Nonlinearity testing and modeling for threshold moving average models

In this paper, we suggest a simple test and an easily applicable modeling procedure for threshold moving average (TMA) models. Firstly, based on the fitted residuals by maximum likelihood estimate (MLE) for MA models, we construct a simple statistic, which is obtained by linear arrange regression and follows F-distribution approximately, to test for threshold nonlinearity and specify the threshold variables. And then, we use some scatterplots to identify the number and locations of the potential thresholds. Finally, with the statistic and Akaike information criterion, we propose the procedure to build TMA models. Both the power of test statistic and the convenience of modeling procedure can work very well demonstrated by simulation experiments and the application to a real example.     

Transformed symmetric generalized autoregressive moving average models

Cordeiro and Andrade [Transformed generalized linear models. J Stat Plan Inference. 2009;139:2970–2987] incorporated the idea of transforming the response variable to the generalized autoregressive moving average (GARMA) model, introduced by Benjamin et al. [Generalized autoregressive moving average models. J Am Stat Assoc. 2003;98:214–223], thus developing the transformed generalized autoregressive moving average (TGARMA) model. The goal of this article is to develop the TGARMA model for symmetric continuous conditional distributions with a possible nonlinear structure for the mean that enables the fitting of a wide range of models to several time series data types. We derive an iterative process for estimating the parameters of the new model by maximum likelihood and obtain a simple formula to estimate the parameter that defines the transformation of the response variable. Furthermore, we determine the moments of the original dependent variable which generalize previous published results. We illustrate the theory by means of real data sets and evaluate the results developed through simulation studies.     

Bayesian analysis of generalized odds-rate hazards models for survival data

In the analysis of censored survival data Cox proportional hazards model (1972) is extremely popular among the practitioners. However, in many real-life situations the proportionality of the hazard ratios does not seem to be an appropriate assumption. To overcome such a problem, we consider a class of nonproportional hazards models known as generalized odds-rate class of regression models. The class is general enough to include several commonly used models, such as proportional hazards model, proportional odds model, and accelerated life time model. The theoretical and computational properties of these models have been re-examined. The propriety of the posterior has been established under some mild conditions. A simulation study is conducted and a detailed analysis of the data from a prostate cancer study is presented to further illustrate the proposed methodology.     

Censored time series analysis with autoregressive moving average models

The authors consider time series observations with data irregularities such as censoring due to a detection limit. Practitioners commonly disregard censored data cases which often result in biased estimates. The authors present an attractive remedy for handling autocorrelated censored data based on a class of autoregressive and moving average (ARMA) models. In particular, they introduce an imputation method well suited for fitting ARMA models in the presence of censored data. They demonstrate the effectiveness of their technique in terms of bias, efficiency, and information loss. They also describe its adaptation to a specific context of meteorological time series data on cloud ceiling height, which are measured subject to the detection limit of the recording device.     

Bayesian identification of double seasonal autoregressive time series models

ABSTRACT

Seasonal autoregressive (SAR) models have been modified and extended to model high frequency time series characterized by exhibiting double seasonal patterns. Some researchers have introduced Bayesian inference for double seasonal autoregressive (DSAR) models; however, none has tackled the problem of Bayesian identification of DSAR models. Therefore, in order to fill this gap, we present a Bayesian methodology to identify the order of DSAR models. Assuming the model errors are normally distributed and using three priors, i.e. natural conjugate, g, and Jeffreys’ priors, on the model parameters, we derive the joint posterior mass function of the model order in a closed-form. Accordingly, the posterior mass function can be investigated and the best order of DSAR model is chosen as a value with the highest posterior probability for the time series being analyzed. We evaluate the proposed Bayesian methodology using simulation study, and we then apply it to real-world hourly internet amount of traffic dataset.     

An algorithm for the exact likelihood of periodic autoregressive moving average models

An algorithm to compute the autocovariance functions of periodic autoregressive moving average models is proposed. As a result, an easily implemented algorithm for the exact likelihood of these models is rendered possible.     

Sample moments of the autocorrelations of moving average processes and a modification to bartlett'sasymptotic variance formula

In this paper we express the sample autocorrelations for a moving average process of order q as a function of its own theoretical autocorrelations and the sample autocorrelations for the generating white noise series. Approximate analytic expressions are then obtained forthe moments of the sample autocorrelations of the moving average process.

Using these expressions, together with numerical evidence, we show that Bartlett's asymptotic formula for the variance of the sample autocorrelations of moving average processes, which is used widely in identifying these processes, is a large overestimate when considering finitesample sizes.

Our approach is for motivational purposes and so is purely formal, the amount of mathematics presented being kept to a minimum.     

Estimating multivariate autoregressive moving average models by fitting long autoregressions

Some simple methods for the estimation of mixed multivariate autoregressive moving average time series models are introduced. The methods require the fitting of a long autoregression to the data and the computation of consistent initial estimates for the parameters of the model. After these preliminaries the estimators of the paper are obtained by applying weighted least squares to a multivariate auxiliary regression model. Two types of weight matrices are considered. Both of them yield estimators which are strongly consistent and asymptotically normally distributed. The first estimators are also asymptotically efficient while the second ones are not fully efficient but computationally simple. A simulation study is performed to illustrate the behaviour of the estimators in finite samples.     

The specification of vector autoregressive moving average models

In this paper we propose a new identification method based on the residual white noise autoregressive criterion (Pukkila et al., 1990) to select the order of VARMA structures. Results from extensive simulation experiments based on different model structures with varying number of observations and number of component series are used to demonstrate the performance of this new procedure. We also use economic and business data to compare the model structures selected by this order selection method with those identified in other published studies.     

Considering Taguchi loss function on statistically constrained economic sum of squares exponentially weighted moving average charts

In this paper, Duncan's cost model combined Taguchi's quadratic loss function is applied to develop the economic-statistical design of the sum of squares exponentially weighted moving average (SS-EWMA) chart. The genetic algorithm is applied to search for the optimal decision variables of SS-EWMA chart such that the expected cost is minimized. Sensitivity analysis reveals that the optimal sample size and sampling interval decrease; optimal smoothing constant and control limit increase as the mean and/or variance increases. Moreover, the combination of optimal parameter levels in orthogonal array experiment plays an important guideline for monitoring the process mean and/or variance.     

Derivation of the theoretical autocovariance and autocorrelation function of autoregessive moving average processes

Closed form expressions for the theoretical autocovariance and autocorrelation function of mixed autoregressive moving average processes are presented. The results provide insight into the construction of autocovariances and autocorrelatians and are useful in theoretical analysis, model identification as well as in implementing maximum likelihood estimation algorithms.     

Following a moving target—Monte Carlo inference for dynamic Bayesian models

Markov chain Monte Carlo (MCMC) sampling is a numerically intensive simulation technique which has greatly improved the practicality of Bayesian inference and prediction. However, MCMC sampling is too slow to be of practical use in problems involving a large number of posterior (target) distributions, as in dynamic modelling and predictive model selection. Alternative simulation techniques for tracking moving target distributions, known as particle filters, which combine importance sampling, importance resampling and MCMC sampling, tend to suffer from a progressive degeneration as the target sequence evolves. We propose a new technique, based on these same simulation methodologies, which does not suffer from this progressive degeneration.     

A Universal Prior Distribution for Bayesian Consistency of Non parametric Procedures

The introduction of the Hausdorff α-entropy in Xing (2008a Xing, Y. (2008a). Convergence rates of posterior distributions for observations without the iid structure, 38 pages. Available at: www.arxiv.org:0811.4677v1. [Google Scholar]), Xing (2008b Xing, Y. (2008b). On adaptive Bayesian inference. Electron. J. Stat. 2:848–862.[Crossref] , [Google Scholar]), Xing (2010 Xing, Y. (2010). Rates of posterior convergence for iid Observations. Commun. Stat. Theory Methods. 39(19):3389–3398.[Taylor & Francis Online] , [Google Scholar]), Xing (2011 Xing, Y. (2011). Convergence rates of nonparametric posterior distributions. J. Stat. Plann. Inference 141:3382–3390.[Crossref], [Web of Science ®] , [Google Scholar]), and Xing and Ranneby (2009 Xing, Y., Ranneby, B. (2009). Sufficient conditions for Bayesian consistency. J. Stat. Plann. Inference. 139:2479–2489.[Crossref], [Web of Science ®] , [Google Scholar]) has lead a series of improvements of well-known results on posterior consistency. In this paper we discuss an application of the Hausdorff α-entropy. We construct a universal prior distribution such that the corresponding posterior distribution is almost surely consistent. The approach of the construction of this type of prior distribution is natural, but it works very well for all separable models. We illustrate such prior distributions by examples. In particular, we obtain that if the true density function is known to be some normal probability density function with unknown mean and unknown variance then without any additional assumption one can construct a prior distribution which leads to posterior consistency.     

Bayesian inference and model comparison for asymmetric smooth transition heteroskedastic models

Inference, quantile forecasting and model comparison for an asymmetric double smooth transition heteroskedastic model is investigated. A Bayesian framework in employed and an adaptive Markov chain Monte Carlo scheme is designed. A mixture prior is proposed that alleviates the usual identifiability problem as the speed of transition parameter tends to zero, and an informative prior for this parameter is suggested, that allows for reliable inference and a proper posterior, despite the non-integrability of the likelihood function. A formal Bayesian posterior model comparison procedure is employed to compare the proposed model with its two limiting cases: the double threshold GARCH and symmetric ARX GARCH models. The proposed methods are illustrated using both simulated and international stock market return series. Some illustrations of the advantages of an adaptive sampling scheme for these models are also provided. Finally, Bayesian forecasting methods are employed in a Value-at-Risk study of the international return series. The results generally favour the proposed smooth transition model and highlight explosive and smooth nonlinear behaviour in financial markets.     

Bayesian dynamic models for survival data with a cure fraction

In this paper, we propose a new class of semi-parametric cure rate models. Specifically, we construct dynamic models for piecewise hazard functions over a finite partition of the time axis. Allowing the size of partition and the levels of baseline hazard to be random, our proposed models provide a great flexibility in controlling the degree of parametricity in the right tail of the survival distribution and the amount of correlations among the log-baseline hazard levels. Several properties of the proposed models are derived, and propriety of the implied posteriors with improper noninformative priors for regression coefficients based on the proposed models is established for the fixed partition of the time axis. In addition, an efficient reversible jump computational algorithm is developed for carrying out posterior computation. A real data set from a melanoma clinical trial is analyzed in detail to further demonstrate the proposed methodology.     

Evaluation of quadratic forms and traces for iterative estimation in first-order moving average models

This paper deals with the evaluation of certain quadratic forms and traces associated with the irst-order moving average model. The problem arose while considering the maximum likelihood estimation under normality of the parameters of this model. The quadratic forms are y¹R-^jy, where y is a vector of observations generated by the models and R is the correlation matrix of the model; the traces are trR-^j j cam be any natural number, but emphasis is placed on small Yalues, j = 1,2,3. Procedures in the time and frequency domains are studied, and the amount of computations needed in each case are considered and compared, from which a preferred approach emerges. The computations are compared with several alternative procedures suggested in the literature.     

Sensitivity analysis and choosing between alternative polytomous IRT models using Bayesian model comparison criteria

Polytomous Item Response Theory (IRT) models are used by specialists to score assessments and questionnaires that have items with multiple response categories. In this article, we study the performance of five model comparison criteria for comparing fit of the graded response and generalized partial credit models using the same dataset when the choice between the two is unclear. Simulation study is conducted to analyze the sensitivity of priors and compare the performance of the criteria using the No-U-Turn Sampler algorithm, under a Bayesian approach. The results were used to select a model for an application in mental health data.     

Bayesian computation for the superposition of nonhomogeneous poisson processes

Bayesian inference for the superposition of nonhomogeneous Poisson processes is studied. A Markov-chain Monte Carlo method with data augmentation is developed to compute the features of the posterior distribution. For each observed failure epoch, a latent variable is introduced that indicates which component of the superposition model gives rise to the failure. This data-augmentation approach facilitates specification of the transitional kernel in the Markov chain. Moreover, new Bayesian tests are developed for the full superposition model against simpler submodels. Model determination by a predictive likelihood approach is studied. A numerical example based on a real data set is given.