Sufficient conditions for Bayesian consistency

We introduce the Hausdorff α$\alpha$-entropy to study the strong Hellinger consistency of posterior distributions. We obtain general Bayesian consistency theorems which extend the well-known results of Barron et al. [1999. The consistency of posterior distributions in nonparametric problems. Ann. Statist. 27, 536–561] and Ghosal et al. [1999. Posterior consistency of Dirichlet mixtures in density estimation. Ann. Statist. 27, 143–158] and Walker [2004. New approaches to Bayesian consistency. Ann. Statist. 32, 2028–2043]. As an application we strengthen previous results on Bayesian consistency of the (normal) mixture models.     

On the detection of changes in autoregressive time series,II. Resampling procedures

We study an autoregressive time series model with a possible change in the regression parameters. Approximations to the critical values for change-point tests are obtained through various bootstrapping methods. Theoretical results show that the bootstrapping procedures have the same limiting behavior as their asymptotic counterparts discussed in Hušková et al. [2007. On the detection of changes in autoregressive time series, I. Asymptotics. J. Statist. Plann. Inference 137, 1243–1259]. In fact, a small simulation study illustrates that the bootstrap tests behave better than the original asymptotic tests if performance is measured by the α$\alpha$- and β$\beta$-errors, respectively.     

Linear sufficiency and completeness in the context of estimating the parametric function in the general Gauss–Markov model

In this paper we consider linear sufficiency and linear completeness in the context of estimating the estimable parametric function K^′β${\mathbf{K}}^{\prime }\mathit{\beta }$ under the general Gauss–Markov model {y,Xβ,σ²V}$\{\mathbf{y},\mathbf{X}\mathit{\beta },{\sigma }^2\mathbf{V}\}$. We give new characterizations for linear sufficiency, and define and characterize linear completeness in a case of estimation of K^′β${\mathbf{K}}^{\prime }\mathit{\beta }$. Also, we consider a predictive approach for obtaining the best linear unbiased estimator of K^′β${\mathbf{K}}^{\prime }\mathit{\beta }$, and subsequently, we give the linear analogues of the Rao–Blackwell and Lehmann–Scheffé Theorems in the context of estimating K^′β${\mathbf{K}}^{\prime }\mathit{\beta }$.     

Asymptotics of the Lp-Norms of Density Estimators in the Nonlinear Autoregressive Models

This article investigates the asymptotic behavior of the error density function in nonlinear autoregressive stationary time series regression models. For any 1 ? p < ∞, the kernel density estimator of residuals is shown to be consistent for the error estimator concerning the L_p-distance, which extends the result developed by Cheng and Sun (2008 Cheng, F. X. (2005). Asymptotic distributions of error density estimators in first-order autoregression models. Sankhy Ind. J. Statist. 67:553–567. [Google Scholar]) in L₂-norm. Moreover, the result developed in this article is extended the results of Horváth and Zitikis (2003 Horváth, L., Zitikis, R. (2003). Asymptotics of the L_p-norms of density estimators in the first-order autoregressive models. Statist. Probab. Lett. 65:331342.[Crossref], [Web of Science ®] , [Google Scholar]) to nonlinear autoregressive models.     

Robust estimation of the correlation matrix of longitudinal data

We propose a double-robust procedure for modeling the correlation matrix of a longitudinal dataset. It is based on an alternative Cholesky decomposition of the form Σ=DLL ^? D where D is a diagonal matrix proportional to the square roots of the diagonal entries of Σ and L is a unit lower-triangular matrix determining solely the correlation matrix. The first robustness is with respect to model misspecification for the innovation variances in D, and the second is robustness to outliers in the data. The latter is handled using heavy-tailed multivariate t-distributions with unknown degrees of freedom. We develop a Fisher scoring algorithm for computing the maximum likelihood estimator of the parameters when the nonredundant and unconstrained entries of (L,D) are modeled parsimoniously using covariates. We compare our results with those based on the modified Cholesky decomposition of the form LD ² L ^? using simulations and a real dataset.     

Maximum modulus confidence bands

A family of confidence bands (simultaneous confidence regions) is given for EY = x′β that are piecewise-linear in x. Normality is assumed. These confidence bands are advocated over the usual hyperbolic band when the region of prime interest is distant from ${\overline{\bf x}}$ . In particular, this is the case when x?=?x(t) for time t and future time is of primary interest, that is for the prediction problem. For the case x′?=?(1, t), the family of bands includes that of Graybill and Bowden (J Am Stat Assoc 62:403–408, 1967).     

Transformed generalized linear models

The estimation of data transformation is very useful to yield response variables satisfying closely a normal linear model. Generalized linear models enable the fitting of models to a wide range of data types. These models are based on exponential dispersion models. We propose a new class of transformed generalized linear models to extend the Box and Cox models and the generalized linear models. We use the generalized linear model framework to fit these models and discuss maximum likelihood estimation and inference. We give a simple formula to estimate the parameter that index the transformation of the response variable for a subclass of models. We also give a simple formula to estimate the r$r$th moment of the original dependent variable. We explore the possibility of using these models to time series data to extend the generalized autoregressive moving average models discussed by Benjamin et al. [Generalized autoregressive moving average models. J. Amer. Statist. Assoc. 98, 214–223]. The usefulness of these models is illustrated in a simulation study and in applications to three real data sets.     

Robust estimation of multivariate location and shape

In this paper, we describe an overall strategy for robust estimation of multivariate location and shape, and the consequent identification of outliers and leverage points. Parts of this strategy have been described in a series of previous papers (Rocke, Ann. Statist., in press; Rocke and Woodruff, Statist. Neerlandica 47 (1993), 27–42, J. Amer. Statist. Assoc., in press; Woodruff and Rocke, J. Comput. Graphical Statist. 2 (1993), 69–95; J. Amer. Statist. Assoc. 89 (1994), 888–896) but the overall structure is presented here for the first time. After describing the first-level architecture of a class of algorithms for this problem, we review available information about possible tactics for each major step in the process. The major steps that we have found to be necessary are as follows: (1) partition the data into groups of perhaps five times the dimension; (2) for each group, search for the best available solution to a combinatorial estimator such as the Minimum Covariance Determinant (MCD) — these are the preliminary estimates; (3) for each preliminary estimate, iterate to the solution of a smooth estimator chosen for robustness and outlier resistance; and (4) choose among the final iterates based on a robust criterion, such as minimum volume. Use of this algorithm architecture can enable reliable, fast, robust estimation of heavily contaminated multivariate data in high (> 20) dimension even with large quantities of data. A computer program implementing the algorithm is available from the authors.     

Modeling Electricity Price Using A Threshold Conditional Autoregressive Geometric Process Jump Model

Electricity market prices are highly volatile and often have high spikes. Both government authorities and market participants require sophisticated models and techniques for forecasting future prices and managing relevant financial risks in such a volatile market. This article extends the conditional autoregressive geometric process (CARGP) model (Chan et al., 2012 Chan, J. S.K., Lam, C. P.Y., Yu, P. L.H., Choy, S. T.B., Chen, C. W.S. (2012). A Bayesian conditional autoregressive geometric process model for range data. Computat. Statist. Data Anal. 56:30063019.[Crossref], [Web of Science ®] , [Google Scholar]) to the CARGP model with thresholds and jumps, which is abbreviated as CARGP-TJ model in this article. We will demonstrate that the proposed CARGP-TJ model not only captures the unique features of the electricity price but also performs better than other existing models. For robustness consideration, a heavy-tailed error distribution is adopted. Model implementation relies on the powerful Bayesian Markov chain Monte Carlo simulation techniques via WinBUGS software. The analysis of the daily maximum electricity prices of the New South Wales, Australia reveals that the proposed CARGP-TJ model captures the price spikes well for both in-sample estimation and out-of-sample forecast.     

Flexible univariate and multivariate models based on hidden truncation

A broad spectrum of flexible univariate and multivariate models can be constructed by using a hidden truncation paradigm. Such models can be viewed as being characterized by a basic marginal density, a family of conditional densities and a specified hidden truncation point, or points. The resulting class of distributions includes the basic marginal density as a special case (or as a limiting case), but also includes an array of models that may unexpectedly include many well known densities. Most of the well known skew-normal models (developed from the seed distribution popularized by Azzalini [(1985). A class of distributions which includes the normal ones. Scand. J. Statist. 12(2), 171–178]) can be viewed as being products of such a hidden truncation construction. However, the many hidden truncation models with non-normal component densities undoubtedly deserve further attention.     

Some properties of the generalized TTT transform

Recently Li and Shaked [2007. A general family of univariate stochastic orders. J. Statist. Plann. Inference 137, 3601–3610] introduced the generalized total time on test (GTTT) transform with respect to a given function ?$?$. In this paper we study some properties of it which are related with stochastic orderings. A concept of Lehmann and Rojo [1992. Invariant directional orderings. Ann. Statist. 20, 2100–2110] is applied to a new setting and the GTTT transform is used to define invariance properties and distances of some stochastic orders. Iterations of the GTTT transforms are also studied and their relations with exponential mixtures of gamma distributions are established.     

Empirical Bayes regression analysis with many regressors but fewer observations

In this paper, we consider the prediction problem in multiple linear regression model in which the number of predictor variables, p, is extremely large compared to the number of available observations, n  . The least-squares predictor based on a generalized inverse is not efficient. We propose six empirical Bayes estimators of the regression parameters. Three of them are shown to have uniformly lower prediction error than the least-squares predictors when the vector of regressor variables are assumed to be random with mean vector zero and the covariance matrix (1/n)X^tX$(1/n){\mathit{X}}^{\mathrm{t}}\mathit{X}$ where X^t=(_x1,…,_xn)${\mathit{X}}^{\mathrm{t}}=({\mathit{x}}_1,\dots ,{\mathit{x}}_n)$ is the p×n$p\times n$ matrix of observations on the regressor vector centered from their sample means. For other estimators, we use simulation to show its superiority over the least-squares predictor.     

Asymptotic expansions of the null distributions of some test statistics for profile analysis in general distributions

This paper discusses asymptotic expansions for the null distributions of some test statistics for profile analysis under non-normality. It is known that the null distributions of these statistics converge to chi-square distribution under normality [Siotani, M., 1956. On the distributions of the Hotelling's T²$T^2$-statistics. Ann. Inst. Statist. Math. Tokyo 8, 1–14; Siotani, M., 1971. An asymptotic expansion of the non-null distributions of Hotelling's generalized T²$T^2$-statistic. Ann. Math. Statist. 42, 560–571]. We extend this result by obtaining asymptotic expansions under general distributions. Moreover, the effect of non-normality is also considered. In order to obtain all the results, we make use of matrix manipulations such as direct products and symmetric tensor, rather than usual elementwise tensor notation.     

L models and multiple regressions designs

Given an orthogonal model

${{\bf \lambda}}=\sum_{i=1}^m{{{\bf X}}_i}{\boldsymbol{\alpha}}_i$

an L model

${{\bf y}}={\bf L}\left(\sum_{i=1}^m{{{\bf X}}_i}{\boldsymbol{\alpha}}_i\right)+{\bf e}$

is obtained, and the only restriction is the linear independency of the column vectors of matrix L. Special cases of the L models correspond to blockwise diagonal matrices L = D(L ₁, . . . , L _c ). In multiple regression designs this matrix will be of the form

${\bf L}={\bf D}(\check{{\bf X}}_1,\ldots,\check{{\bf X}}_{c})$

with \({\check{{\bf X}}_j, j=1,\ldots,c}\) the model matrices of the individual regressions, while the original model will have fixed effects. In this way, we overcome the usual restriction of requiring all regressions to have the same model matrix.

     

Adaptive LASSO for linear regression models with ARMA-GARCH errors

The linear regression models with the autoregressive moving average (ARMA) errors (REGARMA models) are often considered, in order to reflect a serial correlation among observations. In this article, we focus on an adaptive least absolute shrinkage and selection operator (LASSO) (ALASSO) method for the variable selection of the REGARMA models and extend it to the linear regression models with the ARMA-generalized autoregressive conditional heteroskedasticity (ARMA-GARCH) errors (REGARMA-GARCH models). This attempt is an extension of the existing ALASSO method for the linear regression models with the AR errors (REGAR models) proposed by Wang et al. in 2007 Wang, H., Li, G., Tsai, C. (2007). Regression coefficient and autoregressive order shrinkage and selection via the lasso. Journal of the Royal Statistical Society: Series B 69:63–78. [Google Scholar]. New ALASSO algorithms are proposed to determine important predictors for the REGARMA and REGARMA-GARCH models. Finally, we provide the simulation results and real data analysis to illustrate our findings.     

On the Rounded Integer-Valued Autoregressive Process

In recent years, there has been a growing interest in modelling integred-valued time series. In this article, we propose a modified and generalized version of the first order rounded integer-valued autoregressive RINAR(1) model, originally introduced by Kachour and Yao (2009 Kachour , M. , Yao , J. F. ( 2009 ). First-order rounded integer-valued autoregressive (RINAR(1)) process . Journal of Time Series Analysis 30 ( 4 ): 417 – 448 .[Crossref], [Web of Science ®] , [Google Scholar]). Indeed, this class can be considered as an alternative of classical models based on the thinning operators. Using a Markov chain method, conditions for stationarity and the existence of moments are investigated. Least squares estimator of the model parameters is considered and its consistence is established. Finally, we describe the price change data using a model of the new class.     

On the distribution of the adaptive LASSO estimator

We study the distribution of the adaptive LASSO estimator [Zou, H., 2006. The adaptive LASSO and its oracle properties. J. Amer. Statist. Assoc. 101, 1418–1429] in finite samples as well as in the large-sample limit. The large-sample distributions are derived both for the case where the adaptive LASSO estimator is tuned to perform conservative model selection as well as for the case where the tuning results in consistent model selection. We show that the finite-sample as well as the large-sample distributions are typically highly nonnormal, regardless of the choice of the tuning parameter. The uniform convergence rate is also obtained, and is shown to be slower than n^-1/2$n^{-1/2}$ in case the estimator is tuned to perform consistent model selection. In particular, these results question the statistical relevance of the ‘oracle’ property of the adaptive LASSO estimator established in Zou [2006. The adaptive LASSO and its oracle properties. J. Amer. Statist. Assoc. 101, 1418–1429]. Moreover, we also provide an impossibility result regarding the estimation of the distribution function of the adaptive LASSO estimator. The theoretical results, which are obtained for a regression model with orthogonal design, are complemented by a Monte Carlo study using nonorthogonal regressors.