Evaluating the relative merits of competing models based on empirical likelihood ratio test

Competing models arise naturally in many research fields, such as survival analysis and economics, when the same phenomenon of interest is explained by different researcher using different theories or according to different experiences. The model selection problem is therefore remarkably important because of its great importance to the subsequent inference; Inference under a misspecified or inappropriate model will be risky. Existing model selection tests such as Vuong's tests [26 Q.H. Vuong, Likelihood ratio test for model selection and non-nested hypothesis, Econometrica 57 (1989), pp. 307–333. doi: 10.2307/1912557[Crossref], [Web of Science ®] , [Google Scholar]] and Shi's non-degenerate tests [21 X. Shi, A non-degenerate Vuong test, Quant. Econ. 6 (2015), pp. 85–121. doi: 10.3982/QE382[Crossref], [Web of Science ®] , [Google Scholar]] suffer from the variance estimation and the departure of the normality of the likelihood ratios. To circumvent these dilemmas, we propose in this paper an empirical likelihood ratio (ELR) tests for model selection. Following Shi [21 X. Shi, A non-degenerate Vuong test, Quant. Econ. 6 (2015), pp. 85–121. doi: 10.3982/QE382[Crossref], [Web of Science ®] , [Google Scholar]], a bias correction method is proposed for the ELR tests to enhance its performance. A simulation study and a real-data analysis are provided to illustrate the performance of the proposed ELR tests.     

Longitudinal profiles of bounded outcome scores as predictors for disease activity in rheumatoid arthritis patients: a joint modeling approach

A variety of statistical approaches have been suggested in the literature for the analysis of bounded outcome scores (BOS). In this paper, we suggest a statistical approach when BOSs are repeatedly measured over time and used as predictors in a regression model. Instead of directly using the BOS as a predictor, we propose to extend the approaches suggested in [16 E. Lesaffre, D. Rizopoulos, and R. Tsonaka, The logistics-transform for bounded outcome scores, Biostatistics 8 (2007), pp. 72–85. doi: 10.1093/biostatistics/kxj034[Crossref], [PubMed], [Web of Science ®] , [Google Scholar],21 M. Molas and E. Lesaffre, A comparison of the three random effects approaches to analyse repeated bounded outcome scores with an application in a stroke revalidation study, Stat. Med. 27 (2008), pp. 6612–6633. doi: 10.1002/sim.3432[Crossref], [PubMed], [Web of Science ®] , [Google Scholar],28 R. Tsonaka, D. Rizopoulos, and E. Lesaffre, Power and sample size calculations for discrete bounded outcome scores, Stat. Med. 25 (2006), pp. 4241–4252. doi: 10.1002/sim.2679[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]] to a joint modeling setting. Our approach is illustrated on longitudinal profiles of multiple patients’ reported outcomes to predict the current clinical status of rheumatoid arthritis patients by a disease activities score of 28 joints (DAS28). Both a maximum likelihood as well as a Bayesian approach is developed.     

Optimal network design for Bayesian spatial prediction of multivariate non-Gaussian environmental data

This paper deals with the problem of increasing air pollution monitoring stations in Tehran city for efficient spatial prediction. As the data are multivariate and skewed, we introduce two multivariate skew models through developing the univariate skew Gaussian random field proposed by Zareifard and Jafari Khaledi [21 H. Zareifard and M. Jafari Khaledi, Non-Gaussian modeling of spatial data using scale mixing of a unified skew Gaussian process, J. Multivariate Anal. 114 (2013), pp. 16–28. doi: 10.1016/j.jmva.2012.07.003[Crossref], [Web of Science ®] , [Google Scholar]]. These models provide extensions of the linear model of coregionalization for non-Gaussian data. In the Bayesian framework, the optimal network design is found based on the maximum entropy criterion. A Markov chain Monte Carlo algorithm is developed to implement posterior inference. Finally, the applicability of two proposed models is demonstrated by analyzing an air pollution data set.     

Random Effects,Fixed Effects and Hausman's Test for the Generalized Mixed Regressive Spatial Autoregressive Panel Data Model

This article suggests random and fixed effects spatial two-stage least squares estimators for the generalized mixed regressive spatial autoregressive panel data model. This extends the generalized spatial panel model of Baltagi et al. (2013 Baltagi, B. H., Egger, P., Pfaffermayr, M. (2013). A generalized spatial panel data model with random effects. Econometric Reviews 32:650–685.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) by the inclusion of a spatial lag term. The estimation method utilizes the Generalized Moments method suggested by Kapoor et al. (2007 Kapoor, M., Kelejian, H. H., Prucha, I. R. (2007). Panel data models with spatially correlated error components. Journal of Econometrics 127(1):97–130.[Crossref], [Web of Science ®] , [Google Scholar]) for a spatial autoregressive panel data model. We derive the asymptotic distributions of these estimators and suggest a Hausman test a la Mutl and Pfaffermayr (2011 Mutl, J., Pfaffermayr, M. (2011). The Hausman test in a Cliff and Ord panel model. Econometrics Journal 14:48–76.[Crossref], [Web of Science ®] , [Google Scholar]) based on the difference between these estimators. Monte Carlo experiments are performed to investigate the performance of these estimators as well as the corresponding Hausman test.     

Analysis of discrete lifetime data under middle-censoring and in the presence of covariates

‘Middle censoring’ is a very general censoring scheme where the actual value of an observation in the data becomes unobservable if it falls inside a random interval (L, R) and includes both left and right censoring. In this paper, we consider discrete lifetime data that follow a geometric distribution that is subject to middle censoring. Two major innovations in this paper, compared to the earlier work of Davarzani and Parsian [3 N. Davarzani and A. Parsian, Statistical inference for discrete middle-censored data, J. Statist. Plan. Inference 141 (2011), pp. 1455–1462. doi: 10.1016/j.jspi.2010.10.012[Crossref], [Web of Science ®] , [Google Scholar]], include (i) an extension and generalization to the case where covariates are present along with the data and (ii) an alternate approach and proofs which exploit the simple relationship between the geometric and the exponential distributions, so that the theory is more in line with the work of Iyer et al. [6 S.K. Iyer, S.R. Jammalamadaka, and D. Kundu, Analysis of middle censored data with exponential lifetime distributions, J. Statist. Plan. Inference 138 (2008), pp. 3550–3560. doi: 10.1016/j.jspi.2007.03.062[Crossref], [Web of Science ®] , [Google Scholar]]. It is also demonstrated that this kind of discretization of life times gives results that are close to the original data involving exponential life times. Maximum likelihood estimation of the parameters is studied for this middle-censoring scheme with covariates and their large sample distributions discussed. Simulation results indicate how well the proposed estimation methods work and an illustrative example using time-to-pregnancy data from Baird and Wilcox [1 D.D. Baird and A.J. Wilcox, Cigarette smoking associated with delayed conception, J, Am. Med. Assoc. 253 (1985), pp. 2979–2983. doi: 10.1001/jama.1985.03350440057031[Crossref], [Web of Science ®] , [Google Scholar]] is included.     

Weighting variables in Kohonen competitive learning algorithms

This paper presents a new variable weight method, called the singular value decomposition (SVD) approach, for Kohonen competitive learning (KCL) algorithms based on the concept of Varshavsky et al. [18 R. Varshavsky, A. Gottlieb, M. Linial, and D. Horn, Novel unsupervised feature filtering of bilogical data, Bioinformatics 22 (2006), pp. 507–513.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]]. Integrating the weighted fuzzy c-means (FCM) algorithm with KCL, in this paper, we propose a weighted fuzzy KCL (WFKCL) algorithm. The goal of the proposed WFKCL algorithm is to reduce the clustering error rate when data contain some noise variables. Compared with the k-means, FCM and KCL with existing variable-weight methods, the proposed WFKCL algorithm with the proposed SVD's weight method provides a better clustering performance based on the error rate criterion. Furthermore, the complexity of the proposed SVD's approach is less than Pal et al. [17 S.K. Pal, R.K. De, and J. Basak, Unsupervised feature evaluation: a neuro-fuzzy approach, IEEE. Trans. Neural Netw. 11 (2000), pp. 366–376.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]], Wang et al. [19 X.Z. Wang, Y.D. Wang, and L.J. Wang, Improving fuzzy c-means clustering based on feature-weight learning, Pattern Recognit. Lett. 25 (2004), pp. 1123–1132.[Crossref], [Web of Science ®] , [Google Scholar]] and Hung et al. [9 W. -L. Hung, M. -S. Yang, and D. -H. Chen, Bootstrapping approach to feature-weight selection in fuzzy c-means algorithms with an application in color image segmentation, Pattern Recognit. Lett. 29 (2008), pp. 1317–1325.[Crossref], [Web of Science ®] , [Google Scholar]].     

Local influence diagnostics for generalized linear mixed models with overdispersion

Since the seminal paper by Cook and Weisberg [9 R.D. Cook and S. Weisberg, Residuals and Influence in Regression, Chapman &; Hall, London, 1982. [Google Scholar]], local influence, next to case deletion, has gained popularity as a tool to detect influential subjects and measurements for a variety of statistical models. For the linear mixed model the approach leads to easily interpretable and computationally convenient expressions, not only highlighting influential subjects, but also which aspect of their profile leads to undue influence on the model's fit [17 E. Lesaffre and G. Verbeke, Local influence in linear mixed models, Biometrics 54 (1998), pp. 570–582. doi: 10.2307/3109764[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]]. Ouwens et al. [24 M.J.N.M. Ouwens, F.E.S. Tan, and M.P.F. Berger, Local influence to detect influential data structures for generalized linear mixed models, Biometrics 57 (2001), pp. 1166–1172. doi: 10.1111/j.0006-341X.2001.01166.x[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]] applied the method to the Poisson-normal generalized linear mixed model (GLMM). Given the model's nonlinear structure, these authors did not derive interpretable components but rather focused on a graphical depiction of influence. In this paper, we consider GLMMs for binary, count, and time-to-event data, with the additional feature of accommodating overdispersion whenever necessary. For each situation, three approaches are considered, based on: (1) purely numerical derivations; (2) using a closed-form expression of the marginal likelihood function; and (3) using an integral representation of this likelihood. Unlike when case deletion is used, this leads to interpretable components, allowing not only to identify influential subjects, but also to study the cause thereof. The methodology is illustrated in case studies that range over the three data types mentioned.     

A new cure rate model based on the Yule–Simon distribution with application to a melanoma data set

In this paper, a new survival cure rate model is introduced considering the Yule–Simon distribution [12 H.A. Simon, On a class of skew distribution functions, Biometrika 42 (1955), pp. 425–440.[Crossref], [Web of Science ®] , [Google Scholar]] to model the number of concurrent causes. We study some properties of this distribution and the model arising when the distribution of the competing causes is the Weibull model. We call this distribution the Weibull–Yule–Simon distribution. Maximum likelihood estimation is conducted for model parameters. A small scale simulation study is conducted indicating satisfactory parameter recovery by the estimation approach. Results are applied to a real data set (melanoma) illustrating the fact that the model proposed can outperform traditional alternative models in terms of model fitting.     

A comparison of alternative methods to construct confidence intervals for the estimate of a break date in linear regression models

This article considers constructing confidence intervals for the date of a structural break in linear regression models. Using extensive simulations, we compare the performance of various procedures in terms of exact coverage rates and lengths of the confidence intervals. These include the procedures of Bai (1997 Bai, J. (1997). Estimation of a change point in multiple regressions. Review of Economics and Statistics 79:551–563.[Crossref], [Web of Science ®] , [Google Scholar]) based on the asymptotic distribution under a shrinking shift framework, Elliott and Müller (2007 Elliott, G., Müller, U. (2007). Confidence sets for the date of a single break in linear time series regressions. Journal of Econometrics 141:1196–1218.[Crossref], [Web of Science ®] , [Google Scholar]) based on inverting a test locally invariant to the magnitude of break, Eo and Morley (2015 Eo, Y., Morley, J. (2015). Likelihood-ratio-based confidence sets for the timing of structural breaks. Quantitative Economics 6:463–497.[Crossref], [Web of Science ®] , [Google Scholar]) based on inverting a likelihood ratio test, and various bootstrap procedures. On the basis of achieving an exact coverage rate that is closest to the nominal level, Elliott and Müller's (2007 Elliott, G., Müller, U. (2007). Confidence sets for the date of a single break in linear time series regressions. Journal of Econometrics 141:1196–1218.[Crossref], [Web of Science ®] , [Google Scholar]) approach is by far the best one. However, this comes with a very high cost in terms of the length of the confidence intervals. When the errors are serially correlated and dealing with a change in intercept or a change in the coefficient of a stationary regressor with a high signal-to-noise ratio, the length of the confidence interval increases and approaches the whole sample as the magnitude of the change increases. The same problem occurs in models with a lagged dependent variable, a common case in practice. This drawback is not present for the other methods, which have similar properties. Theoretical results are provided to explain the drawbacks of Elliott and Müller's (2007 Elliott, G., Müller, U. (2007). Confidence sets for the date of a single break in linear time series regressions. Journal of Econometrics 141:1196–1218.[Crossref], [Web of Science ®] , [Google Scholar]) method.     

Testing for sphericity in a two-way error components panel data model

This article is concerned with sphericity test for the two-way error components panel data model. It is found that the John statistic and the bias-corrected LM statistic recently developed by Baltagi et al. (2011 Baltagi, B. H., Feng, Q., Kao, C. (2011). Testing for sphericity in a fixed effects panel data model. Econometrics Journal 14:25–47.[Crossref], [Web of Science ®] , [Google Scholar])Baltagi et al. (2012 Baltagi, B. H., Feng, Q., Kao, C. (2012). A Lagrange multiplier test for cross-sectional dependence in a fixed effects panel data model. Journal of Econometrics 170:164–177.[Crossref], [Web of Science ®] , [Google Scholar], which are based on the within residuals, are not helpful under the present circumstances even though they are in the one-way fixed effects model. However, we prove that when the within residuals are properly transformed, the resulting residuals can serve to construct useful statistics that are similar to those of Baltagi et al. (2011 Baltagi, B. H., Feng, Q., Kao, C. (2011). Testing for sphericity in a fixed effects panel data model. Econometrics Journal 14:25–47.[Crossref], [Web of Science ®] , [Google Scholar])Baltagi et al. (2012 Baltagi, B. H., Feng, Q., Kao, C. (2012). A Lagrange multiplier test for cross-sectional dependence in a fixed effects panel data model. Journal of Econometrics 170:164–177.[Crossref], [Web of Science ®] , [Google Scholar]). Simulation results show that the newly proposed statistics perform well under the null hypothesis and several typical alternatives.     

Bivariate Continuous Distributions with Specified Conditional Hazard Functions

Analysis of covariance (ANCOVA) is the standard procedure for comparing several treatments when the response variable depends on one or more covariates. We consider the problem of testing the equality of treatment effects when the variances are not assumed to be equal. It is well known that classical F test is not robust with respect to the assumption of equal variances and may lead to misleading conclusions if the variances are not equal. Ananda (1998 Ananda , M. M. A. ( 1998 ). Bayesian and non-Bayesian solutions to analysis of covariance models under heteroscedasticity . J. Econometrics 86 : 177 – 192 .[Crossref], [Web of Science ®] , [Google Scholar]) developed a generalized F test for testing the equality of treatment effects. However, simulation studies show that the actual size of this test can be much higher than the nominal level when the sample sizes are small, particularly when the number of treatments is large. In this article, we develop a test using the parametric bootstrap approach of Krishnamoorthy et al. (2007 Krishnamoorthy , K. , Lu , F. , Mathew , T. ( 2007 ). A parametric bootstrap approach for ANOVA with unequal variances: Fixed and random models . Computat. Statist. Data Anal. 51 : 5731 – 5742 .[Crossref], [Web of Science ®] , [Google Scholar]). Our simulations show that the actual size of our proposed test is close to the nominal level, irrespective of the number of treatments and sample sizes. Our simulations also indicate that our proposed PB test is more robust, with respect to the assumption of normality, than the generalized F test. Therefore, our proposed PB test provides a satisfactory alternative to the generalized F test.     

Flexible parametric modelling of the hazard function in breast cancer studies

In cancer research, study of the hazard function provides useful insights into disease dynamics, as it describes the way in which the (conditional) probability of death changes with time. The widely utilized Cox proportional hazard model uses a stepwise nonparametric estimator for the baseline hazard function, and therefore has a limited utility. The use of parametric models and/or other approaches that enables direct estimation of the hazard function is often invoked. A recent work by Cox et al. [6 Cox, C., Chu, H., Schneider, M. F. and Munoz, A. 2007. Parametric survival analysis and taxonomy of hazard functions for the generalized gamma distribution. Stat. Med., 26: 4352–4374. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]] has stimulated the use of the flexible parametric model based on the Generalized Gamma (GG) distribution, supported by the development of optimization software. The GG distribution allows estimation of different hazard shapes in a single framework. We use the GG model to investigate the shape of the hazard function in early breast cancer patients. The flexible approach based on a piecewise exponential model and the nonparametric additive hazards model are also considered.     

First difference transformation in panel VAR models: Robustness,estimation, and inference

This article considers estimation of Panel Vector Autoregressive Models of order 1 (PVAR(1)) with focus on fixed T consistent estimation methods in First Differences (FD) with additional strictly exogenous regressors. Additional results for the Panel FD ordinary least squares (OLS) estimator and the FDLS type estimator of Han and Phillips (2010 Han, C., Phillips, P. C. B. (2010). Gmm estimation for dynamic panels with fixed effects and strong instruments at unity. Econometric Theory 26:119–151.[Crossref], [Web of Science ®] , [Google Scholar]) are provided. Furthermore, we simplify the analysis of Binder et al. (2005 Binder, M., Hsiao, C., Pesaran, M. H. (2005). Estimation and inference in short panel vector autoregressions with unit root and cointegration. Econometric Theory 21:795–837.[Crossref], [Web of Science ®] , [Google Scholar]) by providing additional analytical results and extend the original model by taking into account possible cross-sectional heteroscedasticity and presence of strictly exogenous regressors. We show that in the three wave panel the log-likelihood function of the unrestricted Transformed Maximum Likelihood (TML) estimator might violate the global identification assumption. The finite-sample performance of the analyzed methods is investigated in a Monte Carlo study.     

Heteroskedasticity Robust Panel Unit Root Testing Under Variance Breaks in Pooled Regressions

Noting that many economic variables display occasional shifts in their second order moments, we investigate the performance of homogenous panel unit root tests in the presence of permanent volatility shifts. It is shown that in this case the test statistic proposed by Herwartz and Siedenburg (2008 Herwartz, H., Siedenburg, F. (2008). Homogenous panel unit root tests under cross-sectional dependence: Finite sample modifications and the wild bootstrap. Computational Statistics and Data Analysis 53(1):137–150.[Crossref], [Web of Science ®] , [Google Scholar]) is asymptotically standard Gaussian. By means of a simulation study we illustrate the performance of first and second generation panel unit root tests and undertake a more detailed comparison of the test in Herwartz and Siedenburg (2008 Herwartz, H., Siedenburg, F. (2008). Homogenous panel unit root tests under cross-sectional dependence: Finite sample modifications and the wild bootstrap. Computational Statistics and Data Analysis 53(1):137–150.[Crossref], [Web of Science ®] , [Google Scholar]) and its heteroskedasticity consistent Cauchy counterpart introduced in Demetrescu and Hanck (2012a Demetrescu, M., Hanck, C. (2012a). A simple nonstationary-volatility robust panel unit root test. Economics Letters 117(2):10–13.[Crossref], [Web of Science ®] , [Google Scholar]). As an empirical illustration, we reassess evidence on the Fisher hypothesis with data from nine countries over the period 1961Q2–2011Q2. Empirical evidence supports panel stationarity of the real interest rate for the entire subperiod. With regard to the most recent two decades, the test results cast doubts on market integration, since the real interest rate is diagnosed nonstationary.     

Inference on locally ordered breaks in multiple regressions

We consider issues related to inference about locally ordered breaks in a system of equations, as originally proposed by Qu and Perron (2007 Qu, Z., Perron, P. (2007). Estimating and testing structural changes in multivariate regressions. Econometrica 75:459–502.[Crossref], [Web of Science ®] , [Google Scholar]). These apply when break dates in different equations within the system are not separated by a positive fraction of the sample size. This allows constructing joint confidence intervals of all such locally ordered break dates. We extend the results of Qu and Perron (2007 Qu, Z., Perron, P. (2007). Estimating and testing structural changes in multivariate regressions. Econometrica 75:459–502.[Crossref], [Web of Science ®] , [Google Scholar]) in several directions. First, we allow the covariates to be any mix of trends and stationary or integrated regressors. Second, we allow for breaks in the variance-covariance matrix of the errors. Third, we allow for multiple locally ordered breaks, each occurring in a different equation within a subset of equations in the system. Via some simulation experiments, we show first that the limit distributions derived provide good approximations to the finite sample distributions. Second, we show that forming confidence intervals in such a joint fashion allows more precision (tighter intervals) compared to the standard approach of forming confidence intervals using the method of Bai and Perron (1998 Bai, J., Perron, P. (1998). Estimating and testing linear models with multiple structural changes. Econometrica 66:47–78.[Crossref], [Web of Science ®] , [Google Scholar]) applied to a single equation. Simulations also indicate that using the locally ordered break confidence intervals yields better coverage rates than using the framework for globally distinct breaks when the break dates are separated by roughly 10% of the total sample size.     

Weighted simulated integrated conditional moment tests for parametric conditional distributions of stationary time series processes

In this article, we propose a weighted simulated integrated conditional moment (WSICM) test of the validity of parametric specifications of conditional distribution models for stationary time series data, by combining the weighted integrated conditional moment (ICM) test of Bierens (1984 Bierens, H. J. (1984). Model specification testing of time series regressions. Journal of Econometrics 26:323–353.[Crossref], [Web of Science ®] , [Google Scholar]) for time series regression models with the simulated ICM test of Bierens and Wang (2012 Bierens, H. J., Wang, L. (2012). Integrated conditional moment tests for parametric conditional distributions. Econometric Theory 28:328–362.[Crossref], [Web of Science ®] , [Google Scholar]) of conditional distribution models for cross-section data. To the best of our knowledge, no other consistent test for parametric conditional time series distributions has been proposed yet in the literature, despite consistency claims made by some authors.     

Bimodal symmetric-asymmetric power-normal families

This article proposes new symmetric and asymmetric distributions applying methods analogous as the ones in Kim (2005 Kim, H.J. (2005). On a class of two-piece skew-normal distributions. Statist.: J. Theoret. Appl. Statist. 39:537–553.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and Arnold et al. (2009 Arnold, B.C., H.W. Gómez, and H.S. Salinas. (2009). On multiple constraint skewed models. Statist. J. Theoret. Appl. Statist. 43: 279–293.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) to the exponentiated normal distribution studied in Durrans (1992 Durrans, S.R. (1992). Distributions of fractional order statistics in hydrology. Water Resour. Res. 28:1649–1655.[Crossref], [Web of Science ®] , [Google Scholar]), that we call the power-normal (PN) distribution. The proposed bimodal extension, the main focus of the paper, is called the bimodal power-normal model and is denoted by BPN(α) model, where α is the asymmetry parameter. The authors give some properties including moments and maximum likelihood estimation. Two important features of the model proposed is that its normalizing constant has closed and simple form and that the Fisher information matrix is nonsingular, guaranteeing large sample properties of the maximum likelihood estimators. Finally, simulation studies and real applications reveal that the proposed model can perform well in both situations.     

A robust statistical approach to select adequate error distributions for financial returns

In this article, we propose a robust statistical approach to select an appropriate error distribution, in a classical multiplicative heteroscedastic model. In a first step, unlike to the traditional approach, we do not use any GARCH-type estimation of the conditional variance. Instead, we propose to use a recently developed nonparametric procedure [31 D. Mercurio and V. Spokoiny, Statistical inference for time-inhomogeneous volatility models, Ann. Stat. 32 (2004), pp. 577–602.[Crossref], [Web of Science ®] , [Google Scholar]]: the local adaptive volatility estimation. The motivation for using this method is to avoid a possible model misspecification for the conditional variance. In a second step, we suggest a set of estimation and model selection procedures (Berk–Jones tests, kernel density-based selection, censored likelihood score, and coverage probability) based on the so-obtained residuals. These methods enable to assess the global fit of a set of distributions as well as to focus on their behaviour in the tails, giving us the capacity to map the strengths and weaknesses of the candidate distributions. A bootstrap procedure is provided to compute the rejection regions in this semiparametric context. Finally, we illustrate our methodology throughout a small simulation study and an application on three time series of daily returns (UBS stock returns, BOVESPA returns and EUR/USD exchange rates).     

Records Properties of Non Stationary Time Series

This article proposes an asymptotic expansion for the Studentized linear discriminant function using two-step monotone missing samples under multivariate normality. The asymptotic expansions related to discriminant function have been obtained for complete data under multivariate normality. The result derived by Anderson (1973 Anderson , T. W. ( 1973 ). An asymptotic expansion of the distribution of the Studentized classification statistic W . The Annals of Statistics 1 : 964 – 972 .[Crossref], [Web of Science ®] , [Google Scholar]) plays an important role in deciding the cut-off point that controls the probabilities of misclassification. This article provides an extension of the result derived by Anderson (1973 Anderson , T. W. ( 1973 ). An asymptotic expansion of the distribution of the Studentized classification statistic W . The Annals of Statistics 1 : 964 – 972 .[Crossref], [Web of Science ®] , [Google Scholar]) in the case of two-step monotone missing samples under multivariate normality. Finally, numerical evaluations by Monte Carlo simulations were also presented.     

Forward Moving Average Representations for MA Processes of Finite Order: Multivariate Stationary and Periodically Correlated

Soltani and Mohammadpour (2006 Soltani , A. R. , Mohammadpour , M. (2006). Moving average representations for multivariate stationary processes. J. Time Ser. Anal. 27(6):831–841.[Crossref], [Web of Science ®] , [Google Scholar]) observed that in general the backward and forward moving average coefficients, correspondingly, for the multivariate stationary processes, unlike the univariate processes, are different. This has stimulated researches concerning derivations of forward moving average coefficients in terms of the backward moving average coefficients. In this article we develop a practical procedure whenever the underlying process is a multivariate moving average (or univariate periodically correlated) process of finite order. Our procedure is based on two key observations: order reduction (Li, 2005 Li , L. M. ( 2005 ). Factorization of moving average spectral densities by state space representations and stacking . J. Multivariate Anal. 96 : 425 – 438 .[Crossref], [Web of Science ®] , [Google Scholar]) and first-order analysis (Mohammadpour and Soltani, 2010 Mohammadpour , M. , Soltani , A. R. ( 2010 ). Forward moving average representation for multivariate MA(1) processes . Commun. Statist. Theory Meth. 39 : 729 – 737 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]).