A noise estimation method for corrupted correlated data

Data acquisition, both in time and in spatial domains, in many cases yields observations with a measurement error. The identification of such a component, that masks the phenomenon under study (signal), must be carried out before the model of interest is specified. The objective of the paper is to propose an estimator for the parameters of an additive noise and compare it with existing methods by applications to both simulated and real data sets.     

SIMEX estimation in case of correlated measurement errors

The simulation-extrapolation (SIMEX) approach of Cook and Stefanski (J. Am. Stat. Assoc. 89:1314–1328, 1994) has proved to be successful in obtaining reliable estimates if variables are measured with (additive) errors. In particular for nonlinear models, this approach has advantages compared to other procedures such as the instrumental variable approach if only variables measured with error are available. However, it has always been assumed that measurement errors for the dependent variable are not correlated with those related to the explanatory variables although such scenario is quite likely. In such a case the (standard) SIMEX suffers from misspecification even for the simple linear regression model. Our paper reports first results from a generalized SIMEX (GSIMEX) approach which takes account of this correlation. We also demonstrate in our simulation study that neglect of the correlation will lead to estimates which may be worse than those from the naive estimator which completely disregards measurement errors.     

Optimum designs for parameter estimation in a mixture experiment with two correlated responses

In this paper, we investigate a mixture problem with two responses, which are functions of the mixing proportions, and are correlated with known dispersion matrix. We obtain D- and A-optimal designs for estimating the parameters of the response functions, when none or some of the regression coefficients of the two functions are the same. It is shown that when no prior knowledge about the regression coefficients is available, the D-optimal design is independent of the dispersion matrix, while the A-optimal design depends on it, provided the response functions are of different degree. On the other hand, when some of the regression coefficients are known to be the same for both the functions, the D-optimal design depends on the dispersion matrix when the two response functions are not of the same degree.     

Rate efficient estimation of realized Laplace transform of volatility with microstructure noise

In this paper, we consider the problem of estimating the Laplace transform of volatility within a fixed time interval [0,T] using high‐frequency sampling, where we assume that the discretized observations of the latent process are contaminated by microstructure noise. We use the pre‐averaging approach to deal with the effect of microstructure noise. Under the high‐frequency scenario, we obtain a consistent estimator whose convergence rate is , which is known as the optimal convergence rate of the estimation of integrated volatility functionals under the presence of microstructure noise. The related central limit theorem is established. The simulation studies justify the finite‐sample performance of the proposed estimator.     

Bayesian estimation of log odds ratios over two-way contingency tables with intraclass correlated cells

In this article, a Bayesian approach is proposed for the estimation of log odds ratios and intraclass correlations over a two-way contingency table, including intraclass correlated cells. Required likelihood functions of log odds ratios are obtained, and determination of prior structures is discussed. Hypothesis testing for log odds ratios and intraclass correlations by using the posterior simulations is outlined. Because the proposed approach includes no asymptotic theory, it is useful for the estimation and hypothesis testing of log odds ratios in the presence of certain intraclass correlation patterns. A family health status and limitations data set is analyzed by using the proposed approach in order to figure out the impact of intraclass correlations on the estimates and hypothesis tests of log odds ratios. Although intraclass correlations are small in the data set, we obtain that even small intraclass correlations can significantly affect the estimates and test results, and our approach is useful for the estimation and testing of log odds ratios in the presence of intraclass correlations.     

Least squares parameter estimation in multiplicative noise models

A simple multiplicative noise model with a constant signal has become a basic mathematical model in processing synthetic aperture radar images. The purpose of this paper is to examine a general multiplicative noise model with linear signals represented by a number of unknown parameters. The ordinary least squares (LS) and weighted LS methods are used to estimate the model parameters. The biases of the weighted LS estimates of the parameters are derived. The biases are then corrected to obtain a second-order unbiased estimator, which is shown to be exactly equivalent to the maximum log quasi-likelihood estimation, though the quasi-likelihood function is founded on a completely different theoretical consideration and is known, at the present time, to be a uniquely acceptable theory for multiplicative noise models. Synthetic simulations are carried out to confirm theoretical results and to illustrate problems in processing data contaminated by multiplicative noises. The sensitivity of the LS and weighted LS methods to extremely noisy data is analysed through the simulated examples.     

Maximum likelihood estimation for ordered expectations of correlated binary variables

A multivariate binary distribution that incorporates the correlation between individual variables is considered. The availability of auxiliary information taking the form of simple ordering constraints on their expected values is assumed. The problem of constructing constraint-preserving estimates for expectations is formulated as conditional maximization of convex likelihood function for corresponding multinomial distribution with suitably chosen restrictions. Starting values for convex optimization algorithms are proposed. The proposed estimator is consistent under mild assumptions.     

Bayesian estimation of an AR(1) process with exponential white noise

The object of this paper is a Bayesian analysis of the autoregressive model X _t ?=?ρX _t?1?+?Y _t where 0?Y _t are independent random variables with an exponential distribution of parameter θ. Our study generalizes some results obtained by Turkmann (1990 Amaral Turkmann, M. A. (1990). Bayesian analysis of an autoregressive process with exponential white noise. Statistics, 4: 601–608.  [Google Scholar]). Our analysis is based on a more general non-informative prior which allows us to improve the estimators of ρ and θ.     

Nonparametric estimation of jump characteristics under market microstructure noise

This article proposes a simple nonparametric method to estimate the jump characteristics in asset price with noisy high-frequency data. We combine the pre-averaging approach and the threshold technique to identify the jumps, and then propose the pre-averaging threshold estimators for the number and sizes of jumps occurred. We further present the asymptotic properties of the proposed estimators. The Monte Carlo simulation shows that the estimators are robust to microstructure noise and work very well especially when the data frequency is ultra-high. Finally, an empirical example further demonstrates the power of the proposed method.     

A robust estimation method for the linear regression model parameters with correlated error terms and outliers

Independence of error terms in a linear regression model, often not established. So a linear regression model with correlated error terms appears in many applications. According to the earlier studies, this kind of error terms, basically can affect the robustness of the linear regression model analysis. It is also shown that the robustness of the parameters estimators of a linear regression model can stay using the M-estimator. But considering that, it acquires this feature as the result of establishment of its efficiency. Whereas, it has been shown that the minimum Matusita distance estimators, has both features robustness and efficiency at the same time. On the other hand, because the Cochrane and Orcutt adjusted least squares estimators are not affected by the dependence of the error terms, so they are efficient estimators. Here we are using of a non-parametric kernel density estimation method, to give a new method of obtaining the minimum Matusita distance estimators for the linear regression model with correlated error terms in the presence of outliers. Also, simulation and real data study both are done for the introduced estimation method. In each case, the proposed method represents lower biases and mean squared errors than the other two methods.KEYWORDS: Robust estimation method, minimum Matusita distance estimation method, non-parametric kernel density estimation method, correlated error terms, outliers     

Regression Kink With an Unknown Threshold

This article explores estimation and inference in a regression kink model with an unknown threshold. A regression kink model (or continuous threshold model) is a threshold regression constrained to be everywhere continuous with a kink at an unknown threshold. We present methods for estimation, to test for the presence of the threshold, for inference on the regression parameters, and for inference on the regression function. A novel finding is that inference on the regression function is nonstandard since the regression function is a nondifferentiable function of the parameters. We apply recently developed methods for inference on nondifferentiable functions. The theory is illustrated by an application to the growth and debt problem introduced by Reinhart and Rogoff, using their long-span time-series for the United States.     

The estimation of derivatives of a nonparametric regression function when the data are correlated

A kernel estimator of a derivative of arbitrary order of a nonparametric average population curve is considered for a correlated-errors model with balanced replicate measurements at each design point. Asymptotic expansions of the mean squared error are derived for two classes of correlation functions in the model. Consistency, choice of smoothing parameter, and rates of convergence are examined for the important special cases of estimating the first and second derivatives.     

Small area estimation: the EBLUP estimator based on spatially correlated random area effects

This paper deals with small area indirect estimators under area level random effect models when only area level data are available and the random effects are correlated. The performance of the Spatial Empirical Best Linear Unbiased Predictor (SEBLUP) is explored with a Monte Carlo simulation study on lattice data and it is applied to the results of the sample survey on Life Conditions in Tuscany (Italy). The mean squared error (MSE) problem is discussed illustrating the MSE estimator in comparison with the MSE of the empirical sampling distribution of SEBLUP estimator. A clear tendency in our empirical findings is that the introduction of spatially correlated random area effects reduce both the variance and the bias of the EBLUP estimator. Despite some residual bias, the coverage rate of our confidence intervals comes close to a nominal 95%.     

ML estimation and an efficiency study for mean estimators in spatially correlated repeated arrays

For spatially correlated repeated arrays, a simple method is proposed for maximum likelihood (ML) estimation of the mean parameters. Efficiency of the sample mean over the maximum likelihood estimator (MLE) is analyzed. Spatial correlations combined with heterogeneity of spatial correlations or heterogeneity of error variances are shown to have adverse effect on efficiency of the sample mean. Therefore, in such spatially correlated and heterogeneous situations, it is recommended that spatial correlations should be properly addressed in estimating mean parameters.     

On maximum likelihood estimation of the long-memory parameter in fractional Gaussian noise

Approximate normality and unbiasedness of the maximum likelihood estimate (MLE) of the long-memory parameter H of a fractional Brownian motion hold reasonably well for sample sizes as small as 20 if the mean and scale parameter are known. We show in a Monte Carlo study that if the latter two parameters are unknown the bias and variance of the MLE of H both increase substantially. We also show that the bias can be reduced by using a parametric bootstrap procedure. In very large samples, maximum likelihood estimation becomes problematic because of the large dimension of the covariance matrix that must be inverted. To overcome this difficulty, we propose a maximum likelihood method based upon first differences of the data. These first differences form a short-memory process. We split the data into a number of contiguous blocks consisting of a relatively small number of observations. Computation of the likelihood function in a block then presents no computational problem. We form a pseudo-likelihood function consisting of the product of the likelihood functions in each of the blocks and provide a formula for the standard error of the resulting estimator of H. This formula is shown in a Monte Carlo study to provide a good approximation to the true standard error. The computation time required to obtain the estimate and its standard error from large data sets is an order of magnitude less than that required to obtain the widely used Whittle estimator. Application of the methodology is illustrated on two data sets.     

Monitoring correlated processes with binomial marginals

Few approaches for monitoring autocorrelated attribute data have been proposed in the literature. If the marginal process distribution is binomial, then the binomial AR(1) model as a realistic and well-interpretable process model may be adequate. Based on known and newly derived statistical properties of this model, we shall develop approaches to monitor a binomial AR(1) process, and investigate their performance in a simulation study. A case study demonstrates the applicability of the binomial AR(1) model and of the proposed control charts to problems from statistical process control.     

Forecasting with serially correlated regression models

In this article we investigate the asymptotic and finite-sample properties of predictors of regression models with autocorrelated errors. We prove new theorems associated with the predictive efficiency of generalized least squares (GLS) and incorrectly structured GLS predictors. We also establish the form associated with their predictive mean squared errors as well as the magnitude of these errors relative to each other and to those generated from the ordinary least squares (OLS) predictor. A large simulation study is used to evaluate the finite-sample performance of forecasts generated from models using different corrections for the serial correlation.     

Unobserved components models with correlated disturbances

The paper deals with the decomposition of a time series process admitting an ARIMA representation into permanent and transitory components, with the intent of investigating whether the introduction of correlated disturbances provides meaningful extensions of the admissible parameter range. The main points are illustrated with reference to ARIMA(2,1,0) and IMA(2,2) models. It is argued that there is very little reason for such extensions, and that the restrictions implied by the assumption of uncorrelated components are sound.This research was supported by the MURST Cofin2000. The paper was presented at the XL Scientific Meeting of the Italian Statistical Society (Florence 2000), ISF 2000 (Lisbon), and at the Europaeisches Heimbildungswerk workshop, Helenau-Bernau (Berlin). I thank participants for their comments and in particular Jörg Breitung for very stimulating discussion. I also wish to thank the associate editor and the referee for their comments.