Chain graph models: topological sorting of meta-arrows and efficient construction of $${\mathcal{B}}$$ -essential graphs

Essential graphs and largest chain graphs are well-established graphical representations of equivalence classes of directed acyclic graphs and chain graphs respectively, especially useful in the context of model selection. Recently, the notion of a labelled block ordering of vertices was introduced as a flexible tool for specifying subfamilies of chain graphs. In particular, both the family of directed acyclic graphs and the family of “unconstrained” chain graphs can be specified in this way, for the appropriate choice of . The family of chain graphs identified by a labelled block ordering of vertices is partitioned into equivalence classes each represented by means of a -essential graph. In this paper, we introduce a topological ordering of meta-arrows and use this concept to devise an efficient procedure for the construction of -essential graphs. In this way we also provide an efficient procedure for the construction of both largest chain graphs and essential graphs. The key feature of the proposed procedure is that every meta-arrow needs to be processed only once.     

Multi-level Monte Carlo methods for the approximation of invariant measures of stochastic differential equations

We develop a framework that allows the use of the multi-level Monte Carlo (MLMC) methodology (Giles in Acta Numer. 24:259–328, 2015. https://doi.org/10.1017/S096249291500001X) to calculate expectations with respect to the invariant measure of an ergodic SDE. In that context, we study the (over-damped) Langevin equations with a strongly concave potential. We show that when appropriate contracting couplings for the numerical integrators are available, one can obtain a uniform-in-time estimate of the MLMC variance in contrast to the majority of the results in the MLMC literature. As a consequence, a root mean square error of $$\mathcal {O}(\varepsilon )$$ is achieved with $$\mathcal {O}(\varepsilon ^{-2})$$ complexity on par with Markov Chain Monte Carlo (MCMC) methods, which, however, can be computationally intensive when applied to large datasets. Finally, we present a multi-level version of the recently introduced stochastic gradient Langevin dynamics method (Welling and Teh, in: Proceedings of the 28th ICML, 2011) built for large datasets applications. We show that this is the first stochastic gradient MCMC method with complexity $$\mathcal {O}(\varepsilon ^{-2}|\log {\varepsilon }|^{3})$$, in contrast to the complexity $$\mathcal {O}(\varepsilon ^{-3})$$ of currently available methods. Numerical experiments confirm our theoretical findings.     

Locally best invariant and locally minimax test of independence

Let X₁ X₂ … X_N be independent normal p-vectors with common mean vector $$ = ($$) and common nonsingular covariance matrix $$ = Diag ($sG_i) [(1–p) I + pE] Diag ($sG_i), $sG_i> 0, i = 1… p, 1>p>=1/p–1. Write r_ij = sample correlation between the i th and the j th variable i j = 1,… p. It has been proved that for testing the hypothesis H₀ : p = 0 against the alternative H₁ : p>0 where $$ and $sG₁,…, $sG_p are unknown, the test which rejects H₀ for large value of $$ r_ij is locally best invariant for every $aL: 0 > $aL > 1 and locally minimax as p $$ 0 in the sense of Giri and Kiefer, 1964, for every $aL: 0 > $aL $$ $aL₀ > 1 where$aL₀ = P_p=0 $$.     

Comparing several exponential populations with more than one control

Suppose there are k ₁ (k ₁ ≥ 1) test treatments that we wish to compare with k ₂ (k ₂ ≥ 1) control treatments. Assume that the observations from the ith test treatment and the jth control treatment follow a two-parameter exponential distribution and , where θ is a common scale parameter and and are the location parameters of the ith test and the jth control treatment, respectively, i = 1, . . . ,k ₁; j = 1, . . . ,k ₂. In this paper, simultaneous one-sided and two-sided confidence intervals are proposed for all k ₁ k ₂ differences between the test treatment location and control treatment location parameters, namely , and the required critical points are provided. Discussions of multiple comparisons of all test treatments with the best control treatment and an optimal sample size allocation are given. Finally, it is shown that the critical points obtained can be used to construct simultaneous confidence intervals for Pareto distribution location parameters.     

Prevalent cohort studies and unobserved heterogeneity

Lifetime Data Analysis - Consider lifetimes originating at a series of calendar times $$ t_{1} ,t_{2} , \ldots $$ . At a certain time $$ t_{0} $$ a cross-sectional sample is taken, generating a...     

What finite-additivity can add to decision theory

Statistical Methods & Applications - We examine general decision problems with loss functions that are bounded below. We allow the loss function to assume the value $$\infty $$ . No other...     

Functional single-index quantile regression models

It is known that functional single-index regression models can achieve better prediction accuracy than functional linear models or fully nonparametric models, when the target is to predict a scalar response using a function-valued covariate. However, the performance of these models may be adversely affected by extremely large values or skewness in the response. In addition, they are not able to offer a full picture of the conditional distribution of the response. Motivated by using trajectories of $$\hbox {PM}_{{10}}$$ concentrations of last day to predict the maximum $$\hbox {PM}_{{10}}$$ concentration of the current day, a functional single-index quantile regression model is proposed to address those issues. A generalized profiling method is employed to estimate the model. Simulation studies are conducted to investigate the finite sample performance of the proposed estimator. We apply the proposed framework to predict the maximal value of $$\hbox {PM}_{{10}}$$ concentrations based on the intraday $$\hbox {PM}_{{10}}$$ concentrations of the previous day.     

Weight-preserving simulated tempering

Statistics and Computing - Simulated tempering is a popular method of allowing MCMC algorithms to move between modes of a multimodal target density $$\pi $$. One problem with simulated tempering...     

A consistent and numerically efficient variable selection method for sparse Poisson regression with applications to learning and signal recovery

Statistics and Computing - We propose an adaptive $$\ell _1$$ -penalized estimator in the framework of Generalized Linear Models with identity-link and Poisson data, by taking advantage of a...     

Selection of sparse vine copulas in high dimensions with the Lasso

Statistics and Computing - We propose a novel structure selection method for high-dimensional ( $$d > 100$$ ) sparse vine copulas. Current sequential greedy approaches for structure...     

Estimating drift parameters in a non-ergodic Gaussian Vasicek-type model

Statistical Methods & Applications - We study a problem of parameter estimation for a non-ergodic Gaussian Vasicek-type model defined as $$dX_t=\theta (\mu + X_t)dt+dG_t,\ t\ge 0$$ with unknown...     

Estimation of the finite population distribution function using a global penalized calibration method

AStA Advances in Statistical Analysis - Auxiliary information $${\varvec{x}}$$ is commonly used in survey sampling at the estimation stage. We propose an estimator of the finite population...     

On an exponential bound for the Kaplan–Meier estimator

We review limit theory and inequalities for the Kaplan–Meier Kaplan and Meier (J Am Stat Assoc 53:457–481, 1958) product limit estimator of a survival function on the whole line . Along the way we provide bounds for the constant in an interesting inequality due to Biotouzé et al. (Ann Inst H Poincaré Probab Stat 35:735–763, 1999), and provide some numerical evidence in support of one of their conjectures. Supported in part by NSF grant DMS-0503822 and by NI-AID grant 2R01 AI291968-04.     

Laplace approximation and natural gradient for Gaussian process regression with heteroscedastic student-t model

Statistics and Computing - We propose the Laplace method to derive approximate inference for Gaussian process (GP) regression in the location and scale parameters of the student- $$ {t}$$...     

Properties of the bridge sampler with a focus on splitting the MCMC sample

Computation of normalizing constants is a fundamental mathematical problem in various disciplines, particularly in Bayesian model selection problems. A sampling-based technique known as bridge sampling (Meng and Wong in Stat Sin 6(4):831–860, 1996) has been found to produce accurate estimates of normalizing constants and is shown to possess good asymptotic properties. For small to moderate sample sizes (as in situations with limited computational resources), we demonstrate that the (optimal) bridge sampler produces biased estimates. Specifically, when one density (we denote as $$p_2$$) is constructed to be close to the target density (we denote as $$p_1$$) using method of moments, our simulation-based results indicate that the correlation-induced bias through the moment-matching procedure is non-negligible. More crucially, the bias amplifies as the dimensionality of the problem increases. Thus, a series of theoretical as well as empirical investigations is carried out to identify the nature and origin of the bias. We then examine the effect of sample size allocation on the accuracy of bridge sampling estimates and discovered that one possibility of reducing both the bias and standard error with a small increase in computational effort is by drawing extra samples from the moment-matched density $$p_2$$ (which we assume easy to sample from), provided that the evaluation of $$p_1$$ is not too expensive. We proceed to show how the simple adaptive approach we termed “splitting” manages to alleviate the correlation-induced bias at the expense of a higher standard error, irrespective of the dimensionality involved. We also slightly modified the strategy suggested by Wang et al. (Warp bridge sampling: the next generation, Preprint, 2019. arXiv:1609.07690) to address the issue of the increase in standard error due to splitting, which is later generalized to further improve the efficiency. We conclude the paper by offering our insights of the application of a combination of these adaptive methods to improve the accuracy of bridge sampling estimates in Bayesian applications (where posterior samples are typically expensive to generate) based on the preceding investigations, with an application to a practical example.     

Improvement of the Liu estimator in linear regression model

In the presence of stochastic prior information, in addition to the sample, Theil and Goldberger (1961) introduced a Mixed Estimator for the parameter vector β in the standard multiple linear regression model (T,Xβ,σ² I). Recently, the Liu estimator which is an alternative biased estimator for β has been proposed by Liu (1993). In this paper we introduce another new Liu type biased estimator called Stochastic restricted Liu estimator for β, and discuss its efficiency. The necessary and sufficient conditions for mean squared error matrix of the Stochastic restricted Liu estimator to exceed the mean squared error matrix of the mixed estimator will be derived for the two cases in which the parametric restrictions are correct and are not correct. In particular we show that this new biased estimator is superior in the mean squared error matrix sense to both the Mixed estimator and to the biased estimator introduced by Liu (1993).     

The type I distribution of the ratio of independent “Weibullized” generalized beta-prime variables

In this paper we introduce the distribution of , with c >  0, where X _i , i =  1, 2, are independent generalized beta-prime-distributed random variables, and establish a closed form expression of its density. This distribution has as its limiting case the generalized beta type I distribution recently introduced by Nadarajah and Kotz (2004). Due to the presence of several parameters the density can take a wide variety of shapes.      

Central quantile subspace

Quantile regression (QR) is becoming increasingly popular due to its relevance in many scientific investigations. There is a great amount of work about linear and nonlinear QR models. Specifically, nonparametric estimation of the conditional quantiles received particular attention, due to its model flexibility. However, nonparametric QR techniques are limited in the number of covariates. Dimension reduction offers a solution to this problem by considering low-dimensional smoothing without specifying any parametric or nonparametric regression relation. The existing dimension reduction techniques focus on the entire conditional distribution. We, on the other hand, turn our attention to dimension reduction techniques for conditional quantiles and introduce a new method for reducing the dimension of the predictor $$\mathbf {X}$$. The novelty of this paper is threefold. We start by considering a single index quantile regression model, which assumes that the conditional quantile depends on $$\mathbf {X}$$ through a single linear combination of the predictors, then extend to a multi-index quantile regression model, and finally, generalize the proposed methodology to any statistical functional of the conditional distribution. The performance of the methodology is demonstrated through simulation examples and real data applications. Our results suggest that this method has a good finite sample performance and often outperforms the existing methods.