance matrix. In this way, efficient numerical calculations become possible. To illustrate the method, we discuss its application to an isotropic generalized normal target density, a non-Gaussian state space model, and the Bayesian lasso. For heavy-tailed distributions, the examples show that the mixture approach indeed leads to improved approximations in the sense of a reduced Kullback-Leibler distance. From a more practical point of view, mixtures can improve estimates of posterior marginal variances. Furthermore, they provide an initial estimate of posterior skewness which is not possible with single Gaussians. We also discuss general sufficient conditions under which mixtures are guaranteed to provide improvements over single-component approximations.

In time series analysis, statistics based on collections of estimators computed from subsamples play a crucial role in an increasing variety of important applications. Proving results about the joint asymptotic distribution of such statistics is challenging, since it typically involves a nontrivial verification of technical conditions and tedious case-by-case asymptotic analysis. In this paper, we provide a novel technique that allows to circumvent those problems in a general setting. Our approach consists of two major steps: a probabilistic part which is mainly concerned with weak convergence of sequential empirical processes, and an analytic part providing general ways to extend this weak convergence to functionals of the sequential empirical process. Our theory provides a unified treatment of asymptotic distributions for a large class of statistics, including recently proposed self-normalized statistics and sub-sampling based p-values. In addition, we comment on the consistency of bootstrap procedures and obtain general results on compact differentiability of certain mappings that are of independent interest.

We consider a Gaussian sequence space model $X_{\lambda}=f_{\lambda}+\xi_{\lambda},$ where the noise variables $(\xi_{\lambda})_{\lambda}$ are independent, but with heterogeneous variances $(\sigma_{\lambda}^{2})_{\lambda}$. Our goal is to estimate the unknown signal vector $(f_{\lambda})$ by a model selection approach. We focus on the situation where the non-zero entries $f_{\lambda}$ are sparse. Then the heterogenous case is much more involved than the homogeneous model where $\sigma_{\lambda}^{2}=\sigma^{2}$ is constant. Indeed, we can no longer profit from symmetry inside the stochastic process that one needs to control. The problem and the penalty do not only depend on the number of coefficients that one selects, but also on their position. This appears also in the minimax bounds where the worst coefficients will go to the larger variances. With a careful and explicit choice of the penalty, however, we are able to select the correct coefficients and get a sharp non-asymptotic control of the risk of our procedure. Some finite sample results from simulations are provided.

The Bayesian Information Criterion (BIC) is widely used for variable selection in mixed effects models. However, its expression is unclear in typical situations of mixed effects models, where simple definition of the sample size is not meaningful. We derive an appropriate BIC expression that is consistent with the random effect structure of the mixed effects model. We illustrate the behavior of the proposed criterion through a simulation experiment and a case study and we recommend its use as an alternative to various existing BIC versions that are implemented in available software.

Partially identified models are characterized by the distribution of observables being compatible with a set of values for the target parameter, rather than a single value. This set is often referred to as an identification region. From a non-Bayesian point of view, the identification region is the object revealed to the investigator in the limit of increasing sample size. Conversely, a Bayesian analysis provides the identification region plus the limiting posterior distribution over this region. This purports to convey varying plausibility of values across the region. Taking a decision-theoretic view, we investigate the extent to which having a distribution across the identification region is indeed helpful.

We derive the exact joint asymptotic distribution for multiple Box-Pierce statistics, and use these results to determine appropriate critical values in joint testing problems of time series goodness-of-fit. By sequentially testing at various lags, we can identify specific problems with a model, and identify superior models. A novel $\alpha$-rationing scheme, motivated by the sequence of conditional probabilities for the statistical tests, is developed and implemented. This method can be used to produce critical values and p-values both for each step of the sequential testing procedure, and for the procedure as a whole. Efficient computational algorithms are discussed. Simulation studies assess the impact of finite samples on the real Type I error. It is also demonstrated empirically that the conventional $\chi^{2}$ critical values for the Box-Pierce statistics are too small, with a Type I error rate greater than the nominal; the new method does not suffer from this defect, and allows for more rigorous model-building. We illustrate on several time series how model defects can be identified and ameliorated.

Studying model checking problems for partially linear single-index models, we propose a variant of the integrated conditional moment test using a linear projection weighting function, which gains dimension reduction and makes the proposed method act as if there exists only one covariate even in the presence of multiple dimensional regressors. We derive asymptotic distributions of the proposed test; i.e., an integral of a centered Gaussian process under the null hypothesis and an integral of a non-centered one under Pitman local alternatives. We also suggest a consistent bootstrap procedure for calculating the critical values. Simulation studies are conducted to demonstrate the performance of the proposed procedure and a real example is analyzed for an illustration.

Uniform convergence of empirical norms - empirical measures of squared functions - is a topic which has received considerable attention in the literature on empirical processes. The results are relevant as empirical norms occur due to symmetrization. They also play a prominent role in statistical applications. The contraction inequality has been a main tool but recently other approaches have shown to lead to better results in important cases. We present an overview including the linear (anisotropic) case, and give new results for inner products of functions. Our main application will be the estimation of the parental structure in a directed acyclic graph. As intermediate result we establish convergence of the least squares estimator when the model is wrong.

Let $X:=(X_{1},\ldots,X_{p})$ be random objects (the inputs), defined on some probability space $(\Omega,\mathcal{F},\mathbb{P})$ and valued in some measurable space $E=E_{1}\times\ldots\times E_{p}$. Further, let $Y:=Y=f(X_{1},\ldots,X_{p})$ be the output. Here, $f$ is a measurable function from $E$ to some Hilbert space $\mathbb{H}$ ($\mathbb{H}$ could be either of finite or infinite dimension). In this work, we give a natural generalization of the Sobol indices (that are classically defined when $Y\in\mathbb{R}$), when the output belongs to $\mathbb{H}$. These indices have very nice properties. First, they are invariant under isometry and scaling. Further they can be, as in dimension $1$, easily estimated by using the so-called Pick and Freeze method. We investigate the asymptotic behaviour of such an estimation scheme.

In recent years, a large variety of continuous shrinkage priors have been developed for a Bayesian analysis of the standard linear regression model in high dimensional settings. We consider two such priors, the Dirichlet-Laplace prior (developed in Bhattacharya et al. (2013)), and the Normal-Gamma prior (developed in (Griffin and Brown, 2010)). For both Dirichlet-Laplace and Normal-Gamma priors, Gibbs sampling Markov chains have been developed to generate approximate samples from the corresponding posterior distributions. We show by using a drift and minorization based analysis that the Gibbs sampling Markov chains corresponding to the aforementioned models are geometrically ergodic. Establishing geometric ergodicity of these Markov chains is crucial, as it provides theoretical justification for the use of Markov chain CLT, which can then be used to obtain asymptotic standard errors for Markov chain based estimates of posterior quantities. Both Gibbs samplers in the paper use the Generalized Inverse Gaussian (GIG) distribution, as one of the conditional distributions. A novel contribution of our convergence analysis is the use of drift functions which include terms that are negative fractional powers of normal random variables, to tackle the presence of the GIG distribution.

Robust methods are little applied (although much studied by statisticians). We monitor very robust regression by looking at the behaviour of residuals and test statistics as we smoothly change the robustness of parameter estimation from a breakdown point of 50% to non-robust least squares. The resulting procedure provides insight into the structure of the data including outliers and the presence of more than one population. Monitoring overcomes the hindrances to the routine adoption of robust methods, being informative about the choice between the various robust procedures. Methods tuned to give nominal high efficiency fail with our most complicated example. We find that the most informative analyses come from S estimates combined with Tukey’s biweight or with the optimal $\rho$ functions.

For our major example with 1,949 observations and 13 explanatory variables, we combine robust S estimation with regression using the forward search, so obtaining an understanding of the importance of individual observations, which is missing from standard robust procedures. We discover that the data come from two different populations. They also contain six outliers.

Our analyses are accompanied by numerous graphs. Algebraic results are contained in two appendices, the second of which provides useful new results on the absolute odd moments of elliptically truncated multivariate normal random variables.

We consider an acceptance-rejection sampler based on a deterministic driver sequence. The deterministic sequence is chosen such that the discrepancy between the empirical target distribution and the target distribution is small. We use quasi-Monte Carlo (QMC) point sets for this purpose. The empirical evidence shows convergence rates beyond the crude Monte Carlo rate of $N^{-1/2}$. We prove that the discrepancy of samples generated by the QMC acceptance-rejection sampler is bounded from above by $N^{-1/s}$. A lower bound shows that for any given driver sequence, there always exists a target density such that the star discrepancy is at most $N^{-2/(s+1)}$. For a general density, whose domain is the real state space $\mathbb{R}^{s-1}$, the inverse Rosenblatt transformation can be used to convert samples from the $(s-1)$-dimensional cube to $\mathbb{R}^{s-1}$. We show that this transformation is measure preserving. This way, under certain conditions, we obtain the same convergence rate for a general target density defined in $\mathbb{R}^{s-1}$. Moreover, we also consider a deterministic reduced acceptance-rejection algorithm recently introduced by Barekat and Caflisch [F. Barekat and R. Caflisch, Simulation with Fluctuation and Singular Rates. ArXiv:1310.4555 [math.NA], 2013.]

This paper deals with the statistical properties of a response adaptive design, described in terms of a two colors urn model, targeting prespecified asymptotic allocations. Results on the rate of divergence of number of patients assigned to each treatment are proved as well as on the asymptotic behavior of the urn composition. Suitable statistics are introduced and studied to test the hypothesis on treatments’ difference.

We study nonparametric estimation of an unknown density function $f$ based on the ranked-based observations obtained from a partially rank-ordered set (PROS) sampling design. PROS sampling design has many applications in environmental, ecological and medical studies where the exact measurement of the variable of interest is costly but a small number of sampling units can be ordered with respect to the variable of interest by any means other than actual measurements and this can be done at low cost. PROS observations involve independent order statistics which are not identically distributed and most of the commonly used nonparametric techniques are not directly applicable to them. We first develop a kernel density estimator of $f$ based on an imperfect PROS sampling procedure and study its theoretical properties. Then, we consider the problem when the underlying distribution is assumed to be symmetric and introduce some plug-in kernel density estimators of $f$. We use an EM type algorithm to estimate misplacement probabilities associated with an imperfect PROS design. Finally, we expand on various numerical illustrations of our results via several simulation studies and a case study to estimate the distribution of wheat yield using the total acreage of land which is planted in wheat as an easily obtained auxiliary information. Our results show that the PROS density estimate performs better than its SRS and RSS counterparts.

Networks are a commonly used mathematical model to describe the rich set of interactions between objects of interest. Many clustering methods have been developed in order to partition such structures, among which several rely on underlying probabilistic models, typically mixture models. The relevant hidden structure may however show overlapping groups in several applications. The Overlapping Stochastic Block Model (Latouche, Birmelé and Ambroise (2011)) has been developed to take this phenomenon into account. Nevertheless, the problem of the choice of the number of classes in the inference step is still open. To tackle this issue, we consider the proposed model in a Bayesian framework and develop a new criterion based on a non asymptotic approximation of the marginal log-likelihood. We describe how the criterion can be computed through a variational Bayes EM algorithm, and demonstrate its efficiency by running it on both simulated and real data.

In this paper, we introduce a new concept of quantiles and depth for directional (circular and spherical) data. In view of the similarities with the classical Mahalanobis depth for multivariate data, we call it the angular Mahalanobis depth. Our unique concept combines the advantages of both the depth and quantile settings: appealing depth-based geometric properties of the contours (convexity, nestedness, rotation-equivariance) and typical quantile-asymptotics, namely we establish a Bahadur-type representation and asymptotic normality (these results are corroborated by a Monte Carlo simulation study). We introduce new user-friendly statistical tools such as directional DD- and QQ-plots and a quantile-based goodness-of-fit test. We illustrate the power of our new procedures by analyzing a cosmic rays data set.

When computing a confidence interval for a binomial proportion $p$ one must choose between using an exact interval, which has a coverage probability of at least $1-\alpha$ for all values of $p$, and a shorter approximate interval, which may have lower coverage for some $p$ but that on average has coverage equal to $1-\alpha$. We investigate the cost of using the exact one and two-sided Clopper–Pearson confidence intervals rather than shorter approximate intervals, first in terms of increased expected length and then in terms of the increase in sample size required to obtain a desired expected length. Using asymptotic expansions, we also give a closed-form formula for determining the sample size for the exact Clopper–Pearson methods. For two-sided intervals, our investigation reveals an interesting connection between the frequentist Clopper–Pearson interval and Bayesian intervals based on noninformative priors.

The Fleming-Harrington and constant piecewise wei- ghted logrank tests for late effects in clinical trials are considered. Both tests depend on a parameter ($q$ for the Fleming-Harrington and $t^{*}$ for the constant piecewise weighted logrank) that has to be chosen before the trial analysis. The problem of choosing the most appropriate test and associated parameter value for a given trial is addressed. For this purpose, the tests are compared in terms of their sensitivity to $q$ and $t^{*}$ and of their asymptotic relative efficiency and necessary sample size. Some guidelines for choosing the most appropriate weight for testing late effects are provided. The methodology is illustrated on a medical dataset.

In Bayesian nonparametric statistics, it is crucial that the support of the prior is very large. Here, we consider species sampling priors. Such priors are widely used within mixture models and it has been shown in the literature that a large support for the mixing prior is essential to ensure the consistency of the posterior. In this paper, simple conditions are given that are necessary and sufficient for the support of a species sampling prior to be full. In particular, for proper species sampling priors, the condition is that the maximum size of the atoms of the corresponding process is small with positive probability. We apply this result to show that the main classes of species sampling priors known in literature have full support under mild conditions. Moreover, we find priors with a very simple construction still having full support.

Uniform random generation of Latin squares is a classical problem. In this paper we prove that both Latin squares and Sudoku designs are maximum cliques of properly defined graphs. We have developed a simple algorithm for uniform random sampling of Latin squares and Sudoku designs. The corresponding SAS code is available in the supplementary material.

It is well known that the sum $S$ of $n$ independent gamma variables—which occurs often, in particular in practical applications—can typically be well approximated by a single gamma variable with the same mean and variance (the distribution of $S$ being quite complicated in general). In this paper, we propose an alternative (and apparently at least as good) single-gamma approximation to $S$. The methodology used to derive it is based on the observation that the jump density of $S$ bears an evident similarity to that of a generic gamma variable, $S$ being viewed as a sum of $n$ independent gamma processes evaluated at time $1$. This observation motivates the idea of a gamma approximation to $S$ in the first place, and, in principle, a variety of such approximations can be made based on it. The same methodology can be applied to obtain gamma approximations to a wide variety of important infinitely divisible distributions on $\mathbb{R}_{+}$ or at least predict/confirm the appropriateness of the moment-matching method (where the first two moments are matched); this is demonstrated neatly in the cases of negative binomial and generalized Dickman distributions, thus highlighting the paper’s contribution to the overall topic.

In a recent paper on sensitivity of subspaces spanned by principal components, [5] introduces an influence measure based on second order expansion of the RV and GCD coefficients which are commonly used as measures of similarity between two matrices. The goal of this short note is to point out that the paper of [2] is based on a similar approach. However this work seems unknown to Prendergast since it is missing in his references. A comparison of the two papers is provided together with a brief review of some related works.

It is well known that changes in mean superimposed by a short-range dependent series can be confused easily with long-range dependence. A procedure to distinguish the two phenomena is introduced. The proposed procedure is based on the local Whittle estimation of the long-range dependence parameter applied to the series after removing changes in mean, and comparing the results to those obtained through the available CUSUM-like approaches. According to the proposed procedure, for example, volatility series in finance seem more consistent with the changes-in-mean models whereas hydrology and telecommunication series are more in line with long-range dependence. As part of this work, a new method based on the local Whittle estimation to find the number of breaks is also introduced and its consistency is proved for the changes-in-mean models.

We introduce a subclass of chain event graphs that we call stratified chain event graphs, and present a dynamic programming algorithm for the optimal selection of such chain event graphs that maximizes a decomposable score derived from a complete independent sample. We apply the algorithm to such a dataset, with a view to deducing the causal structure of the variables under the hypothesis that there are no unobserved confounders. We show that the algorithm is suitable for small problems. Similarities with and differences to a dynamic programming algorithm for MAP learning of Bayesian networks are highlighted, as are the relations to causal discovery using Bayesian networks.

This paper is devoted to the estimation of the shift parameter in a semiparametric regression model when the distribution of the observation times is unknown. Hence, we propose to use a stochastic algorithm which takes into account the estimation of the distribution of the observation times. We establish the almost sure convergence of our estimator and the asymptotic normality. The main result of the paper is that, with little assumptions on the regularity of the regression function, the asymptotic variance obtained is the same as when the distribution is known. In that sense, we improve the recent work of Bercu and Fraysse [1].

Lévy-Driven Ornstein-Uhlenbeck (or CAR(1)) processes were introduced by Barndorff-Nielsen and Shephard [1] as a model for stochastic volatility. Pham [17] developed a general formula to recover the unobserved driving process from the continuously observed CAR(1) process. When the CAR(1) process is observed at discrete times $0$, $h$, $2h$, $...$ $[T/h]h$ the driving process must be approximated. Approximated increments of the driving process are used to test the assumption that the CAR(1) process is Lévy-driven. Asymptotic behavior of the test statistic is investigated. Performance of the test is illustrated through simulation.

In this paper we investigate the stick-breaking representation for the class of $\sigma$-stable Poisson-Kingman models, also known as Gibbs-type random probability measures. This class includes as special cases most of the discrete priors commonly used in Bayesian nonparametrics, such as the two parameter Poisson-Dirichlet process and the normalized generalized Gamma process. Under the assumption $\sigma=u/v$, for any coprime integers $1\leq u<v$ such that $u/v\leq1/2$, we show that a $\sigma$-stable Poisson-Kingman model admits an explicit stick-breaking representation in terms of random variables which are obtained by suitably transforming Gamma random variables and products of independent Beta and Gamma random variables.

This note corrects a minor error in the statement of Theorem 2.3 of EJS533, [1].

We characterize the dynamic properties of generalized autoregressive score models by identifying the regions of the parameter space that imply stationarity and ergodicity of the corresponding nonlinear time series process. We show how these regions are affected by the choice of parameterization and scaling, which are key features for the class of generalized autoregressive score models compared to other observation driven models. All results are illustrated for the case of time-varying means, variances, or higher-order moments.

We investigate mean field variational approximate Bayesian inference for models that use continuous distributions, Horseshoe, Negative-Exponential-Gamma and Generalized Double Pareto, for sparse signal shrinkage. Our principal finding is that the most natural, and simplest, mean field variational Bayes algorithm can perform quite poorly due to posterior dependence among auxiliary variables. More sophisticated algorithms, based on special functions, are shown to be superior. Continued fraction approximations via Lentz’s Algorithm are developed to make the algorithms practical.

We consider a real Gaussian process $X$ with global unknown smoothness $(r_{\scriptscriptstyle 0},\beta _{\scriptscriptstyle 0}$): more precisely $X^{(r_{\scriptscriptstyle 0})}$, $r_{\scriptscriptstyle 0}\in{\mathds {N}}_{0}$, is supposed to be locally stationary with Hölder exponent $\beta _{\scriptscriptstyle 0}$, $\beta _{\scriptscriptstyle 0}\in]0,1[$. For $X$ observed at a finite set of points, we derive estimators of $r_{\scriptscriptstyle 0}$ and $\beta _{\scriptscriptstyle 0}$ based on the quadratic variations for the divided differences of $X$. Under mild conditions, we obtain an exponential bound for estimating $r_{\scriptscriptstyle 0}$, as well as sharp rates of convergence (up to logarithmic factors) for the estimation of $\beta _{\scriptscriptstyle 0}$. An extensive simulation study illustrates the finite-sample properties of both estimators for different types of processes and we also include two real data applications.

We consider the problem of providing nonparametric confidence guarantees — with finite sample Berry-Esseen bounds — for undirected graphs under weak assumptions. We do not assume sparsity or incoherence. We allow the dimension $D$ to increase with the sample size $n$. First, we prove lower bounds that show that if we want accurate inferences with weak assumptions then $D$ must be less than $n$. In that case, we show that methods based on Normal approximations and on the bootstrap lead to valid inferences and we provide new Berry-Esseen bounds on the accuracy of the Normal approximation and the bootstrap. When the dimension is large relative to sample size, accurate inferences for graphs under weak assumptions are not possible. Instead we propose to estimate something less demanding than the entire partial correlation graph. In particular, we consider: cluster graphs, restricted partial correlation graphs and correlation graphs.

Dynamic treatment regimes are of growing interest across the clinical sciences because these regimes provide one way to operationalize and thus inform sequential personalized clinical decision making. Formally, a dynamic treatment regime is a sequence of decision rules, one per stage of clinical intervention. Each decision rule maps up-to-date patient information to a recommended treatment. We briefly review a variety of approaches for using data to construct the decision rules. We then review a critical inferential challenge that results from nonregularity, which often arises in this area. In particular, nonregularity arises in inference for parameters in the optimal dynamic treatment regime; the asymptotic, limiting, distribution of estimators are sensitive to local perturbations. We propose and evaluate a locally consistent Adaptive Confidence Interval (ACI) for the parameters of the optimal dynamic treatment regime. We use data from the Adaptive Pharmacological and Behavioral Treatments for Children with ADHD Trial as an illustrative example. We conclude by highlighting and discussing emerging theoretical problems in this area.

Inference for parameters associated with optimal dynamic treatment regimes is challenging as these estimators are nonregular when there are non-responders to treatments. In this discussion, we comment on three aspects of alleviating this nonregularity. We first discuss an alternative approach for smoothing the quality functions. We then discuss some further details on our existing work to identify non-responders through penalization. Third, we propose a clinically meaningful value assessment whose estimator does not suffer from nonregularity.

The contributions of the review paper on the technical challenges and applications of dynamic treatment regimes are briefly discussed and potential extensions to continuous action spaces and high dimensional problems are indicated.

We develop two analytic approaches to solve D-optimal approximate designs under generalized linear models. The first approach provides analytic D-optimal allocations for generalized linear models with two factors, which include as a special case the $2^{2}$ main-effects model considered by Yang, Mandal and Majumdar [19]. The second approach leads to explicit solutions for a class of generalized linear models with more than two factors. With the aid of the analytic solutions, we provide a necessary and sufficient condition under which a D-optimal design with two quantitative factors could be constructed on the boundary points only. It bridges the gap between D-optimal factorial designs and D-optimal designs with continuous factors.

This paper investigates the (in)-consistency of various bootstrap methods for making inference on a change-point in time in the Cox model with right censored survival data. A criterion is established for the consistency of any bootstrap method. It is shown that the usual nonparametric bootstrap is inconsistent for the maximum partial likelihood estimation of the change-point. A new model-based bootstrap approach is proposed and its consistency established. Simulation studies are carried out to assess the performance of various bootstrap schemes.

In this paper, we consider the well known problem of estimating a density function under qualitative assumptions. More precisely, we estimate monotone non-increasing densities in a Bayesian setting and derive concentration rate for the posterior distribution for a Dirichlet process and finite mixture prior. We prove that the posterior distribution based on both priors concentrates at the rate $(n/\log(n))^{-1/3}$, which is the minimax rate of estimation up to a $\log(n)$ factor. We also study the behaviour of the posterior for the point-wise loss at any fixed point of the support of the density and for the sup-norm. We prove that the posterior distribution is consistent for both loss functions.

We consider nonparametric maximum-likelihood estimation of a log-concave density in case of interval-censored, right-censored and binned data. We allow for the possibility of a subprobability density with an additional mass at $+\infty$, which is estimated simultaneously. The existence of the estimator is proved under mild conditions and various theoretical aspects are given, such as certain shape and consistency properties. An EM algorithm is proposed for the approximate computation of the estimator and its performance is illustrated in two examples.

We are interested in consistent estimation of the mixing matrix in the ICA model, when the error distribution is close to (but different from) Gaussian. In particular, we consider $n$ independent samples from the ICA model $X=A\epsilon$, where we assume that the coordinates of $\epsilon$ are independent and identically distributed according to a contaminated Gaussian distribution, and the amount of contamination is allowed to depend on $n$. We then investigate how the ability to consistently estimate the mixing matrix depends on the amount of contamination. Our results suggest that in an asymptotic sense, if the amount of contamination decreases at rate $1/\sqrt{n}$ or faster, then the mixing matrix is only identifiable up to transpose products. These results also have implications for causal inference from linear structural equation models with near-Gaussian additive noise.

Bartlett correction is one of the desirable features of empirical likelihood (EL) since it allows constructions of confidence regions with improved coverage probabilities. Previous studies demonstrated the Bartlett correction of EL for independent observations and for short-memory time series. By establishing the validity of Edgeworth expansion for the signed root empirical log-likelihood ratio, the validity of Bartlett correction of EL for Gaussian long-memory time series is established. In particular, orders of the coverage error of confidence regions can be reduced from $\log^{6}n/n$ to $\log^{3}n/n$, which is different from the classical rate of reduction from $n^{-1}$ to $n^{-2}$.

Respondent-Driven Sampling (RDS) is n approach to sampling design and inference in hard-to-reach human populations. It is often used in situations where the target population is rare and/or stigmatized in the larger population, so that it is prohibitively expensive to contact them through the available frames. Common examples include injecting drug users, men who have sex with men, and female sex workers. Most analysis of RDS data has focused on estimating aggregate characteristics, such as disease prevalence. However, RDS is often conducted in settings where the population size is unknown and of great independent interest. This paper presents an approach to estimating the size of a target population based on data collected through RDS.

The proposed approach uses a successive sampling approximation to RDS to leverage information in the ordered sequence of observed personal network sizes. The inference uses the Bayesian framework, allowing for the incorporation of prior knowledge. A flexible class of priors for the population size is used that aids elicitation. An extensive simulation study provides insight into the performance of the method for estimating population size under a broad range of conditions. A further study shows the approach also improves estimation of aggregate characteristics. Finally, the method demonstrates sensible results when used to estimate the size of known networked populations from the National Longitudinal Study of Adolescent Health, and when used to estimate the size of a hard-to-reach population at high risk for HIV.

In this paper, we consider the so-called Shape Invariant Model that is used to model a function $f^{0}$ submitted to a random translation of law $g^{0}$ in a white noise. This model is of interest when the law of the deformations is unknown. Our objective is to recover the law of the process $\mathbb{P}_{f^{0},g^{0}}$ as well as $f^{0}$ and $g^{0}$. To do this, we adopt a Bayesian point of view and find priors on $f$ and $g$ so that the posterior distribution concentrates at a polynomial rate around $\mathbb{P}_{f^{0},g^{0}}$ when $n$ goes to $+\infty$. We then derive results on the identifiability of the SIM, as well as results on the functional objects themselves. We intensively use Bayesian non-parametric tools coupled with mixture models, which may be of independent interest in model selection from a frequentist point of view.

Fast accumulation of large amounts of complex data has created a need for more sophisticated statistical methodologies to discover interesting patterns and better extract information from these data. The large scale of the data often results in challenging high-dimensional estimation problems where only a minority of the data shows specific grouping patterns. To address these emerging challenges, we develop a new clustering methodology that introduces the idea of a regularization path into unsupervised learning. A regularization path for a clustering problem is created by varying the degree of sparsity constraint that is imposed on the differences between objects via the minimax concave penalty with adaptive tuning parameters. Instead of providing a single solution represented by a cluster assignment for each object, the method produces a short sequence of solutions that determines not only the cluster assignment but also a corresponding number of clusters for each solution. The optimization of the penalized loss function is carried out through an MM algorithm with block coordinate descent. The advantages of this clustering algorithm compared to other existing methods are as follows: it does not require the input of the number of clusters; it is capable of simultaneously separating irrelevant or noisy observations that show no grouping pattern, which can greatly improve data interpretation; it is a general methodology that can be applied to many clustering problems. We test this method on various simulated datasets and on gene expression data, where it shows better or competitive performance compared against several clustering methods.

Sufficient conditions are derived for the asymptotic efficiency and equivalence of componentwise Bayesian and classical estimators of the infinite-dimensional parameters characterizing $l^{2}$ valued Poisson process, and Hilbert valued Gaussian random variable models. Conjugate families are considered for the Poisson and Gaussian univariate likelihoods, in the Bayesian estimation of the components of such infinite-dimensional parameters. In the estimation of the functional mean of a Hilbert valued Gaussian random variable, sufficient and necessary conditions, that ensure a better performance of the Bayes estimator with respect to the classical one, are also obtained for the finite-sample size case. A simulation study is carried out to provide additional information on the relative efficiency of Bayes and classical estimators in a high-dimensional framework.

Many practical studies analyze semi-continuous longitudinal data using, for example, ZAIG (Zero-Adjusted Inverse Gaussian) or BEZI (Beta Zero-Inflated) models as marginal distributions. We develop herein estimating equations analogous to Liang and Zeger’s estimating equation of independence and related diagnostic techniques for regression models, with zero-inflated response variable and panel data structure. A simulation study to evaluate some properties of the estimators obtained from the estimating equations and two applications with real data are presented.

In the framework of conditional density estimation, we use candidates taking the form of mixtures of Gaussian regressions with logistic weights and means depending on the covariate. We aim at estimating the number of components of this mixture, as well as the other parameters, by a penalized maximum likelihood approach. We provide a lower bound on the penalty that ensures an oracle inequality for our estimator. We perform some numerical experiments that support our theoretical analysis.