Classification using distance nearest neighbours

This paper proposes a new probabilistic classification algorithm using a Markov random field approach. The joint distribution of class labels is explicitly modelled using the distances between feature vectors. Intuitively, a class label should depend more on class labels which are closer in the feature space, than those which are further away. Our approach builds on previous work by Holmes and Adams (J. R. Stat. Soc. Ser. B 64:295–306, 2002; Biometrika 90:99–112, 2003) and Cucala et al. (J. Am. Stat. Assoc. 104:263–273, 2009). Our work shares many of the advantages of these approaches in providing a probabilistic basis for the statistical inference. In comparison to previous work, we present a more efficient computational algorithm to overcome the intractability of the Markov random field model. The results of our algorithm are encouraging in comparison to the k-nearest neighbour algorithm.     

Metropolis–Hastings algorithms with adaptive proposals

Different strategies have been proposed to improve mixing and convergence properties of Markov Chain Monte Carlo algorithms. These are mainly concerned with customizing the proposal density in the Metropolis–Hastings algorithm to the specific target density and require a detailed exploratory analysis of the stationary distribution and/or some preliminary experiments to determine an efficient proposal. Various Metropolis–Hastings algorithms have been suggested that make use of previously sampled states in defining an adaptive proposal density. Here we propose a general class of adaptive Metropolis–Hastings algorithms based on Metropolis–Hastings-within-Gibbs sampling. For the case of a one-dimensional target distribution, we present two novel algorithms using mixtures of triangular and trapezoidal densities. These can also be seen as improved versions of the all-purpose adaptive rejection Metropolis sampling (ARMS) algorithm to sample from non-logconcave univariate densities. Using various different examples, we demonstrate their properties and efficiencies and point out their advantages over ARMS and other adaptive alternatives such as the Normal Kernel Coupler.     

Generating random numbers from a distribution specified by its Laplace transform

This paper discusses simulation from an absolutely continuous distribution on the positive real line when the Laplace transform of the distribution is known but its density and distribution functions may not be available. We advocate simulation by the inversion method using a modified Newton-Raphson method, with values of the distribution and density functions obtained by numerical transform inversion. We show that this algorithm performs well in a series of increasingly complex examples. Caution is needed in some situations when the numerical Laplace transform inversion becomes unreliable. In particular the algorithm should not be used for distributions with finite range. But otherwise, except for rather pathological distributions, the approach offers a rapid way of generating random samples with minimal user effort. We contrast our approach with an alternative algorithm due to Devroye (Comput. Math. Appl. 7, 547–552, 1981).     

A ‘nondecimated’ lifting transform

Classical nondecimated wavelet transforms are attractive for many applications. When the data comes from complex or irregular designs, the use of second generation wavelets in nonparametric regression has proved superior to that of classical wavelets. However, the construction of a nondecimated second generation wavelet transform is not obvious. In this paper we propose a new ‘nondecimated’ lifting transform, based on the lifting algorithm which removes one coefficient at a time, and explore its behavior. Our approach also allows for embedding adaptivity in the transform, i.e. wavelet functions can be constructed such that their smoothness adjusts to the local properties of the signal. We address the problem of nonparametric regression and propose an (averaged) estimator obtained by using our nondecimated lifting technique teamed with empirical Bayes shrinkage. Simulations show that our proposed method has higher performance than competing techniques able to work on irregular data. Our construction also opens avenues for generating a ‘best’ representation, which we shall explore.     

Recent advances to model anisotropic space–time data

Building new and flexible classes of nonseparable spatio-temporal covariances and variograms has resulted a key point of research in the last years. The goal of this paper is to present an up-to-date overview of recent spatio-temporal covariance models taking into account the problem of spatial anisotropy. The resulting structures are proved to have certain interesting mathematical properties, together with a considerable applicability. In particular, we focus on the problem of modelling anisotropy through isotropy within components. We present the Bernstein class, and a generalisation of Gneiting’s approach (2002a) to obtain new classes of space–time covariance functions which are spatially anisotropic. We also discuss some methods for building covariance functions that attain negative values. We finally present several differentiation and integration operators acting on particular space–time covariance classes.      

A convergent algorithm for a generalized multivariate isotonic regression problem

Sasabuchi et al. (Biometrika 70(2):465–472, 1983) introduces a multivariate version of the well-known univariate isotonic regression which plays a key role in the field of statistical inference under order restrictions. His proposed algorithm for computing the multivariate isotonic regression, however, is guaranteed to converge only under special conditions (Sasabuchi et al., J Stat Comput Simul 73(9):619–641, 2003). In this paper, a more general framework for multivariate isotonic regression is given and an algorithm based on Dykstra’s method is used to compute the multivariate isotonic regression. Two numerical examples are given to illustrate the algorithm and to compare the result with the one published by Fernando and Kulatunga (Comput Stat Data Anal 52:702–712, 2007).     

State space mixed models for longitudinal observations with binary and binomial responses

We propose a new class of state space models for longitudinal discrete response data where the observation equation is specified in an additive form involving both deterministic and random linear predictors. These models allow us to explicitly address the effects of trend, seasonal or other time-varying covariates while preserving the power of state space models in modeling serial dependence in the data. We develop a Markov chain Monte Carlo algorithm to carry out statistical inference for models with binary and binomial responses, in which we invoke de Jong and Shephard’s (Biometrika 82(2):339–350, 1995) simulation smoother to establish an efficient sampling procedure for the state variables. To quantify and control the sensitivity of posteriors on the priors of variance parameters, we add a signal-to-noise ratio type parameter in the specification of these priors. Finally, we illustrate the applicability of the proposed state space mixed models for longitudinal binomial response data in both simulation studies and data examples.     

Robust second-order least-squares estimator for regression models

The second-order least-squares estimator (SLSE) was proposed by Wang (Statistica Sinica 13:1201–1210, 2003) for measurement error models. It was extended and applied to linear and nonlinear regression models by Abarin and Wang (Far East J Theor Stat 20:179–196, 2006) and Wang and Leblanc (Ann Inst Stat Math 60:883–900, 2008). The SLSE is asymptotically more efficient than the ordinary least-squares estimator if the error distribution has a nonzero third moment. However, it lacks robustness against outliers in the data. In this paper, we propose a robust second-order least squares estimator (RSLSE) against X-outliers. The RSLSE is highly efficient with high breakdown point and is asymptotically normally distributed. We compare the RSLSE with other estimators through a simulation study. Our results show that the RSLSE performs very well.     

On directional Metropolis–Hastings algorithms

New Metropolis–Hastings algorithms using directional updates are introduced in this paper. Each iteration of a directional Metropolis–Hastings algorithm consists of three steps (i) generate a line by sampling an auxiliary variable, (ii) propose a new state along the line, and (iii) accept/reject according to the Metropolis–Hastings acceptance probability. We consider two classes of directional updates. The first uses a point in ⁿ as auxiliary variable, the second an auxiliary direction vector. The proposed algorithms generalize previous directional updating schemes since we allow the distribution of the auxiliary variable to depend on properties of the target at the current state. By letting the proposal distribution along the line depend on the density of the auxiliary variable, we identify proposal mechanisms that give unit acceptance rate. When we use direction vector as auxiliary variable, we get the advantageous effect of large moves in the Markov chain and hence the autocorrelation length of the samples is small. We apply the directional Metropolis–Hastings algorithms to a Gaussian example, a mixture of Gaussian densities, and a Bayesian model for seismic data.     

Hidden Second‐order Stationary Spatial Point Processes

In the existing statistical literature, the almost default choice for inference on inhomogeneous point processes is the most well‐known model class for inhomogeneous point processes: reweighted second‐order stationary processes. In particular, the K‐function related to this type of inhomogeneity is presented as the inhomogeneous K‐function. In the present paper, we put a number of inhomogeneous model classes (including the class of reweighted second‐order stationary processes) into the common general framework of hidden second‐order stationary processes, allowing for a transfer of statistical inference procedures for second‐order stationary processes based on summary statistics to each of these model classes for inhomogeneous point processes. In particular, a general method to test the hypothesis that a given point pattern can be ascribed to a specific inhomogeneous model class is developed. Using the new theoretical framework, we reanalyse three inhomogeneous point patterns that have earlier been analysed in the statistical literature and show that the conclusions concerning an appropriate model class must be revised for some of the point patterns.     

Indices of Dependence Between Types in Multivariate Point Patterns

We propose new summary statistics quantifying several forms of dependence between points of different types in a multi-type spatial point pattern. These statistics are the multivariate counterparts of the J -function for point processes of a single type, introduced by Lieshout & Baddeley (1996). They are based on comparing distances from a type i point to either the nearest type j point or to the nearest point in the pattern regardless of type to these distances seen from an arbitrary point in space. Information about the range of interaction can also be inferred. Our statistics can be computed explicitly for a range of well-known multivariate point process models. Some applications to bivariate and trivariate data sets are presented as well.     

A generalization of Turnbull’s estimator for nonparametric estimation of the conditional survival function with interval-censored data

Simple nonparametric estimates of the conditional distribution of a response variable given a covariate are often useful for data exploration purposes or to help with the specification or validation of a parametric or semi-parametric regression model. In this paper we propose such an estimator in the case where the response variable is interval-censored and the covariate is continuous. Our approach consists in adding weights that depend on the covariate value in the self-consistency equation proposed by Turnbull (J R Stat Soc Ser B 38:290–295, 1976), which results in an estimator that is no more difficult to implement than Turnbull’s estimator itself. We show the convergence of our algorithm and that our estimator reduces to the generalized Kaplan–Meier estimator (Beran, Nonparametric regression with randomly censored survival data, 1981) when the data are either complete or right-censored. We demonstrate by simulation that the estimator, bootstrap variance estimation and bandwidth selection (by rule of thumb or cross-validation) all perform well in finite samples. We illustrate the method by applying it to a dataset from a study on the incidence of HIV in a group of female sex workers from Kinshasa.     

A computational framework for empirical Bayes inference

In empirical Bayes inference one is typically interested in sampling from the posterior distribution of a parameter with a hyper-parameter set to its maximum likelihood estimate. This is often problematic particularly when the likelihood function of the hyper-parameter is not available in closed form and the posterior distribution is intractable. Previous works have dealt with this problem using a multi-step approach based on the EM algorithm and Markov Chain Monte Carlo (MCMC). We propose a framework based on recent developments in adaptive MCMC, where this problem is addressed more efficiently using a single Monte Carlo run. We discuss the convergence of the algorithm and its connection with the EM algorithm. We apply our algorithm to the Bayesian Lasso of Park and Casella (J. Am. Stat. Assoc. 103:681–686, 2008) and on the empirical Bayes variable selection of George and Foster (J. Am. Stat. Assoc. 87:731–747, 2000).     

Double ASN Minimax sampling plans by variables when the standard deviation is unknown

We deal with the double sampling plans by variables proposed by Bowker and Goode (Sampling Inspection by Variables, McGraw–Hill, New York, 1952) when the standard deviation is unknown. Using the procedure for the calculation of the OC given by Krumbholz and Rohr (Allg. Stat. Arch. 90:233–251, 2006), we present an optimization algorithm allowing to determine the ASN Minimax plan. This plan, among all double plans satisfying the classical two-point-condition on the OC, has the minimal ASN maximum.     

Estimating and testing generalized linear models under inequality restrictions

We consider maximum likelihood estimation and likelihood ratio tests under inequality restrictions on the parameters. A special case are order restrictions, which may appear for example in connection with effects of an ordinal qualitative covariate. Our estimation approach is based on the principle of sequential quadratic programming, where the restricted estimate is computed iteratively and a quadratic optimization problem under inequality restrictions is solved in each iteration. Testing for inequality restrictions is based on the likelihood ratio principle. Under certain regularity assumptions the likelihood ratio test statistic is asymptotically distributed like a mixture of χ², where the weights are a function of the restrictions and the information matrix. A major problem in theory is that in general there is no unique least favourable point. We present some empirical findings on finite-sample behaviour of tests and apply the methods to examples from credit scoring and dentistry.     

On MSE of EBLUP

We consider Best Linear Unbiased Predictors (BLUPs) and Empirical Best Linear Unbiased Predictors (EBLUPs) under the general mixed linear model. The BLUP was proposed by Henderson (Ann Math Stat 21:309–310, 1950). The formula of this BLUP includes unknown elements of the variance-covariance matrix of random variables. If the elements in the formula of the BLUP proposed by Henderson (Ann Math Stat 21:309–310, 1950) are replaced by some type of estimators, we obtain the two-stage predictor called the EBLUP which is model-unbiased (Kackar and Harville in Commun Stat A 10:1249–1261, 1981). Kackar and Harville (J Am Stat Assoc 79:853–862, 1984) show an approximation of the mean square error (the MSE) of the predictor and propose an estimator of the MSE. The MSE and estimators of the MSE are also studied by Prasad and Rao (J Am Stat Assoc 85:163–171, 1990), Datta and Lahiri (Stat Sin 10:613–627, 2000) and Das et al. (Ann Stat 32(2):818–840, 2004). In the paper we consider the BLUP proposed by Royall (J Am Stat Assoc 71:657–473, 1976. Ża̧dło (On unbiasedness of some EBLU predictor. Physica-Verlag, Heidelberg, pp 2019–2026, 2004) shows that the BLUP proposed by Royall (J Am Stat Assoc 71:657–473, 1976) may be treated as a generalisation of the BLUP proposed by Henderson (Ann Math Stat 21:309–310, 1950) and proves model unbiasedness of the EBLUP based on the formula of the BLUP proposed by Royall (J Am Stat Assoc 71:657–473, 1976) under some assumptions. In this paper we derive the formula of the approximate MSE of the EBLUP and its estimators. We prove that the approximation of the MSE is accurate to terms o(D ⁻¹) and that the estimator of the MSE is approximately unbiased in the sense that its bias is o(D ⁻¹) under some assumptions, where D is the number of domains. The proof is based on the results obtained by Datta and Lahiri (Stat Sin 10:613–627, 2000). Using our results we show some EBLUP based on the special case of the general linear model. We also present the formula of its MSE and estimators of its MSE and their performance in Monte Carlo simulation study.      

Spatial Matching of Multiple Configurations of Points with a Bioinformatics Application

In this article, we present a general model to deal with the problem of matching multiple objects or configurations of points from a Bayesian point of view. We study both labeled and non labeled cases. Our model generalizes, in terms of non singular affine transformations and multiple configurations, previous two-terms matching models. As a practical application in Bioinformatics, we consider data from a microarray experiment of gorilla, bonobo, and human-cultured fibroblasts. We find out the matchings and the best affine transformation between the projections of genes on a two-dimensional space, obtained by a multidimensional scaling technique.     

A note on the equality of the OLSE and the BLUE of the parametric function in the general Gauss–Markov model

In this note we consider the equality of the ordinary least squares estimator (OLSE) and the best linear unbiased estimator (BLUE) of the estimable parametric function in the general Gauss–Markov model. Especially we consider the structures of the covariance matrix V for which the OLSE equals the BLUE. Our results are based on the properties of a particular reparametrized version of the original Gauss–Markov model.      

ABC Shadow algorithm: a tool for statistical analysis of spatial patterns

This paper presents an original ABC algorithm, ABC Shadow, that can be applied to sample posterior densities that are continuously differentiable. The proposed algorithm solves the main condition to be fulfilled by any ABC algorithm, in order to be useful in practice. This condition requires enough samples in the parameter space region, induced by the observed statistics. The algorithm is tuned on the posterior of a Gaussian model which is entirely known, and then, it is applied for the statistical analysis of several spatial patterns. These patterns are issued or assumed to be outcomes of point processes. The considered models are: Strauss, Candy and area-interaction.     

A transformation approach for the analysis of interval-censored failure time data

This paper discusses the analysis of interval-censored failure time data, which has recently attracted a great amount of attention (Li and Pu, Lifetime Data Anal 9:57–70, 2003; Sun, The statistical analysis of interval-censored data, 2006; Tian and Cai, Biometrika 93(2):329–342, 2006; Zhang et al., Can J Stat 33:61–70, 2005). Interval-censored data mean that the survival time of interest is observed only to belong to an interval and they occur in many fields including clinical trials, demographical studies, medical follow-up studies, public health studies and tumorgenicity experiments. A major difficulty with the analysis of interval-censored data is that one has to deal with a censoring mechanism that involves two related variables. For the inference, we present a transformation approach that transforms general interval-censored data into current status data, for which one only needs to deal with one censoring variable and the inference is thus much easy. We apply this general idea to regression analysis of interval-censored data using the additive hazards model and numerical studies indicate that the method performs well for practical situations. An illustrative example is provided.