Showing papers in "Journal of The Royal Statistical Society Series B-statistical Methodology in 2004"
TL;DR: In this article, it was shown that the goal of the two approaches are essentially equivalent, and that the FDR point estimates can be used to define valid FDR controlling procedures in both finite sample and asymptotic settings.
Abstract: Summary. The false discovery rate (FDR) is a multiple hypothesis testing quantity that describes the expected proportion of false positive results among all rejected null hypotheses. Benjamini and Hochberg introduced this quantity and proved that a particular step-up p-value method controls the FDR. Storey introduced a point estimate of the FDR for fixed significance regions. The former approach conservatively controls the FDR at a fixed predetermined level, and the latter provides a conservatively biased estimate of the FDR for a fixed predetermined significance region. In this work, we show in both finite sample and asymptotic settings that the goals of the two approaches are essentially equivalent. In particular, the FDR point estimates can be used to define valid FDR controlling procedures. In the asymptotic setting, we also show that the point estimates can be used to estimate the FDR conservatively over all significance regions simultaneously, which is equivalent to controlling the FDR at all levels simultaneously. The main tool that we use is to translate existing FDR methods into procedures involving empirical processes. This simplifies finite sample proofs, provides a framework for asymptotic results and proves that these procedures are valid even under certain forms of dependence.
1,413 citations
TL;DR: In this article, the authors present a Bayesian framework which unifies the various tools of probabilistic sensitivity analysis, which allows effective sensitivity analysis to be achieved by using far smaller numbers of model runs than standard Monte Carlo methods.
Abstract: Summary. In many areas of science and technology, mathematical models are built to simulate complex real world phenomena. Such models are typically implemented in large computer programs and are also very complex, such that the way that the model responds to changes in its inputs is not transparent. Sensitivity analysis is concerned with understanding how changes in the model inputs influence the outputs. This may be motivated simply by a wish to understand the implications of a complex model but often arises because there is uncertainty about the true values of the inputs that should be used for a particular application. A broad range of measures have been advocated in the literature to quantify and describe the sensitivity of a model's output to variation in its inputs. In practice the most commonly used measures are those that are based on formulating uncertainty in the model inputs by a joint probability distribution and then analysing the induced uncertainty in outputs, an approach which is known as probabilistic sensitivity analysis. We present a Bayesian framework which unifies the various tools of prob- abilistic sensitivity analysis. The Bayesian approach is computationally highly efficient. It allows effective sensitivity analysis to be achieved by using far smaller numbers of model runs than standard Monte Carlo methods. Furthermore, all measures of interest may be computed from a single set of runs.
1,074 citations
TL;DR: In this article, a semiparametric approach is proposed to analyze air pollution data and reveal complex extremal dependence behavior that is consistent with scientific understanding of the process. But it is not suitable for applications where the extreme values of all the variables are unlikely to occur together or when interest is in regions of the support of the joint distribution where only a subset of components is extreme.
Abstract: Summary. Multivariate extreme value theory and methods concern the characterization, estimation and extrapolation of the joint tail of the distribution of a d-dimensional random variable. Existing approaches are based on limiting arguments in which all components of the variable become large at the same rate. This limit approach is inappropriate when the extreme values of all the variables are unlikely to occur together or when interest is in regions of the support of the joint distribution where only a subset of components is extreme. In practice this restricts existing methods to applications where d is typically 2 or 3. Under an assumption about the asymptotic form of the joint distribution of a d-dimensional random variable conditional on its having an extreme component, we develop an entirely new semiparametric approach which overcomes these existing restrictions and can be applied to problems of any dimension. We demonstrate the performance of our approach and its advantages over existing methods by using theoretical examples and simulation studies. The approach is used to analyse air pollution data and reveals complex extremal dependence behaviour that is consistent with scientific understanding of the process. We find that the dependence structure exhibits marked seasonality, with ex- tremal dependence between some pollutants being significantly greater than the dependence at non-extreme levels.
588 citations
TL;DR: It is shown how this approach can be adapted to approximate the restricted likelihood and it is demonstrated how an estimating equations approach allows us to judge the efficacy of the resulting approximation.
Abstract: Summary. Likelihood methods are often difficult to use with large, irregularly sited spatial data sets, owing to the computational burden. Even for Gaussian models, exact calculations of the likelihood for n observations require O(n3) operations. Since any joint density can be written as a product of conditional densities based on some ordering of the observations, one way to lessen the computations is to condition on only some of the ‘past’ observations when computing the conditional densities. We show how this approach can be adapted to approximate the restricted likelihood and we demonstrate how an estimating equations approach allows us to judge the efficacy of the resulting approximation. Previous work has suggested conditioning on those past observations that are closest to the observation whose conditional density we are approximating. Through theoretical, numerical and practical examples, we show that there can often be considerable benefit in conditioning on some distant observations as well.
478 citations
TL;DR: A new procedure is proposed for clustering attribute value data that encourages those algorithms to detect automatically subgroups of objects that preferentially cluster on subsets of the attribute variables rather than on all of them simultaneously.
Abstract: [Read before The Royal Statistical Society at a meeting organized by the Research Section on Wednesday, May 5th, 2004, Professor J. T. Kent in the Chair ] Summary. A new procedure is proposed for clustering attribute value data. When used in conjunction with conventional distance-based clustering algorithms this procedure encourages those algorithms to detect automatically subgroups of objects that preferentially cluster on subsets of the attribute variables rather than on all of them simultaneously. The relevant attribute subsets for each individual cluster can be different and partially (or completely) overlap with those of other clusters. Enhancements for increasing sensitivity for detecting especially low cardinality groups clustering on a small subset of variables are discussed. Applications in different domains, including gene expression arrays, are presented.
440 citations
TL;DR: In this article, the authors consider the problem of testing null hypotheses that include restrictions on the variance component in a linear mixed model with one variance component and derive the finite sample and asymptotic distribution of the likelihood ratio test and the restricted likelihood ratios test.
Abstract: Summary. We consider the problem of testing null hypotheses that include restrictions on the variance component in a linear mixed model with one variance component and we derive the finite sample and asymptotic distribution of the likelihood ratio test and the restricted likelihood ratio test. The spectral, representations of the likelihood ratio test and the restricted likelihood ratio test statistics are used as the basis of efficient simulation algorithms of their null distributions. The large sample x2 mixture approximations using the usual asymptotic theory for a null hypothesis on the boundary of the parameter space have been shown to be poor in simulation studies. Our asymptotic calculations explain these empirical results. The theory of Self and Liang applies only to linear mixed models for which the data vector can be partitioned into a large number of independent and identically distributed subvectors. One-way analysis of variance and penalized splines models illustrate the results.
394 citations
TL;DR: A simple modification of the generalized cross-validation method for smoothing parameter selection is evaluated, which to a large extent fixes the occasional undersmoothing problem that is suffered by generalized cross‐validation.
Abstract: Summary. Smoothing splines via the penalized least squares method provide versatile and effective nonparametric models for regression with Gaussian responses. The computation of smoothing splines is generally of the order O(n3), n being the sample size, which severely limits its practical applicability. We study more scalable computation of smoothing spline regression via certain low dimensional approximations that are asymptotically as efficient. A simple algorithm is presented and the Bayes model that is associated with the approximations is derived, with the latter guiding the porting of Bayesian confidence intervals. The practical choice of the dimension of the approximating space is determined through simulation studies, and empirical comparisons of the approximations with the exact solution are presented. Also evaluated is a simple modification of the generalized cross-validation method for smoothing parameter selection, which to a large extent fixes the occasional undersmoothing problem that is suffered by generalized cross-validation.
290 citations
TL;DR: In this paper, a convolution operator can either be smooth (polynomial decay of the Fourier transform) or irregular (such as the convolution with a box-car).
Abstract: In this paper, we present an inverse estimation procedure which combines Fourier analysis with wavelet expansion. In the periodic setting, our method can recover a blurred function observed in white noise. The blurring process is achieved through a convolution operator which can either be smooth (polynomial decay of the Fourier transform) or irregular (such as the convolution with a box-car). The proposal is non-linear and does not require any prior knowledge of the smoothness class; it enjoys fast computation and is spatially adaptive. This contrasts with more traditional ltering methods which demand a certain amount of regularisation and often fail to recover non-homogeneous functions. A ne tuning of our method is derived via asymptotic minimax theory which reveals some key dierences with the direct case of Donoho et al. (1995): (a) band-limited wavelet families have nice theoretical and computing features; (b) the high frequency cut o depends on the spectral characteristics of the convolution kernel; (c) thresholds are level dependent in a geometric fashion. We tested our method using simulated lidar data for underwater remote sensing. Both visual and numerical results show an improvement over existing methods. Finally, the theory behind our estimation paradigm gives a complete characterisation of the ’Maxiset’ of the method i.e. the set of functions where the method attains a near-optimal rate of convergence for a variety of L p loss functions.
191 citations
TL;DR: A Markov chain Monte Carlo scheme is developed to allow efficient implementation of full posterior inference in the given model and includes a regression at the level of the nonparametric model.
Abstract: Summary. We consider the problem of combining inference in related nonparametric Bayes models. Analogous to parametric hierarchical models, the hierarchical extension formalizes borrowing strength across the related submodels. In the nonparametric context, modelling is complicated by the fact that the random quantities over which we define the hierarchy are infinite dimensional. We discuss a formal definition of such a hierarchical model. The approach includes a regression at the level of the nonparametric model. For the special case of Dirichlet process mixtures, we develop a Markov chain Monte Carlo scheme to allow efficient implementation of full posterior inference in the given model.
178 citations
TL;DR: The authors proposed a class of inverse intensity-of-visit process-weighted estimators in marginal regression models for longitudinal responses that may be observed in continuous time, which can handle arbitrary patterns of missing data as embedded in a subject's visit process.
Abstract: Summary. A frequent problem in longitudinal studies is that subjects may miss scheduled visits or be assessed at self-selected points in time. As a result, observed outcome data may be highly unbalanced and the availability of the data may be directly related to the outcome measure and/or some auxiliary factors that are associated with the outcome. If the follow-up visit and outcome processes are correlated, then marginal regression analyses will produce biased estimates. Building on the work of Robins, Rotnitzky and Zhao, we propose a class of inverse intensity-of-visit process-weighted estimators in marginal regression models for longitudinal responses that may be observed in continuous time. This allows us to handle arbitrary patterns of missing data as embedded in a subject's visit process. We derive the large sample distribution for our inverse visit-intensity-weighted estimators and investigate their finite sample behaviour by simulation. Our approach is illustrated with a data set from a health services research study in which homeless people with mental illness were randomized to three different treatments and measures of homelessness (as percentage days homeless in the past 3 months) and other auxiliary factors were recorded at follow-up times that are not fixed by design.
148 citations
TL;DR: A semiparametric model for functional data where the warping functions are assumed to be linear combinations of q common components, which are estimated from the data (hence the name ‘self‐modelling’).
Abstract: The paper introduces a semiparametric model for functional data. The warping functions are assumed to be linear combinations ofqcommon components, which are estimated from the data (hence the name‘self-modelling’). Even small values ofqprovide remarkable model flexibility, comparable with nonparametric methods. At the same time, this approach avoids overfitting because the common components are estimated combining data across individuals. As a convenient by-product, component scores are often interpretable and can be used for statistical inference (an example of classification based on scores is given). [ABSTRACT FROM AUTHOR]
TL;DR: In this paper, a Markov chain Monte Carlo methodology for Bayesian inference for non-Gaussian Ornstein-Uhlenbeck stochastic volatility processes is presented. But the performance of the methods is investigated for different types of simulated data.
Abstract: We develop Markov chain Monte Carlo methodology for Bayesian inference for non-Gaussian Ornstein–Uhlenbeck stochastic volatility processes. The approach introduced involves expressing the unobserved stochastic volatility process in terms of a suitable marked Poisson process. We introduce two specific classes of Metropolis–Hastings algorithms which correspond to different ways of jointly parameterizing the marked point process and the model parameters. The performance of the methods is investigated for different types of simulated data. The approach is extended to consider the case where the volatility process is expressed as a superposition of Ornstein–Uhlenbeck processes. We apply our methodology to the US dollar–Deutschmark exchange rate.
TL;DR: In this article, a modified likelihood ratio (MLR) test is proposed for finite mixture models with normal, binomial and Poisson kernels, where the estimates of the parameters are obtained from a modified probability function.
Abstract: Summary. We consider a finite mixture model with k components and a kernel distribution from a general one-parameter family. The problem of testing the hypothesis k = 2 versus k > 3 is studied. There has been no general statistical testing procedure for this problem. We propose a modified likelihood ratio statistic where under the null and the alternative hypotheses the estimates of the parameters are obtained from a modified likelihood function. It is shown that estimators of the support points are consistent. The asymptotic null distribution of the modified likelihood ratio test proposed is derived and found to be relatively simple and easily applied. Simulation studies for the asymptotic modified likelihood ratio test based on finite mixture models with normal, binomial and Poisson kernels suggest that the test proposed performs well. Simulation studies are also conducted for a bootstrap method with normal kernels. An example involving foetal movement data from a medical study illustrates the testing procedure.
TL;DR: In this article, two characteristics for stationary and isotropic marked point processes, E(r) and V(r), are introduced that allow for investigating mark-point interactions.
Abstract: Two characteristics for stationary and isotropic marked point processes, E(r)and V(r), are introduced that allow for investigating mark-point interactions. These quantities are functions of the inter-point distance rand denote the conditional expectation and the conditional variance of a mark, respectively, given there is a further point of the process a distance rapart. Tests based on Eand Vare presented that help to decide whether the values of the marks can be modelled by random field that is independent of the unmarked point process. We demonstrate the use of our approach for three data sets in forestry.
TL;DR: In this article, a smoothing spline variant of the triogram model based on a roughness penalty adapted to the piecewise linear structure of a triangulation is proposed.
Abstract: Summary. Hansen, Kooperberg and Sardy introduced a family of continuous, piecewise linear functions defined over adaptively selected triangulations of the plane as a general approach to statistical modelling of bivariate densities and regression and hazard functions. These triograms enjoy a natural affine equivariance that offers distinct advantages over competing tensor product methods that are more commonly used in statistical applications. Triograms employ basis functions consisting of linear 'tent functions' defined with respect to a triangulation of a given planar domain. As in knot selection for univariate splines, Hansen and colleagues adopted the regression spline approach of Stone. Vertices of the triangulation are introduced or removed sequentially in an effort to balance fidelity to the data and parsimony. We explore a smoothing spline variant of the triogram model based on a roughness penalty adapted to the piecewise linear structure of the triogram model. We show that the roughness penalty proposed may be interpreted as a total variation penalty on the gradient of the fitted function. The methods are illustrated with real and artificial examples, including an application to estimated quantile surfaces of land value in the Chicago metropolitan area.
TL;DR: In this paper, the authors considered the double-autoregressive model and proved consistency and asymptotic normality of the estimated parameters under the condition E ln jφ C p αηtj 1 as well as E" 2 / D1.
Abstract: Summary. The paper considers the double-autoregressive model yt D φyt� 1 C "t with "t D ηt p .ω C αy 2� 1 /. Consistency and asymptotic normality of the estimated parameters are proved under the condition E ln jφ C p αηtj 1 as well as E." 2 / D1 . It is well known that all kinds of estimators of φ in these cases are not normal when "t are independent and identically distributed. Our result is novel and surprising. Two tests are proposed for testing stationarity of the model and their asymptotic distributions are shown to be a function of bivariate Brownian motions. Critical values of the tests are tabulated and some simulation results are reported. An application to the US 90-day treasury bill rate series is given.
TL;DR: In this paper, the authors propose a method to estimate a parameter θ in models whose likelihood function does not have an analytical closed form, but from which random samples can be drawn for fixed values of θ.
Abstract: Summary. There are models for which the evaluation of the likelihood is infeasible in practice. For these models the Metropolis–Hastings acceptance probability cannot be easily computed. This is the case, for instance, when only departure times from a G/G/1 queue are observed and inference on the arrival and service distributions are required. Indirect inference is a method to estimate a parameter θ in models whose likelihood function does not have an analytical closed form, but from which random samples can be drawn for fixed values of θ. First an auxiliary model is chosen whose parameter β can be directly estimated. Next, the parameters in the auxiliary model are estimated for the original data, leading to an estimate . The parameter β is also estimated by using several sampled data sets, simulated from the original model for different values of the original parameter θ. Finally, the parameter θ which leads to the best match to is chosen as the indirect inference estimate. We analyse which properties an auxiliary model should have to give satisfactory indirect inference. We look at the situation where the data are summarized in a vector statistic T, and the auxiliary model is chosen so that inference on β is drawn from T only. Under appropriate assumptions the asymptotic covariance matrix of the indirect estimators is proportional to the asymptotic covariance matrix of T and componentwise inversely proportional to the square of the derivative, with respect to θ, of the expected value of T. We discuss how these results can be used in selecting good estimating functions. We apply our findings to the queuing problem.
TL;DR: In this article, the authors describe novel Bayesian models for time-frequency inverse modelling of non-stationary signals, based on the idea of a Gabor regression, in which a time series is represented as a superposition of translated, modulated versions of a window function exhibiting good timefrequency concentration.
Abstract: Summary. We describe novel Bayesian models for time–frequency inverse modelling of non-stationary signals. These models are based on the idea of a Gabor regression, in which a time series is represented as a superposition of translated, modulated versions of a window function exhibiting good time–frequency concentration. As a necessary consequence, the resultant set of potential predictors is in general overcomplete—constituting a frame rather than a basis—and hence the resultant models require careful regularization through appropriate choices of variable selection schemes and prior distributions. We introduce prior specifications that are tailored to representative time series, and we develop effective Markov chain Monte Carlo methods for inference. To highlight the potential applications of such methods, we provide examples using two of the most distinctive time–frequency surfaces—speech and music signals—as well as standard test functions from the wavelet regression literature.
TL;DR: In this article, the authors discuss the inversion of the gas profiles (ozone, NO3, NO2, aerosols and neutral density) in the upper atmosphere from the spectral occultation measurements.
Abstract: Summary. We discuss the inversion of the gas profiles (ozone, NO3, NO2, aerosols and neutral density) in the upper atmosphere from the spectral occultation measurements. The data are produced by the ‘Global ozone monitoring of occultation of stars’ instrument on board the Envisat satellite that was launched in March 2002. The instrument measures the attenuation of light spectra at various horizontal paths from about 100 km down to 10–20 km. The new feature is that these data allow the inversion of the gas concentration height profiles. A short introduction is given to the present operational data management procedure with examples of the first real data inversion. Several solution options for a more comprehensive statistical inversion are presented. A direct inversion leads to a non-linear model with hundreds of parameters to be estimated. The problem is solved with an adaptive single-step Markov chain Monte Carlo algorithm. Another approach is to divide the problem into several non-linear smaller dimensional problems, to run parallel adaptive Markov chain Monte Carlo chains for them and to solve the gas profiles in repetitive linear steps. The effect of grid size is discussed, and we present how the prior regularization takes the grid size into account in a way that effectively leads to a grid-independent inversion.
TL;DR: In this article, a spline estimation and the Bayes information criterion are used for non-linear additive autoregressive models to overcome the "curse of dimensionality", whereas the spline estimators effectively take into account such a structure in estimation.
Abstract: Summary. We propose a lag selection method for non-linear additive autoregressive models that is based on spline estimation and the Bayes information criterion. The additive structure of the autoregression function is used to overcome the ‘curse of dimensionality’, whereas the spline estimators effectively take into account such a structure in estimation. A stepwise procedure is suggested to implement the method proposed. A comprehensive Monte Carlo study demonstrates good performance of the method proposed and a substantial computational advantage over existing local-polynomial-based methods. Consistency of the lag selection method based on the Bayes information criterion is established under the assumption that the observations are from a stochastic process that is strictly stationary and strongly mixing, which provides the first theoretical result of this kind for spline smoothing of weakly dependent data.
TL;DR: In this paper, a new estimator for the parameters of a generalized linear latent variable model (GLLVM) based on a Laplace approximation to the likelihood function is proposed, which can be computed even for models with a large number of variables.
Abstract: Generalized linear latent variable models (GLLVMs), as defined by Bartholomew and Knott, enable modelling of relationships between manifest and latent variables. They extend structural equation modelling techniques, which are powerful tools in the social sciences. However, because of the complexity of the log-likelihood function of a GLLVM, an approximation such as numerical integration must be used for inference. This can limit drastically the number of variables in the model and can lead to biased estimators. We propose a new estimator for the parameters of a GLLVM, based on a Laplace approximation to the likelihood function and which can be computed even for models with a large number of variables. The new estimator can be viewed as an M-estimator, leading to readily available asymptotic properties and correct inference. A simulation study shows its excellent finite sample properties, in particular when compared with a well-established approach such as LISREL. A real data example on the measurement of wealth for the computation of multidimensional inequality is analysed to highlight the importance of the methodology.
TL;DR: This work uses a multivariate binary model so that the covariance matrix from estimating equation theory can be compared with the inverse Fisher information matrix and leads to simple rules for the choice of α in an exchangeable or autoregressive AR(1) weight matrix, based on the strength of dependence between the binary variables.
Abstract: Summary. Using standard correlation bounds, we show that in generalized estimation equations (GEEs) the so-called ‘working correlation matrix’R(α) for analysing binary data cannot in general be the true correlation matrix of the data. Methods for estimating the correlation param-eter in current GEE software for binary responses disregard these bounds. To show that the GEE applied on binary data has high efficiency, we use a multivariate binary model so that the covariance matrix from estimating equation theory can be compared with the inverse Fisher information matrix. But R(α) should be viewed as the weight matrix, and it should not be confused with the correlation matrix of the binary responses. We also do a comparison with more general weighted estimating equations by using a matrix Cauchy–Schwarz inequality. Our analysis leads to simple rules for the choice of α in an exchangeable or autoregressive AR(1) weight matrix R(α), based on the strength of dependence between the binary variables. An example is given to illustrate the assessment of dependence and choice of α.
TL;DR: In this paper, Bayesian confidence intervals are derived for the coefficient curves and efficient computational methods for computing the curve estimators, fitted values, posterior variances and data-adaptive methods for selecting the levels of smoothing.
Abstract: Summary. Smoothing spline estimators are considered for inference in varying-coefficient models with one effect modifying covariate. Bayesian ‘confidence intervals’ are developed for the coefficient curves and efficient computational methods are derived for computing the curve estimators, fitted values, posterior variances and data-adaptive methods for selecting the levels of smoothing. The efficacy and utility of the methodology proposed are demonstrated through a small simulation study and the analysis of a real data set.
TL;DR: In this article, the optimal convergence rate of n−1/4 for estimating the mixing distribution is achievable for both the maximum likelihood and the maximum modified likelihood estimators in mixture regression models.
Abstract: Summary. We establish asymptotic theory for both the maximum likelihood and the maximum modified likelihood estimators in mixture regression models. Moreover, under specific and reasonable conditions, we show that the optimal convergence rate of n−1/4 for estimating the mixing distribution is achievable for both the maximum likelihood and the maximum modified likelihood estimators. We also derive the asymptotic distributions of two log-likelihood ratio test statistics for testing homogeneity and we propose a resampling procedure for approximating the p-value. Simulation studies are conducted to investigate the empirical performance of the two test statistics. Finally, two real data sets are analysed to illustrate the application of our theoretical results.
TL;DR: It is demonstrated that, if the original step-up procedure of Benjamini and Hochberg is modified to exercise adaptive control of the false discovery rate, its performance is virtually identical to that of the fixed rejection region approach.
Abstract: Summary. The use of a fixed rejection region for multiple hypothesis testing has been shown to outperform standard fixed error rate approaches when applied to control of the false discovery rate. In this work it is demonstrated that, if the original step-up procedure of Benjamini and Hochberg is modified to exercise adaptive control of the false discovery rate, its performance is virtually identical to that of the fixed rejection region approach. In addition, the dependence of both methods on the proportion of true null hypotheses is explored, with a focus on the difficulties that are involved in the estimation of this quantity.
TL;DR: In this article, the authors provide an overview of available recurrent events analysis methods and present an inverse probability of censoring weighted estimator for the regression parameters in the Andersen-Gill model that is commonly used for recurrent event analysis.
Abstract: Summary. Recurrent events models have had considerable attention recently. The majority of approaches show the consistency of parameter estimates under the assumption that censoring is independent of the recurrent events process of interest conditional on the covariates that are included in the model. We provide an overview of available recurrent events analysis methods and present an inverse probability of censoring weighted estimator for the regression parameters in the Andersen–Gill model that is commonly used for recurrent event analysis. This estimator remains consistent under informative censoring if the censoring mechanism is estimated consistently, and it generally improves on the naive estimator for the Andersen–Gill model in the case of independent censoring. We illustrate the bias of ad hoc estimators in the presence of informative censoring with a simulation study and provide a data analysis of recurrent lung exacerbations in cystic fibrosis patients when some patients are lost to follow-up.
TL;DR: In this article, the authors considered joint probability distributions generated recursively in terms of univariate conditional distributions satisfying conditional independence restrictions, and derived the strength of the associations that are implied by such linear families for chain graph models.
Abstract: Summary. We consider joint probability distributions generated recursively in terms of univariate conditional distributions satisfying conditional independence restrictions. The independences are captured by missing edges in a directed graph.A matrix form of such a graph, called the generating edge matrix, is triangular so the distributions that are generated over such graphs are called triangular systems. We study consequences of triangular systems after grouping or reordering of the variables for analyses as chain graph models, i.e. for alternative recursive factorizations of the given density using joint conditional distributions. For this we introduce families of linear triangular equations which do not require assumptions of distributional form. The strength of the associations that are implied by such linear families for chain graph models is derived. The edge matrices of chain graphs that are implied by any triangular system are obtained by appropriately transforming the generating edge matrix. It is shown how induced independences and dependences can be studied by graphs, by edge matrix calculations and via the properties of densities. Some ways of using the results are illustrated.
TL;DR: In this article, a low-order approximation for nonparametric inference with errors in explanatory variables is proposed. But this approach does not address the problem of consistent estimation of the true target functions consistently.
Abstract: Summary. We suggest two new methods, which are applicable to both deconvolution and regression with errors in explanatory variables, for nonparametric inference.The two approaches involve kernel or orthogonal series methods. They are based on defining a low order approximation to the problem at hand, and proceed by constructing relatively accurate estimators of that quantity rather than attempting to estimate the true target functions consistently. Of course, both techniques could be employed to construct consistent estimators, but in many contexts of importance (e.g. those where the errors are Gaussian) consistency is, from a practical viewpoint, an unattainable goal. We rephrase the problem in a form where an explicit, interpretable, low order approximation is available.The information that we require about the error distribution (the error-in-variables distribution, in the case of regression) is only in the form of low order moments and so is readily obtainable by a rudimentary analysis of indirect measurements of errors, e.g. through repeated measurements. In particular, we do not need to estimate a function, such as a characteristic function, which expresses detailed properties of the error distribution.This feature of our methods, coupled with the fact that all our estimators are explicitly defined in terms of readily computable averages, means that the methods are particularly economical in computing time.
TL;DR: This work proposes two new complex‐valued wavelet shrinkage techniques: one based on multiwavelet style shrinkage and the other using Bayesian methods, which are both simpler and dramatically faster than its competitors.
Abstract: Summary. Wavelet shrinkage is an effective nonparametric regression technique, especially when the underlying curve has irregular features such as spikes or discontinuities. The basic idea is simple: take the discrete wavelet transform (DWT) of data consisting of a signal corrupted by noise; shrink or remove the wavelet coefficients to remove the noise; and then invert the DWT to form an estimate of the true underlying curve. Various authors have proposed increasingly sophisticated methods of doing this using real-valued wavelets. Complex-valued wavelets exist, but are rarely used. We propose two new shrinkage techniques which use complex-valued wavelets. Extensive simulations show that our methods almost always give significantly more accurate estimates when compared with methods based on real-valued wavelets. Further, one of our methods is both simpler and dramatically faster than its competitors. In an attempt to understand the excellent performance of this latter method we present a new risk bound on its hard thresholded coefficients.
TL;DR: The forward-backward algorithm as mentioned in this paper is an exact filtering algorithm which can efficiently calculate likelihoods, and which can be used to simulate from posterior distributions, and it has been used to calculate the distribution of a sum of gamma random variables, and to simulate their joint distribution given their sum.
Abstract: Summary. The forward–backward algorithm is an exact filtering algorithm which can efficiently calculate likelihoods, and which can be used to simulate from posterior distributions. Using a simple result which relates gamma random variables with different rates, we show how the forward–backward algorithm can be used to calculate the distribution of a sum of gamma random variables, and to simulate from their joint distribution given their sum. One application is to calculating the density of the time of a specific event in a Markov process, as this time is the sum of exponentially distributed interevent times. This enables us to apply the forward–backward algorithm to a range of new problems. We demonstrate our method on three problems: calculating likelihoods and simulating allele frequencies under a non-neutral population genetic model, analysing a stochastic epidemic model and simulating speciation times in phylogenetics.