Estimation and testing in generalized mean-reverting processes with change-point

In this paper, we study an inference problem in generalized Ornstein–Uhlenbeck processes with an unknown change-point when the drift parameter is suspected to satisfy a linear restriction. The testing problem studied generalizes a very recent problem about testing the existence of a change-point. To this end, we derive the asymptotic properties of the unrestricted estimator (UE) and the restricted estimator for the drift parameters, and we construct some shrinkage estimators (SEs). Further, we derive a test for testing the uncertain restriction and establish its asymptotic power. Moreover, we derive the asymptotic distributional risk of the proposed estimators and we prove that SEs dominate the UE. Finally, we present some numerical results which confirm the consistency of the proposed test as well as the superiority of the SEs over UE.     

Asymptotic theory of simultaneous estimation of Poisson means

The Poisson distribution is often a good approximation to the underlying sampling distribution and is central to the study of categorical data. In this paper, we propose a new unified approach to an investigation of point properties of simultaneous estimations of Poisson population parameters with general quadratic loss functions. The main accent is made on the shrinkage estimation. We build a series of estimators that could be represented as a convex combination of linear statistics such as maximum likelihood estimator (benchmark estimator), restricted estimator, composite estimator, preliminary test estimator, shrinkage estimator, positive rule shrinkage estimator (James-Stein type estimator). All these estimators are represented in a general integrated estimation approach, which allows us to unify our investigation and order them with respect to the risk. A simulation study with numerical and graphical results is conducted to illustrate the properties of the investigated estimators.     

Double Shrinkage Estimators in the GMANOVA Model

In the GMANOVA model or equivalent growth curve model, shrinkage effects on the MLE (maximum likelihood estimator) are considered under an invariant risk matrix. We first study the fundamental structure of the problem through which we decompose the estimation problem into some conditional problems and then demonstrate some classes of double shrinkage minimax estimators which uniformly dominate the MLE in the matrix risk.     

Robust Wavelet-Domain Estimation of the Fractional Difference Parameter in Heavy-Tailed Time Series: An Empirical Study

We investigate the performance of several wavelet-based estimators of the fractional difference parameter. We consider situations where, in addition to long-range dependence, the time series exhibit heavy tails and are perturbed by polynomial and change-point trends. We make detailed study of a wavelet-domain pseudo Maximum Likelihood Estimator (MLE), for which we provide an asymptotic and finite-sample justification. Using numerical experiments, we show that unlike the traditional time-domain estimators, estimators based on the wavelet transform are robust to additive trends and change points in mean, and produce accurate estimates even under significant departures from normality. The Wavelet-domain MLE appears to dominate a regression-based wavelet estimator in terms of smaller root mean squared error. These findings are derived from a simulation study and application to computer traffic traces.     

Asymptotic theory for estimating the parameters of a Lévy process

Summary We consider consistency and asymptotic normality of maximum likelihood estimators (MLE) for parameters of a Lévy process of the discontinuous type. The MLE are based on a single realization of the process on a given interval [0,t]. Depending on properties of the Lévy measure we either consider the MLE corresponding to jumps of size greater than ε and, keepingt fixed, we let ε tend to 0, or we consider the MLE corresponding to the complete information of the realization of the process on [0,t] and lett tend to ∞. The results of this paper improve in both generality and rigor previous asymptotic estimation results for such processes.     

Analysis of rounded data from dependent sequences

Observations on continuous populations are often rounded when recorded due to the precision of the recording mechanism. However, classical statistical approaches have ignored the effect caused by the rounding errors. When the observations are independent and identically distributed, the exact maximum likelihood estimation (MLE) can be employed. However, if rounded data are from a dependent structure, the MLE of the parameters is difficult to calculate since the integral involved in the likelihood equation is intractable. This paper presents and examines a new approach to the parameter estimation, named as “short, overlapping series” (SOS), to deal with the α-mixing models in presence of rounding errors. We will establish the asymptotic properties of the SOS estimators when the innovations are normally distributed. Comparisons of this new approach with other existing techniques in the literature are also made by simulation with samples of moderate sizes.     

One-step estimation of spatial dependence parameters: Properties and extensions of the APLE statistic

We consider one-step estimation of parameters that represent the strength of spatial dependence in a geostatistical or lattice spatial model. While the maximum likelihood estimators (MLE) of spatial dependence parameters are known to have various desirable properties, they do not have closed-form expressions. Therefore, we consider a one-step alternative to maximum likelihood estimation based on solving an approximate (i.e., one-step) profile likelihood estimating equation. The resulting approximate profile likelihood estimator (APLE) has a closed-form representation, making it a suitable alternative to the widely used Moran’s I statistic. Since the finite-sample and asymptotic properties of one-step estimators of covariance-function parameters have not been studied rigorously, we explore these properties for the APLE of the spatial dependence parameter in the simultaneous autoregressive (SAR) model. Motivated by the APLE statistic’s closed from, we develop exploratory spatial data analysis tools that capture regions of local clustering or the extent to which the strength of spatial dependence varies across space. We illustrate these exploratory tools using both simulated data and observed crime rates in Columbus, OH.     

Random field forward interest rate models,market price of risk and their statistics

In this paper we consider discrete time forward interest rate models. In our approach, unlike in the classical Heath–Jarrow–Morton framework, the forward rate curves are driven by a random field. Hence we get a general interest rate structure. Our aim is to give an overview of our results in such a model on the following questions: no-arbitrage conditions, maximum likelihood estimation of the volatility, as well as the joint estimation of the parameters and the asymptotic behaviour of the estimators, relationship with continuous models. Finally we give discussion on the practical problems of the estimation and we show several numerical results on the statistics of such models. This research has been supported by the Hungarian Scientific Research Fund under Grants No. OTKA–F046061/2004 and OTKA–T048544/2005.     

Estimation of Parameters of the Generalized Pareto\Distribution by the Least Squares

Traditional estimations of parameters of the generalized Pareto distribution (GPD) are generally constrained by the shape parameter of GPD. Such as: the method-of-moments (MOM), the probability-weighted moments (PWM), L-moments (LM), the maximum likelihood estimation (MLE) and so on. In this paper we use the fact that GPD can be transformed into the exponential distribution and use the results of parameters estimation for the exponential distribution, than we propose parameters estimators of the two-parameter or three-parameter GPD by the least squares method. Some asymptotic results are provided and the proposed method not constrained by the shape parameter of GPD. A simulation study is carried out to evaluate the performance of the proposed method and to compare them with other methods suggested in this paper. The simulation results indicate that the proposed method performs better than others in some common situation.     

Shrinkage estimation and variable selection in multiple regression models with random coefficient autoregressive errors

In this paper, we consider improved estimation strategies for the parameter vector in multiple regression models with first-order random coefficient autoregressive errors (RCAR(1)). We propose a shrinkage estimation strategy and implement variable selection methods such as lasso and adaptive lasso strategies. The simulation results reveal that the shrinkage estimators perform better than both lasso and adaptive lasso when and only when there are many nuisance variables in the model.     

Simultaneous Estimation of Several Intraclass Correlation Coefficients

Based on shrinkage and preliminary test rules, various estimators are proposed for estimation of several intraclass correlation coefficients when independent samples are drawn from multivariate normal populations. It is demonstrated that the James-Stein type estimators are asymptotically superior to the usual estimators. Furthermore, it is also indicated through asymptotic results that none of the preliminary test and shrinkage estimators dominate each other, though they perform relatively well as compared to the classical estimator. The relative dominance picture of the estimators is presented. A Monte Carlo study is performed to appraise the properties of the proposed estimators for small samples.     

On an improvement of Hill and some other estimators

In the paper, we propose a new idea in the tail-index estimation. This idea allows us to improve the asymptotic performance of the classical Hill estimator and other most popular estimators over the range of the parameters present in the second-order regular-variation condition. We prove the asymptotic normality of the introduced estimators and provide a comparison (using the asymptotic mean-squared error) with other estimators of the tail index.     

Replicated INAR(1) Processes

Replicated time series are a particular type of repeated measures, which consist of time-sequences of measurements taken from several subjects (experimental units). We consider independent replications of count time series that are modelled by first-order integer-valued autoregressive processes, INAR(1). In this work, we propose several estimation methods using the classical and the Bayesian approaches and both in time and frequency domains. Furthermore, we study the asymptotic properties of the estimators. The methods are illustrated and their performance is compared in a simulation study. Finally, the methods are applied to a set of observations concerning sunspot data. AMS 2000 Subject Classification 62M10, 91B70, 60G10     

Pre-averaged kernel estimators for the drift function of a diffusion process in the presence of microstructure noise

We consider estimation of the drift function of a stationary diffusion process when we observe high-frequency data with microstructure noise over a long time interval. We propose to estimate the drift function at a point by a Nadaraya–Watson estimator that uses observations that have been pre-averaged to reduce the noise. We give conditions under which our estimator is consistent and asympotically normal. Its rate and asymptotic bias and variance are the same as those without microstructure noise. To use our method in data analysis, we propose a data-based cross-validation method to determine the bandwidth in the Nadaraya–Watson estimator. Via simulation, we study several methods of bandwidth choices, and compare our estimator to several existing estimators. In terms of mean squared error, our new estimator outperforms existing estimators.     

Positive shrinkage, improved pretest and absolute penalty estimators in partially linear models

Shrinkage estimators of a partially linear regression parameter vector are constructed by shrinking estimators in the direction of the estimate which is appropriate when the regression parameters are restricted to a linear subspace. We investigate the asymptotic properties of positive Stein-type and improved pretest semiparametric estimators under quadratic loss. Under an asymptotic distributional quadratic risk criterion, their relative dominance picture is explored analytically. It is shown that positive Stein-type semiparametric estimators perform better than the usual Stein-type and least square semiparametric estimators and that an improved pretest semiparametric estimator is superior to the usual pretest semiparametric estimator. We also consider an absolute penalty type estimator for partially linear models and give a Monte Carlo simulation comparisons of positive shrinkage, improved pretest and the absolute penalty type estimators. The comparison shows that the shrinkage method performs better than the absolute penalty type estimation method when the dimension of the parameter space is much larger than that of the linear subspace.     

Statistical inference for coherent systems with Weibull distributed component lifetimes under complete and incomplete information

Point estimators for the parameters of the component lifetime distribution in coherent systems are evolved assuming to be independently and identically Weibull distributed component lifetimes. We study both complete and incomplete information under continuous monitoring of the essential component lifetimes. First, we prove that the maximum likelihood estimator (MLE) under complete information based on progressively Type‐II censored system lifetimes uniquely exists and we present two approaches to compute the estimates. Furthermore, we consider an ad hoc estimator, a max‐probability plan estimator and the MLE for the parameters under incomplete information. In order to compute the MLEs, we consider a direct maximization of the likelihood and an EM‐algorithm–type approach, respectively. In all cases, we illustrate the results by simulations of the five‐component bridge system and the 10‐component parallel system, respectively.     

Filter‐based portfolio strategies in an HMM setting with varying correlation parametrizations

We consider portfolio optimization in a regime‐switching market. The assets of the portfolio are modeled through a hidden Markov model (HMM) in discrete time, where drift and volatility of the single assets are allowed to switch between different states. We consider different parametrizations of the involved asset covariances: statewise uncorrelated assets (though linked through the common Markov chain), assets correlated in a state‐independent way, and assets where the correlation varies from state to state. As a benchmark, we also consider a model without regime switches. We utilize a filter‐based expectation‐maximization (EM) algorithm to obtain optimal parameter estimates within this multivariate HMM and present parameter estimators in all three HMM settings. We discuss the impact of these different models on the performance of several portfolio strategies. Our findings show that for simulated returns, our strategies in many settings outperform naïve investment strategies, like the equal weights strategy. Information criteria can be used to detect the best model for estimation as well as for portfolio optimization. A second study using real data confirms these findings.     

Imperfect repair in degradation processes: A Kijima‐type approach

We consider a Wiener process with linear drift for degradation modeling. Regularly, maintenance actions are carried out which produce a reduction of the degradation level. In this paper, we consider the influence of such maintenance actions to the further development of the degradation process and the resulting lifetime distribution. A connection between virtual age in Kijima‐type models and degradation level in the underlying degradation process is developed. Furthermore, estimators for the process parameters as well as for the degree of repair are developed.     

Adaptive estimation of an ergodic diffusion process based on sampled data

We consider adaptive maximum likelihood type estimation of both drift and diffusion coefficient parameters for an ergodic diffusion process based on discrete observations. Two kinds of adaptive maximum likelihood type estimators are proposed and asymptotic properties of the adaptive estimators, including convergence of moments, are obtained.     

Estimation in the presence of many nuisance parameters: Composite likelihood and plug-in likelihood

We consider the incidental parameters problem in this paper, i.e. the estimation for a small number of parameters of interest in the presence of a large number of nuisance parameters. By assuming that the observations are taken from a multiple strictly stationary process, the two estimation methods, namely the maximum composite quasi-likelihood estimation (MCQLE) and the maximum plug-in quasi-likelihood estimation (MPQLE) are considered. For the MCQLE, we profile out nuisance parameters based on lower-dimensional marginal likelihoods, while the MPQLE is based on some initial estimators for nuisance parameters. The asymptotic normality for both the MCQLE and the MPQLE is established under the assumption that the number of nuisance parameters and the number of observations go to infinity together, and both the estimators for the parameters of interest enjoy the standard root-n$n$ convergence rate. Simulation with a spatial–temporal model illustrates the finite sample properties of the two estimation methods.