Thesis (M.A.)--University of Hawaii, 1962. / Typescript. Bibliography: leaves 186-188.

In this thesis we study the problem of estimation of parametric covariance and variogram functions for spatial and spatio- temporal random processes. It has the following principal parts. Variogram Estimation: We consider the "weighted" least squares criterion of fitting a parametric variogram function to second order stationary geo-statistical processes. Two new weight functions are investigated as alternative to the commonly used weight function proposed by Cressie (1985). We discuss asymptotic convergence properties of the sample variogram estimator and estimators of unknown parameters of parametric variogram functions, under a "mixed increasing domain" sampling design as proposed by Lahiriet al. While empirical results of Mean Square Errors, for parameter estimation, obtained using both the proposed functions are found to be comparatively better, we also theoretically establish that under general conditions one of the proposed weight functions give estimates with better asymptotic effciency. Spatio-Temporal Covariance Estimation: Over the past decade, there have been some important advances in methods for constructing valid spatiotemporal covariance functions; but not much attention has been given - so far - on methods of parameter estimation. In this thesis we propose a new frequency domain approach to estimating parameters of spatio-temporal covariance functions. We derive asymptotic strong consistency properties of the estimators using the concept of stochastic equicontinuity. The theory is illustrated with a simulation. Non-Linearity of Geostatistical Data: Linear prediction theory for spatial data is well established and substantial literature is available on the subject. Relatively less is known about non-linearity. In our final and ongoing, research problem we propose a non-linear predictor for geostatistical data. We demonstrate that the predictor is a function of higher order moments. This leads us to construct spatial bispectra for parametric third order moments.

Thesis (Ed. D.)--Virginia Polytechnic Institute and State University, 1991. / Vita. Abstract. Includes bibliographical references (leaves 260-262). Also available via the Internet.

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The dissertation considers an indirect approach for the estimation of the cointegrating parameters, in the sense that the estimators are jointly constructed along with estimating other nuisance parameters. This approach was proposed by Robinson (2008) where a bivariate local Whittle estimator was developed to jointly estimate a cointegrating parameter along with the memory parameters and the phase parameters (discussed in chapter 2). The main contributions of this dissertation is to establish, similar to Robinson (2008), a joint estimation of the memory, cointegrating and phase parameters in stationary and nonstationary fractionally cointegrated models in a multivariate framework. In order to accomplish such task, a general shape of the spectral density matrix, first noted in Davidson and Hashimzade (2008), is utilised to cover multivariate jointly dependent stationary long memory time series allowing more than one cointegrating relation (discussed in chapter 3). Consequently, the notion of the extended discrete Fourier transform is adopted based on the work of Phillips (1999) to allow for the multivariate estimation to cover the non stationary region (explained in chapter 4). Overall, the estimation methods adopted in this dissertation follows the semiparametric approach, in that the spectral density is only specified in a neighbourhood of zero frequency. The dissertation is organised in four self-contained chapters that are connected to each other, in additional to this introductory chapter: • Chapter 1 discusses the univariate long memory time series analysis covering different definitions, models and estimation methods. Consequently, parametric and semiparametric estimation methods were applied to a univariate series of the daily Egyptian stock returns to examine the presence of long memory properties. The results show strong and significant evidence of long memory in the Egyptian stock market which refutes the hypothesis of market efficiency. • Chapter 2 expands the analysis in the first chapter using a bivariate framework first introduced by Robinson (2008) for long memory time series in stationary system. The bivariate model presents four unknown parameters, including two memory parameters, a phase parameter and a cointegration parameter, which are jointly estimated. The estimation analysis is applied to a bivariate framework includes the US and Canada inflation rates where a linear combination between the US and Canada inflation rates that has a long memory less than the two individual series has been detected. • Chapter 3 introduces a semiparametric local Whittle (LW) estimator for a general multivariate stationary fractional cointegration using a general shape of the spectral density matrix first introduced by Davidson and Hashimzade (2008). The proposed estimator is used to jointly estimate the memory parameters along with the cointegrating and phase parameters. The consistency and asymptotic normality of the proposed estimator is proved. In addition, a Monte Carlo study is conducted to examine the performance of the new proposed estimator for different sample sizes. The multivariate local whittle estimation analysis is applied to three different relevant examples to examine the presence of fractional cointegration relationships. • In the first three chapters, the estimation procedures focused on the stationary case where the memory parameter is between zero and half. On the other hand, the analysis in chapter 4, which is a natural progress to that in chapter 3, adjusts the estimation procedures in order to cover the non-stationary values of the memory parameters. Chapter 4 expands the analysis in chapter 3 using the extended discrete Fourier transform and periodogram to extend the local Whittle estimation to non stationary multivariate systems. As a result, the new extended local Whittle (XLW) estimator can be applied throughout the stationary and non stationary zones. The XLW estimator is identical to the LW estimator in the stationary region, introduced in chapter 3. Application to a trivariate series of US money aggregates is employed.

ii Abstract This thesis focuses on one of the attractive topics of current financial literature, the application of wavelet-based methods in volatility modeling. It introduces a new, wavelet-based estimator (wavelet Whittle estimator) of a FIEGARCH model, ARCH- family model capturing long-memory and asymmetry in volatility, and studies its properties. Based on an extensive Monte Carlo experiment, both the behavior of the new estimator in various situations and its relative performance with respect to two more traditional estimators (maximum likelihood estimator and Fourier-based Whittle estimator) are assessed, along with practical aspects of its application. Possible solutions are proposed for most of the issues detected, including suggestion of a new specification of the estimator. This uses maximal overlap discrete wavelet transform instead of the traditionally used discrete wavelet transform, which should improve the estimator performance in all its applications, not only in the case of FIEGARCH model estimation. The thesis concludes that, after optimization of the estimation setup, the wavelet-based estimator may become an attractive robust alternative to the traditional methods.

We analyze by simulation the properties of two time domain and two frequency domain estimators for low order autoregressive fractionally integrated moving average Gaussian models, ARFIMA (p,d,q). The estimators considered are the exact maximum likelihood for demeaned data, EML, the associated modified profile likelihood, MPL, and the Whittle estimator with, WLT, and without tapered data, WL. Length of the series is 100. The estimators are compared in terms of pile-up effect, mean square error, bias, and empirical confidence level. The tapered version of the Whittle likelihood turns out to be a reliable estimator for ARMA and ARFIMA models. Its small losses in performance in case of ``well-behaved" models are compensated sufficiently in more ``difficult" models. The modified profile likelihood is an alternative to the WLT but is computationally more demanding. It is either equivalent to the EML or more favorable than the EML. For fractionally integrated models, particularly, it dominates clearly the EML. The WL has serious deficiencies for large ranges of parameters, and so cannot be recommended in general. The EML, on the other hand, should only be used with care for fractionally integrated models due to its potential large negative bias of the fractional integration parameter. In general, one should proceed with caution for ARMA(1,1) models with almost canceling roots, and, in particular, in case of the EML and the MPL for inference in the vicinity of a moving average root of +1. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

La première partie de cette thèse est consacrée à l'étude de la dispersion du pollen de maïs. Le grain de pollen est vu comme une particule soumise à un champ de forces et sa trajectoire est modélisée à l'aide de différents processus de diffusion. Lorsque deux champs sont contigüs (milieu homogène), différentes fonctions de dispersion individuelles paramétriques sont alors obtenues, différentes hypothèses étant faites sur des temps d'atteinte de processus stochastiques. A partir d'expériences, les paramètres sont alors estimés en considérant un modèle de régression non linéaire. Le choix du modèle le mieux adapté se fait à l'aide d'un critère de type Akaïke et de méthodes graphiques. Par ailleurs ces modèles permettent d'effectuer des prédictions. Les résultats sont alors appliqués lorsque deux champs sont séparés par une autre culture (milieu hétérogène), afin d'étudier l'effet d'une discontinuité sur la dispersion. <br />Dans la seconde partie, on s'intéresse à des modèles à volatilité stochastique «mean-reverting», souvent utilisés en économie. Le processus observé est fonction d'une diffusion non observable dont on souhaite estimer les paramètres. Une méthode d'estimation à deux pas basée sur la structure ARMA(1,1) du processus est proposée, en utilisant un estimateur de moments et un contraste de Whittle. Des simulations sont réalisées afin de comparer cette méthode avec d'autres méthodes existantes. Ensuite un paramètre dit «leverage» est ajouté et un modèle discrétisé est étudié. Un critère auxiliaire est proposé pour estimer les paramètres à l'aide d'une méthode d'inférence indirecte. Enfin des simulations sont réalisées pour évaluer leurs performances.

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Dans ce travail, nous étudions de manière approfondie les processus à mémoire longue saisonniers avec variance infinie des innovations. Dans le premier chapitre, nous rappelons les différentes propriétés des lois -stables univariées (stabilité, calcul des moments, simulation, : : :). Nous introduisons ensuite deux modèles à variance infinie largement utilisés dans la littérature statistique : les modèles ARMA -stables et les modèles ARFIMA -stables développés respectivement par Mikosch et al. [57] et Kokoszka et Taqqu [45]. Constatant les limites de ces modèles, nous construisons dans le second chapitre de nouveaux modèles appelés processus ARFISMA symétriques -stables. Ces modèles nous permettent de prendre en compte dans une modélisation la présence éventuelle des trois éléments caractéristiques suivants : mémoire longue, saisonnalité et variance infinie, que l'on rencontre souvent en finance, en télécommunication ou en hydrologie. Après avoir conclu le chapitre par l'étude du comportement asymptotique du modèle par des simulations, nous abordons dans le troisième chapitre, le problème d'estimation des paramètres d'un processus ARFISMA -stable. Nous présentons plusieurs méthodes d'estimation : une méthode semiparamétrique développée par Reisen et al.[67], une méthode de Whittle classique utilisée par Mikosch et al. [57] et par Kokoszka et Taqqu [45], et une autre approche de la méthode de Whittle basée sur l'évaluation de la vraisemblance de Whittle par une méthode de Monte Carlo par chaînes de Markov (MCMC). De nombreuses simulations, effectuées avec le logiciel R [64], permettent de comparer ces méthodes d'estimation. Cependant, ces méthodes ne permettent pas d'estimer le paramètre d'innovation . Ainsi, nous introduisons, dans le quatrième chapitre deux méthodes d'estimation : la méthode de la fonction caractéristique empirique développée par Knight et Yu [43] et la méthode des moments généralisés basée sur des moments conditionnels continus, suggérée par Carrasco et Florens [16]. De plus, afin de comparer les propriétés asymptotiques des estimateurs, des simulations de Monte Carlo sont effectuées. Enfin, dans le cinquième chapitre, nous appliquons ce modèle sur des données de débits du fleuve Sénégal à la station de Bakel. En guise de comparaison, nous considérons le modèle linéaire classique de Box et Jenkins [11], et nous comparons leurs capacités prédictives.