Characteristics of some classes of space–time covariance functions

Although a wide list of classes of space–time covariance functions is now available, selecting an appropriate class of models for a variable under study is still difficult and it represents a priority problem with respect to the choice of a particular model of a specified class. Then, knowing the characteristics of various classes of covariances, and their auxiliary functions, and matching those with the characteristics of the empirical space–time covariance surface might be helpful in the selection of a suitable class. In this paper some characteristics, such as behavior at the origin, asymptotic behavior, nonseparability and anisotropy aspects, are studied for some well known classes of covariance models of stationary space–time random fields. Moreover, some important issues related to modeling choices are described and a case study is presented.     

Positive and negative non-separability for space–time covariance models

Separable spatio-temporal covariance models, defined as the product of purely spatial and purely temporal covariance functions, are often used in practice, but frequently they only represent a convenient assumption. On the other hand, non-separable models are receiving a lot of attention, since they are more flexible to handle empirical covariances showed up in applications. Different forms of non-separability for space–time covariance functions have been recently defined in the literature. In this paper, the notion of positive and negative non-separability is further formalized in order to distinguish between pointwise and uniform non-separability. Various well-known non-separable space–time stationary covariance models are analyzed and classified by using the new definition of non-separability. In particular, wide classes of non-separable spatio-temporal covariance functions, able to capture positive and negative non-separability, are proposed and some examples of these classes are given. General results concerning the non-separability of spatial–temporal covariance functions obtained by a linear combination of spatial–temporal covariance functions and some stability properties are also presented. These results can be helpful to generate as well as to select appropriate covariance models for describing space–time data.     

Semi-parametric estimation of the autoregressive parameter in non-Gaussian Ornstein–Uhlenbeck processes

ABSTRACT

This paper considers the problem of estimating the autoregressive parameter in discretely observed Ornstein–Uhlenbeck processes. Two consistent estimators are proposed: one obtained by maximizing a kernel-based likelihood function, and another by minimizing a Kolmogorov-type distance from independence. After establishing the consistency of these estimators, their finite-sample performance and possible normality in large samples, is investigated by means of extensive simulations. An illustrative example to credit rating is discussed.     

Space–time correlation analysis: a comparative study

Space–time correlation modelling is one of the crucial steps of traditional structural analysis, since space–time models are used for prediction purposes. A comparative study among some classes of space–time covariance functions is proposed. The relevance of choosing a suitable model by taking into account the characteristic behaviour of the models is proved by using a space–time data set of ozone daily averages and the flexibility of the product-sum model is also highlighted through simulated data sets.     

Generation of d–optimal designs in a finite design space

This paper presents some considerations about the numerical procedures for generating D–optimal design in a finite design space. The influence of starting procedures and the finite set of points on the design efficiency is considered. Some modifications of the existing procedures for D–optimal designs generation are described. It is shown that for large number of factors the sequential procedures are more appropriate than the nonsequential ones     

Fitting semiparametric accelerated failure time models for nested case–control data

A nested case–control (NCC) study is an efficient cohort-sampling design in which a subset of controls are sampled from the risk set at each event time. Since covariate measurements are taken only for the sampled subjects, time and efforts of conducting a full scale cohort study can be saved. In this paper, we consider fitting a semiparametric accelerated failure time model to failure time data from a NCC study. We propose to employ an efficient induced smoothing procedure for rank-based estimating method for regression parameters estimation. For variance estimation, we propose to use an efficient resampling method that utilizes the robust sandwich form. We extend our proposed methods to a generalized NCC study that allows a sampling of cases. Finite sample properties of the proposed estimators are investigated via an extensive stimulation study. An application to a tumor study illustrates the utility of the proposed method in routine data analysis.     

Hidden Markov model for discrete circular–linear wind data time series

ABSTRACT

In this work, we deal with a bivariate time series of wind speed and direction. Our observed data have peculiar features, such as informative missing values, non-reliable measures under a specific condition and interval-censored data, that we take into account in the model specification. We analyse the time series with a non-parametric Bayesian hidden Markov model, introducing a new emission distribution, suitable to model our data, based on the invariant wrapped Poisson, the Poisson and the hurdle density. The model is estimated on simulated datasets and on the real data example that motivated this work.     

On testes for threshold–type nonlinearity in irregulaly spaced time series

Formulating the model first in continuous time, we have developed a state space approach to the problem of testing for threshold-type nonlinearity when the data are irregularly spaced.     

Spatial–temporal variation of casuarina spread in Cauvery delta and north eastern zone of Tamil Nadu,India: a spatial autoregressive model

The low forest cover and productivity are the major obstacles for mitigating the demand supply gap of raw material for forest-based industries, which could be fulfilled from a tree outside forest area. Casuarina is a multi-utile, short rotation tree which adapts to all ecosystems. The casuarina wood is predominantly demanded for fuel, construction and paper industries which is mostly preferred by farmers, traders and industries. This study explores the spatial and temporal variability of casuarina spread in mitigating the gap of demand and supply in Tamil Nadu using a spatial autoregressive model. The spread of casuarina was spatially and temporally significant, which was negatively influenced by the gross area irrigated as main and direct effects and positively in an indirect effect. An assured irrigation forces the farmers to choose traditional agricultural crops for their livelihood in their own district. The increase in the price of casuarina would increase the spread of casuarina in both own district and neighbouring districts. The spread of casuarina would augment the supply of raw material for forest-based industries.     

Some applications of nonlinear and non-Gaussian state–space modelling by means of hidden Markov models

Nonlinear and non-Gaussian state–space models (SSMs) are fitted to different types of time series. The applications include homogeneous and seasonal time series, in particular earthquake counts, polio counts, rainfall occurrence data, glacial varve data and daily returns on a share. The considered SSMs comprise Poisson, Bernoulli, gamma and Student-t distributions at the observation level. Parameter estimations for the SSMs are carried out using a likelihood approximation that is obtained after discretization of the state space. The approximation can be made arbitrarily accurate, and the approximated likelihood is precisely that of a finite-state hidden Markov model (HMM). The proposed method enables us to apply standard HMM techniques. It is easy to implement and can be extended to all kinds of SSMs in a straightforward manner.     

A note on an integer valued time series model with Poisson–negative binomial marginal distribution

Abstract

This paper proposes a new model for autoregressive time series of counts in terms of a convolution of Poisson and negative binomial random variables, known as Poisson–negative binomial (PNB) distribution. The corresponding first-order integer valued time series models are developed and their properties are discussed. The geometric PNB and the geometric semi PNB distributions are also introduced and studied.     

Do we need the constant term in the heterogenous autoregressive model for forecasting realized volatilities?

No-constant strategy is considered for the heterogenous autoregressive (HAR) model of Corsi, which is motivated by smaller biases of its estimated HAR coefficients than those of the constant HAR model. The no-constant model produces better forecasts than the constant model for four real datasets of the realized volatilities (RVs) of some major assets. Robustness of forecast improvement is verified for other functions of realized variance and log RV and for the extended datasets of all 20 RVs of Oxford-Man realized library. A Monte Carlo simulation also reveals improved forecasts for some historic HAR model estimated by Corsi.     

Statistical considerations in bioequivalence of two area under the concentration–time curves obtained from serial sampling data

In this paper, we study the bioequivalence (BE) inference problem motivated by pharmacokinetic data that were collected using the serial sampling technique. In serial sampling designs, subjects are independently assigned to one of the two drugs; each subject can be sampled only once, and data are collected at K distinct timepoints from multiple subjects. We consider design and hypothesis testing for the parameter of interest: the area under the concentration–time curve (AUC). Decision rules in demonstrating BE were established using an equivalence test for either the ratio or logarithmic difference of two AUCs. The proposed t-test can deal with cases where two AUCs have unequal variances. To control for the type I error rate, the involved degrees-of-freedom were adjusted using Satterthwaite's approximation. A power formula was derived to allow the determination of necessary sample sizes. Simulation results show that, when the two AUCs have unequal variances, the type I error rate is better controlled by the proposed method compared with a method that only handles equal variances. We also propose an unequal subject allocation method that improves the power relative to that of the equal and symmetric allocation. The methods are illustrated using practical examples.     

A two-stage estimation in the Clayton–Oakes model with marginal linear transformation models for multivariate failure time data

This paper considers the analysis of multivariate survival data where the marginal distributions are specified by semiparametric transformation models, a general class including the Cox model and the proportional odds model as special cases. First, consideration is given to the situation where the joint distribution of all failure times within the same cluster is specified by the Clayton–Oakes model (Clayton, Biometrika 65:141–151, l978; Oakes, J R Stat Soc B 44:412–422, 1982). A two-stage estimation procedure is adopted by first estimating the marginal parameters under the independence working assumption, and then the association parameter is estimated from the maximization of the full likelihood function with the estimators of the marginal parameters plugged in. The asymptotic properties of all estimators in the semiparametric model are derived. For the second situation, the third and higher order dependency structures are left unspecified, and interest focuses on the pairwise correlation between any two failure times. Thus, the pairwise association estimate can be obtained in the second stage by maximizing the pairwise likelihood function. Large sample properties for the pairwise association are also derived. Simulation studies show that the proposed approach is appropriate for practical use. To illustrate, a subset of the data from the Diabetic Retinopathy Study is used.     

Functional principal component analysis for the explorative analysis of multisite–multivariate air pollution time series with long gaps

The knowledge of the urban air quality represents the first step to face air pollution issues. For the last decades many cities can rely on a network of monitoring stations recording concentration values for the main pollutants. This paper focuses on functional principal component analysis (FPCA) to investigate multiple pollutant datasets measured over time at multiple sites within a given urban area. Our purpose is to extend what has been proposed in the literature to data that are multisite and multivariate at the same time. The approach results to be effective to highlight some relevant statistical features of the time series, giving the opportunity to identify significant pollutants and to know the evolution of their variability along time. The paper also deals with missing value issue. As it is known, very long gap sequences can often occur in air quality datasets, due to long time failures not easily solvable or to data coming from a mobile monitoring station. In the considered dataset, large and continuous gaps are imputed by empirical orthogonal function procedure, after denoising raw data by functional data analysis and before performing FPCA, in order to further improve the reconstruction.     

Spatial modelling for binary data using␣a␣hidden conditional autoregressive Gaussian process: a multivariate extension of the probit model

A Bayesian approach to modelling binary data on a regular lattice is introduced. The method uses a hierarchical model where the observed data is the sign of a hidden conditional autoregressive Gaussian process. This approach essentially extends the familiar probit model to dependent data. Markov chain Monte Carlo simulations are used on real and simulated data to estimate the posterior distribution of the spatial dependency parameters and the method is shown to work well. The method can be straightforwardly extended to regression models.     

Moving average models—time series in m-dimensions

Stochastic models for discrete time series in the time domain are well known but such models lack consideration of spatial dependency I We expand on their work by constructing spatially dependent moving average models. Definitions of order, stationarity, invertibility, autocorrelation function, and spectrum are made as natural extensions of those in zero dimensions and are implemented in the one and two-space dimensional models.     

SUB-SAMPLE MANN–WHITNEY–WILCOXON TEST

ABSTRACT

This paper presents a class of sub-sample rank-sum statistics to test the stochastic ordering between two distributions. The proposed class includes as special case the min–max test of Öztürk.[1] Öztürk, Ö. 2001. A Generalization of Ahmad's Class of Mann–Whitney–Wilcoxon Statistics. Austrl. and New Zealand J. Statist., 43: 67–74.  [Google Scholar] It is shown that the asymptotic distribution of the test statistic is normal and its Pitman asymptotic efficiency is as good as or higher than the competitors in the class of sub-sample Mann–Whitney–Wilcoxon statistics.     

Functional coefficient time series models with trending regressors

This paper studies a functional coe?cient time series model with trending regressors, where the coe?cients are unknown functions of time and random variables. We propose a local linear estimation method to estimate the unknown coe?cient functions, and establish the corresponding asymptotic theory under mild conditions. We also develop a test procedure to see if the functional coe?cients take particular parametric forms. For practical use, we further propose a Bayesian approach to select the bandwidths, and conduct several numerical experiments to examine the finite sample performance of our proposed local linear estimator and the test procedure. The results show that the local linear estimator works well and the proposed test has satisfactory size and power. In addition, our simulation studies show that the Bayesian bandwidth selection method performs better than the cross-validation method. Furthermore, we use the functional coe?cient model to study the relationship between consumption per capita and income per capita in United States, and it was shown that the functional coe?cient model with our proposed local linear estimator and Bayesian bandwidth selection method performs well in both in-sample fitting and out-of-sample forecasting.