Bootstrapping regression models with BLUS residuals

To bootstrap a regression problem, pairs of response and explanatory variables or residuals can be resam‐pled, according to whether we believe that the explanatory variables are random or fixed. In the latter case, different residuals have been proposed in the literature, including the ordinary residuals (Efron 1979), standardized residuals (Bickel & Freedman 1983) and Studentized residuals (Weber 1984). Freedman (1981) has shown that the bootstrap from ordinary residuals is asymptotically valid when the number of cases increases and the number of variables is fixed. Bickel & Freedman (1983) have shown the asymptotic validity for ordinary residuals when the number of variables and the number of cases both increase, provided that the ratio of the two converges to zero at an appropriate rate. In this paper, the authors introduce the use of BLUS (Best Linear Unbiased with Scalar covariance matrix) residuals in bootstrapping regression models. The main advantage of the BLUS residuals, introduced in Theil (1965), is that they are uncorrelated. The main disadvantage is that only n —p residuals can be computed for a regression problem with n cases and p variables. The asymptotic results of Freedman (1981) and Bickel & Freedman (1983) for the ordinary (and standardized) residuals are generalized to the BLUS residuals. A small simulation study shows that even though only n — p residuals are available, in small samples bootstrapping BLUS residuals can be as good as, and sometimes better than, bootstrapping from standardized or Studentized residuals.     

Estimation of parameters of bivariate normal distribution using concomitants of record values

In this paper, we discuss the concomitants of record values arising from the well-known bivariate normal distribution BVND(μ₁, μ₂,σ₁,σ₂, ρ). We have obtained the best linear unbiased estimators of μ₂ and σ₂ when ρ is known and derived some unbiased linear estimators of ρ when μ₂ and σ₂ are known, based on the concomitants of first n record values. The variances of these estimators have been obtained.     

Bayesian and Robust Bayesian analysis under a general class of balanced loss functions

For estimating an unknown parameter θ, we introduce and motivate the use of balanced loss functions of the form L_{r, w, d0}(q, d)=wr(d₀, d)+ (1-w) r(q, d){L_{\rho, \omega, \delta_0}(\theta, \delta)=\omega \rho(\delta_0, \delta)+ (1-\omega) \rho(\theta, \delta)}, as well as the weighted version q(q) L_{r, w, d0}(q, d){q(\theta) L_{\rho, \omega, \delta_0}(\theta, \delta)}, where ρ(θ, δ) is an arbitrary loss function, δ ₀ is a chosen a priori “target” estimator of q, w ? [0,1){\theta, \omega \in[0,1)}, and q(·) is a positive weight function. we develop Bayesian estimators under L_{r, w, d0}{L_{\rho, \omega, \delta_0}} with ω > 0 by relating such estimators to Bayesian solutions under L_{r, w, d0}{L_{\rho, \omega, \delta_0}} with ω = 0. Illustrations are given for various choices of ρ, such as absolute value, entropy, linex, and squared error type losses. Finally, under various robust Bayesian analysis criteria including posterior regret gamma-minimaxity, conditional gamma-minimaxity, and most stable, we establish explicit connections between optimal actions derived under balanced and unbalanced losses.     

Consistency of completely outlier-adjusted simultaneous redescending M-estimators of location and scale

This paper gives conditions for the consistency of simultaneous redescending M-estimators for location and scale. The consistency postulates the uniqueness of the parameters μ and σ, which are defined analogously to the estimations by using the population distribution function instead of the empirical one. The uniqueness of these parameters is no matter of course, because redescending ψ- and χ-functions, which define the parameters, cannot be chosen in a way that the parameters can be considered as the result of a common minimizing problem where the sum of ρ-functions of standardized residuals is to be minimized. The parameters arise from two minimizing problems where the result of one problem is a parameter of the other one. This can give different solutions. Proceeding from a symmetrical unimodal distribution and the usual symmetry assumptions for ψ and χ leads, in most but not in all cases, to the uniqueness of the parameters. Under this and some other assumptions, we can also prove the consistency of the according M-estimators, although these estimators are usually not unique even when the parameters are. The present article also serves as a basis for a forthcoming paper, which is concerned with a completely outlier-adjusted confidence interval for μ. So we introduce a ñ where data points far away from the bulk of the data are not counted at all.     

Note on the bias in the estimation of the serial correlation coefficient of AR(1) processes

We derive approximating formulas for the mean and the variance of an autocorrelation estimator which are of practical use over the entire range of the autocorrelation coefficient ρ. The least-squares estimator ∑ ^{n −1} _{i =1}ε _i ε _{i +1} / ∑ ^{n −1} _{i =1}ε² _i is studied for a stationary AR(1) process with known mean. We use the second order Taylor expansion of a ratio, and employ the arithmetic-geometric series instead of replacing partial Cesàro sums. In case of the mean we derive Marriott and Pope's (1954) formula, with (n− 1)⁻¹ instead of (n)⁻¹, and an additional term α (n− 1)⁻². This new formula produces the expected decline to zero negative bias as ρ approaches unity. In case of the variance Bartlett's (1946) formula results, with (n− 1)⁻¹ instead of (n)⁻¹. The theoretical expressions are corroborated with a simulation experiment. A comparison shows that our formula for the mean is more accurate than the higher-order approximation of White (1961), for |ρ| > 0.88 and n≥ 20. In principal, the presented method can be used to derive approximating formulas for other estimators and processes. Received: November 30, 1999; revised version: July 3, 2000     

The multivariate linear model with multivariatet and intra-class covariance structure

The prediction distribution of future responses from a multivariate linear model with error having a multivariatet-distribution and intra-class covariance structure has been derived. The distribution depends on ρ, the intra-class correlation coefficient. For unknown ρ, the marginal likelihood function of ρ has been obtained and the prediction distribution has been approximated by the estimate of ρ. As an application, a β-expectation tolerance region for the model has been constructed.     

Model fitting and inference under latent equilibrium processes

This paper presents a methodology for model fitting and inference in the context of Bayesian models of the type f(Y | X,θ)f(X|θ)f(θ), where Y is the (set of) observed data, θ is a set of model parameters and X is an unobserved (latent) stationary stochastic process induced by the first order transition model f(X ^(t+1)|X ^(t),θ), where X ^(t) denotes the state of the process at time (or generation) t. The crucial feature of the above type of model is that, given θ, the transition model f(X ^(t+1)|X ^(t),θ) is known but the distribution of the stochastic process in equilibrium, that is f(X|θ), is, except in very special cases, intractable, hence unknown. A further point to note is that the data Y has been assumed to be observed when the underlying process is in equilibrium. In other words, the data is not collected dynamically over time. We refer to such specification as a latent equilibrium process (LEP) model. It is motivated by problems in population genetics (though other applications are discussed), where it is of interest to learn about parameters such as mutation and migration rates and population sizes, given a sample of allele frequencies at one or more loci. In such problems it is natural to assume that the distribution of the observed allele frequencies depends on the true (unobserved) population allele frequencies, whereas the distribution of the true allele frequencies is only indirectly specified through a transition model. As a hierarchical specification, it is natural to fit the LEP within a Bayesian framework. Fitting such models is usually done via Markov chain Monte Carlo (MCMC). However, we demonstrate that, in the case of LEP models, implementation of MCMC is far from straightforward. The main contribution of this paper is to provide a methodology to implement MCMC for LEP models. We demonstrate our approach in population genetics problems with both simulated and real data sets. The resultant model fitting is computationally intensive and thus, we also discuss parallel implementation of the procedure in special cases.     

Choosing Prior Hyperparameters: With Applications to Time-Varying Parameter Models

Time-varying parameter models with stochastic volatility are widely used to study macroeconomic and financial data. These models are almost exclusively estimated using Bayesian methods. A common practice is to focus on prior distributions that themselves depend on relatively few hyperparameters such as the scaling factor for the prior covariance matrix of the residuals governing time variation in the parameters. The choice of these hyperparameters is crucial because their influence is sizeable for standard sample sizes. In this article, we treat the hyperparameters as part of a hierarchical model and propose a fast, tractable, easy-to-implement, and fully Bayesian approach to estimate those hyperparameters jointly with all other parameters in the model. We show via Monte Carlo simulations that, in this class of models, our approach can drastically improve on using fixed hyperparameters previously proposed in the literature. Supplementary materials for this article are available online.     

Pseudo latent models: Goodness of fit measures, residuals, estimation, testing, and simulation

Binary response models consider pseudo-R ² measures which are not based on residuals while several concepts of residuals were developed for tests. In this paper the endogenous variable of the latent model corresponding to the binary observable model is substituted by a pseudo variable. Then goodness of fit measures and tests can be based on a joint concept of residuals as for linear models. Different kinds of residuals based on probit ML estimates are employed. The analytical investigations and the simulation results lead to the recommendation to use standardized residuals where there is no difference between observed and generalized residuals. In none of the investigated situations this estimator is far away from the best result. While in large samples all considered estimators are very similar, small sample properties speak in favour of residuals which are modifications of those suggested in the literature. An empirical application demonstrates that it is not necessary to develop new testing procedures for the observable models with dichotomous regressands. Well-know approaches for linear models with continuous endogenous variables which are implemented in usual econometric packages can be used for pseudo latent models. An erratum to this article is available at .     

A generalized skew two-piece skew-normal distribution

In this paper, we discuss a general class of skew two-piece skew-normal distributions, denoted by GSTPSN(λ₁, λ₂, ρ). We derive its moment generating function and discuss some simple and interesting properties of this distribution. We then discuss the modes of these distributions and present a useful representation theorem as well. Next, we focus on a different generalization of the two-piece skew-normal distribution which is a symmetric family of distributions and discuss some of its properties. Finally, three well-known examples are used to illustrate the practical usefulness of this family of distributions.     

Quantity quantiles linear regression

We show that the definition of the θth sample quantile as the solution to a minimization problem introduced by Koenker and Bassett (Econometrica 46(1):33–50, 1978) can be easily extended to obtain an analogous definition for the θth sample quantity quantile widely investigated and applied in the Italian literature. The key point is the use of the first-moment distribution of the variable instead of its distribution function. By means of this definition we introduce a linear regression model for quantity quantiles and analyze some properties of the residuals. In Sect. 4 we show a brief application of the methodology proposed. This research was partially supported by Fondo d’Ateneo per la Ricerca anno 2005—Università degli Studi di Milano-Bicocca. The paper is the result of the common work of the authors; in particular M. Zenga has written Sects. 1 and 5 while P. Radaelli has written the remaining sections.     

Adjusted Pearson residuals in exponential family nonlinear models

In this paper, we give matrix formulae of order 𝒪(n ^?1), where n is the sample size, for the first two moments of Pearson residuals in exponential family nonlinear regression models [G.M. Cordeiro and G.A. Paula, Improved likelihood ratio statistic for exponential family nonlinear models, Biometrika 76 (1989), pp. 93–100.]. The formulae are applicable to many regression models in common use and generalize the results by Cordeiro [G.M. Cordeiro, On Pearson's residuals in generalized linear models, Statist. Prob. Lett. 66 (2004), pp. 213–219.] and Cook and Tsai [R.D. Cook and C.L. Tsai, Residuals in nonlinear regression, Biometrika 72(1985), pp. 23–29.]. We suggest adjusted Pearson residuals for these models having, to this order, the expected value zero and variance one. We show that the adjusted Pearson residuals can be easily computed by weighted linear regressions. Some numerical results from simulations indicate that the adjusted Pearson residuals are better approximated by the standard normal distribution than the Pearson residuals.     

Effect of individual observations on the Box–Cox transformation

In this paper, we consider the influence of individual observations on inferences about the Box–Cox power transformation parameter from a Bayesian point of view. We compare Bayesian diagnostic measures with the ‘forward’ method of analysis due to Riani and Atkinson. In particular, we look at the effect of omitting observations on the inference by comparing particular choices of transformation using the conditional predictive ordinate and the k _d measure of Pettit and Young. We illustrate the methods using a designed experiment. We show that a group of masked outliers can be detected using these single deletion diagnostics. Also, we show that Bayesian diagnostic measures are simpler to use to investigate the effect of observations on transformations than the forward search method.     

A Bayesian analysis of a change in the parameters of autoregressive time series

In this article, we consider a Bayesian analysis of a possible change in the parameters of autoregressive time series of known order p, AR(p). An unconditional Bayesian test based on highest posterior density (HPD) credible sets is determined. The test is useful to detect a change in any one of the parameters separately. Using the Gibbs sampler algorithm, we approximate the posterior densities of the change point and other parameters to calculate the p-values that define our test.     

Wavelet Shrinkage with Double Weibull Prior

In this article, we propose a denoising methodology in the wavelet domain based on a Bayesian hierarchical model using Double Weibull prior. We propose two estimators, one based on posterior mean (Double Weibull Wavelet Shrinker, DWWS) and the other based on larger posterior mode (DWWS-LPM), and show how to calculate them efficiently. Traditionally, mixture priors have been used for modeling sparse wavelet coefficients. The interesting feature of this article is the use of non-mixture prior. We show that the methodology provides good denoising performance, comparable even to state-of-the-art methods that use mixture priors and empirical Bayes setting of hyperparameters, which is demonstrated by extensive simulations on standardly used test functions. An application to real-word dataset is also considered.     

A hierarchical Bayesian regression model for the uncertain functional constraint using screened scale mixtures of Gaussian distributions

This paper considers a hierarchical Bayesian analysis of regression models using a class of Gaussian scale mixtures. This class provides a robust alternative to the common use of the Gaussian distribution as a prior distribution in particular for estimating the regression function subject to uncertainty about the constraint. For this purpose, we use a family of rectangular screened multivariate scale mixtures of Gaussian distribution as a prior for the regression function, which is flexible enough to reflect the degrees of uncertainty about the functional constraint. Specifically, we propose a hierarchical Bayesian regression model for the constrained regression function with uncertainty on the basis of three stages of a prior hierarchy with Gaussian scale mixtures, referred to as a hierarchical screened scale mixture of Gaussian regression models (HSMGRM). We describe distributional properties of HSMGRM and an efficient Markov chain Monte Carlo algorithm for posterior inference, and apply the proposed model to real applications with constrained regression models subject to uncertainty.     

An efficient Monte Carlo EM algorithm for Bayesian lasso

The lasso is a popular technique of simultaneous estimation and variable selection in many research areas. The marginal posterior mode of the regression coefficients is equivalent to estimates given by the non-Bayesian lasso when the regression coefficients have independent Laplace priors. Because of its flexibility of statistical inferences, the Bayesian approach is attracting a growing body of research in recent years. Current approaches are primarily to either do a fully Bayesian analysis using Markov chain Monte Carlo (MCMC) algorithm or use Monte Carlo expectation maximization (MCEM) methods with an MCMC algorithm in each E-step. However, MCMC-based Bayesian method has much computational burden and slow convergence. Tan et al. [An efficient MCEM algorithm for fitting generalized linear mixed models for correlated binary data. J Stat Comput Simul. 2007;77:929–943] proposed a non-iterative sampling approach, the inverse Bayes formula (IBF) sampler, for computing posteriors of a hierarchical model in the structure of MCEM. Motivated by their paper, we develop this IBF sampler in the structure of MCEM to give the marginal posterior mode of the regression coefficients for the Bayesian lasso, by adjusting the weights of importance sampling, when the full conditional distribution is not explicit. Simulation experiments show that the computational time is much reduced with our method based on the expectation maximization algorithm and our algorithms and our methods behave comparably with other Bayesian lasso methods not only in prediction accuracy but also in variable selection accuracy and even better especially when the sample size is relatively large.     

Parameter recovery for a skew-normal IRT model under a Bayesian approach: hierarchical framework,prior and kernel sensitivity and sample size

Item response theory (IRT) comprises a set of statistical models which are useful in many fields, especially when there is an interest in studying latent variables (or latent traits). Usually such latent traits are assumed to be random variables and a convenient distribution is assigned to them. A very common choice for such a distribution has been the standard normal. Recently, Azevedo et al. [Bayesian inference for a skew-normal IRT model under the centred parameterization, Comput. Stat. Data Anal. 55 (2011), pp. 353–365] proposed a skew-normal distribution under the centred parameterization (SNCP) as had been studied in [R.B. Arellano-Valle and A. Azzalini, The centred parametrization for the multivariate skew-normal distribution, J. Multivariate Anal. 99(7) (2008), pp. 1362–1382], to model the latent trait distribution. This approach allows one to represent any asymmetric behaviour concerning the latent trait distribution. Also, they developed a Metropolis–Hastings within the Gibbs sampling (MHWGS) algorithm based on the density of the SNCP. They showed that the algorithm recovers all parameters properly. Their results indicated that, in the presence of asymmetry, the proposed model and the estimation algorithm perform better than the usual model and estimation methods. Our main goal in this paper is to propose another type of MHWGS algorithm based on a stochastic representation (hierarchical structure) of the SNCP studied in [N. Henze, A probabilistic representation of the skew-normal distribution, Scand. J. Statist. 13 (1986), pp. 271–275]. Our algorithm has only one Metropolis–Hastings step, in opposition to the algorithm developed by Azevedo et al., which has two such steps. This not only makes the implementation easier but also reduces the number of proposal densities to be used, which can be a problem in the implementation of MHWGS algorithms, as can be seen in [R.J. Patz and B.W. Junker, A straightforward approach to Markov Chain Monte Carlo methods for item response models, J. Educ. Behav. Stat. 24(2) (1999), pp. 146–178; R.J. Patz and B.W. Junker, The applications and extensions of MCMC in IRT: Multiple item types, missing data, and rated responses, J. Educ. Behav. Stat. 24(4) (1999), pp. 342–366; A. Gelman, G.O. Roberts, and W.R. Gilks, Efficient Metropolis jumping rules, Bayesian Stat. 5 (1996), pp. 599–607]. Moreover, we consider a modified beta prior (which generalizes the one considered in [3 Azevedo, C. L.N., Bolfarine, H. and Andrade, D. F. 2011. Bayesian inference for a skew-normal IRT model under the centred parameterization. Comput. Stat. Data Anal., 55: 353–365. [Crossref], [Web of Science ®] , [Google Scholar]]) and a Jeffreys prior for the asymmetry parameter. Furthermore, we study the sensitivity of such priors as well as the use of different kernel densities for this parameter. Finally, we assess the impact of the number of examinees, number of items and the asymmetry level on the parameter recovery. Results of the simulation study indicated that our approach performed equally as well as that in [3 Azevedo, C. L.N., Bolfarine, H. and Andrade, D. F. 2011. Bayesian inference for a skew-normal IRT model under the centred parameterization. Comput. Stat. Data Anal., 55: 353–365. [Crossref], [Web of Science ®] , [Google Scholar]], in terms of parameter recovery, mainly using the Jeffreys prior. Also, they indicated that the asymmetry level has the highest impact on parameter recovery, even though it is relatively small. A real data analysis is considered jointly with the development of model fitting assessment tools. The results are compared with the ones obtained by Azevedo et al. The results indicate that using the hierarchical approach allows us to implement MCMC algorithms more easily, it facilitates diagnosis of the convergence and also it can be very useful to fit more complex skew IRT models.     

Stochastic boosting algorithms

In this article we develop a class of stochastic boosting (SB) algorithms, which build upon the work of Holmes and Pintore (Bayesian Stat. 8, Oxford University Press, Oxford, 2007). They introduce boosting algorithms which correspond to standard boosting (e.g. Bühlmann and Hothorn, Stat. Sci. 22:477–505, 2007) except that the optimization algorithms are randomized; this idea is placed within a Bayesian framework. We show that the inferential procedure in Holmes and Pintore (Bayesian Stat. 8, Oxford University Press, Oxford, 2007) is incorrect and further develop interpretational, computational and theoretical results which allow one to assess SB’s potential for classification and regression problems. To use SB, sequential Monte Carlo (SMC) methods are applied. As a result, it is found that SB can provide better predictions for classification problems than the corresponding boosting algorithm. A theoretical result is also given, which shows that the predictions of SB are not significantly worse than boosting, when the latter provides the best prediction. We also investigate the method on a real case study from machine learning.     

Measuring robustness for weighted distributions: Bayesian perspective

There are many situations where the usual random sample from a population of interest is not available, due to the data having unequal probabilities of entering the sample. The method of weighted distributions models this ascertainment bias by adjusting the probabilities of actual occurrence of events to arrive at a specification of the probabilities of the events as observed and recorded. We consider two different classes of contaminated or mixture of weight functions, Γ _a ={w(x):w(x)=(1−ε)w ₀(x)+εq(x),q∈Q} and Γ _g ={w(x):w(x)=w ₀ ^1−ε (x)q ^ε(x),q∈Q} wherew ₀(x) is the elicited weighted function,Q is a class of positive functions and 0≤ε≤1 is a small number. Also, we study the local variation of ϕ-divergence over classes Γ _a and Γ _g . We devote on measuring robustness using divergence measures which is based on the Bayesian approach. Two examples will be studied.