Rejection sampling for tempered Lévy processes

Statistics and Computing - We extend the idea of tempering stable Lévy processes to tempering more general classes of Lévy processes. We show that the original process can be decomposed...     

Perfect Forward Simulation via Simulated Tempering

Several authors discuss how the simulated tempering scheme provides a very simple mechanism for introducing regenerations within a Markov chain. In this article we explain how regenerative simulated tempering schemes provide a very natural mechanism for perfect simulation. We use this to provide a perfect simulation algorithm, which uses a single-sweep forward-simulation without the need for recursively searching through negative times. We demonstrate this algorithm in the context of several examples.     

Alternative Estimators for Randomized Response Techniques in Multi-Character Surveys

Tail estimates are developed for power law probability distributions with exponential tempering, using a conditional maximum likelihood approach based on the upper-order statistics. Tempered power law distributions are intermediate between heavy power-law tails and Laplace or exponential tails, and are sometimes called “semi-heavy” tailed distributions. The estimation method is demonstrated on simulated data from a tempered stable distribution, and for several data sets from geophysics and finance that show a power law probability tail with some tempering.     

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...     

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.     

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.     

Convergence of the Equi-Energy Sampler and Its Application to the Ising Model

We provide a complete proof of the convergence of a recently developed sampling algorithm called the equi-energy (EE) sampler (Kou, Zhou, and Wong, 2006) in the case that the state space is countable. We show that in a countable state space, each sampling chain in the EE sampler is strongly ergodic a.s. with the desired steady-state distribution. Furthermore, all chains satisfy the individual ergodic property. We apply the EE sampler to the Ising model to test its efficiency, comparing it with the Metropolis algorithm and the parallel tempering algorithm. We observe that the dynamic exponent of the EE sampler is significantly smaller than those of parallel tempering and the Metropolis algorithm, demonstrating the high efficiency of the EE sampler.     

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.     

A unifying view on some problems in probability and statistics

Statistical Methods & Applications - Let $$L$$ be a linear space of real random variables on the measurable space $$(\varOmega ,\mathcal {A})$$ . Conditions for the existence of a probability...     

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...     

Sampling from multimodal distributions using tempered transitions

I present a new Markov chain sampling method appropriate for distributions with isolated modes. Like the recently developed method of simulated tempering, the tempered transition method uses a series of distributions that interpolate between the distribution of interest and a distribution for which sampling is easier. The new method has the advantage that it does not require approximate values for the normalizing constants of these distributions, which are needed for simulated tempering, and can be tedious to estimate. Simulated tempering performs a random walk along the series of distributions used. In contrast, the tempered transitions of the new method move systematically from the desired distribution, to the easily-sampled distribution, and back to the desired distribution. This systematic movement avoids the inefficiency of a random walk, an advantage that is unfortunately cancelled by an increase in the number of interpolating distributions required. Because of this, the sampling efficiency of the tempered transition method in simple problems is similar to that of simulated tempering. On more complex distributions, however, simulated tempering and tempered transitions may perform differently. Which is better depends on the ways in which the interpolating distributions are deceptive.     

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 $$ $aL₀ > 1 where$aL₀ = P_p=0 $$.     

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...     

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- $$ {t}$$...     

Smooth functional tempering for nonlinear differential equation models

Differential equations are used in modeling diverse system behaviors in a wide variety of sciences. Methods for estimating the differential equation parameters traditionally depend on the inclusion of initial system states and numerically solving the equations. This paper presents Smooth Functional Tempering, a new population Markov Chain Monte Carlo approach for posterior estimation of parameters. The proposed method borrows insights from parallel tempering and model based smoothing to define a sequence of approximations to the posterior. The tempered approximations depend on relaxations of the solution to the differential equation model, reducing the need for estimating the initial system states and obtaining a numerical differential equation solution. Rather than tempering via approximations to the posterior that are more heavily rooted in the prior, this new method tempers towards data features. Using our proposed approach, we observed faster convergence and robustness to both initial values and prior distributions that do not reflect the features of the data. Two variations of the method are proposed and their performance is examined through simulation studies and a real application to the chemical reaction dynamics of producing nylon.     

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...     

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.     

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.     

A Bayesian restoration of an ion channel signal

We present a Bayesian method of ion channel analysis and apply it to a simulated data set. An alternating renewal process prior is assigned to the signal, and an autoregressive process is fitted to the noise. After choosing model hyperconstants to yield 'uninformative' priors on the parameters, the joint posterior distribution is computed by using the reversible jump Markov chain Monte Carlo method. A novel form of simulated tempering is used to improve the mixing of the original sampler.