Consistent estimation of percolation quantities

We consider ordinary independent percolation on the integer lattice, and construct consistent estimators for most of the important quantities associated to this model. We also present some simulation results.     

Consistent estimation in the bilinear multivariate errors-in-variables model

A bilinear multivariate errors-in-variables model is considered. It corresponds to an overdetermined set of linear equations AXB=C, A∈ℝ^m×n, B∈ℝ^p×q, in which the data A, B, C are perturbed by errors. The total least squares estimator is inconsistent in this case.  An adjusted least squares estimator is constructed, which converges to the true value X, as m →∞, q →∞. A small sample modification of the estimator is presented, which is more stable for small m and q and is asymptotically equivalent to the adjusted least squares estimator. The theoretical results are confirmed by a simulation study. Acknowledgements. We thank two anonymous reviewers for their suggestions and corrections.? A. Kukush is supported by a postdoctoral research fellowship of the Belgian office for Scientific, Technical and Cultural Affairs, promoting Scientific and Technical Collaboration with Central and Eastern Europe.? S. Van Huffel is a full professor with the Katholieke Universiteit Leuven.? I. Markovsky is a research assistant with the Katholieke Universiteit Leuven.? This paper presents research results of the Belgian Programme on Interuniversity Poles of Attraction (IUAP V-22), initiated by the Belgian State, Prime Minister's Office – Federal Office for Scientific, Technical and Cultural Affairs of the Concerted Research Action (GOA) projects of the Flemish Government MEFISTO-666 (Mathematical Engineering for Information and Communication Systems Technology), of the IDO/99/03 project (K.U. Leuven) “Predictive computer models for medical classification problems using patient data and expert knowledge”, of the FWO projects G.0078.01, G.0200.00, and G0.0270.02.? The scientific responsibility is assumed by its authors.     

Consistent estimation of a regression with errors in the variables

Summary As is well known, least squares estimates of regression coefficients are inconsistent if the variables are measured with random errors. In the classical case of known variances and covariances for these error variables, consistent estimates can be derived. It is shown that these estimators generally have a joint asymptotic normal distribution, the covariance matrix of which is derived. No use is made of normality assumptions, but knowledge of the third and fourth moments of error variables is utilized.     

Consistent estimation of the fixed effects stochastic frontier model

In this paper we consider a fixed-effects stochastic frontier model. That is, we have panel data, fixed individual (firm) effects, and the usual stochastic frontier analysis (SFA) composed error.     

Consistent likelihood-based estimation of a star-shaped distribution

We consider the classical problem of nonparametrically estimating a star-shaped distribution, i.e., a distribution function F on [0,∞) with the property that F(u)/u is nondecreasing on the set {u : F(u) < 1}. This problem is intriguing because of the fact that a well defined maximum likelihood estimator (MLE) exists, but this MLE is inconsistent. In this paper, we argue that the likelihood that is commonly used in this context is somewhat unnatural and propose another, so called ‘smoothed likelihood’. However, also the resulting MLE turns out to be inconsistent. We show that more serious smoothing of the likelihood yields consistent estimators in this model.     

Consistent moment estimators of regression coefficients in the presence of errors in variables

This paper examines the possibilities of moment estimators of regression coefficients in the errors-in-variables problem suggested by Geary (1942) and others [Scott (1950) and Drion (1951)]. This approach yields consistent estimators of regression coefficients based on uni- and bi-variate moments (or cumulants) of third or higher order. These are computationally simple and need milder assumptions than the standard techniques, viz., ML and IV estimation. After a review of past investigations, this paper proposes new moment estimators and compares the asymptotic efficiencies of six estimators proposed earlier or here and of the OLS estimator. The case where the true regressor is lognormally distributed receives considerable attention in this communication.     

Term structure estimation in the presence of autocorrelation

This paper assesses the effects of autocorrelation on parameter estimates of affine term structure models (ATSM) when principal components analysis is used to extract factors. In contrast to recent studies, we design and run a Monte Carlo experiment that relies on the construction of a simulation design that is consistent with the data, rather than theory or observation, and find that parameter estimation from ATSM is precise in the presence of serial correlation in the measurement error term. Our findings show that parameter estimation of ATSM with principal component based factors is robust to autocorrelation misspecification.     

Consistent factor estimation in dynamic factor models with structural instability

This paper considers the estimation of approximate dynamic factor models when there is temporal instability in the factor loadings. We characterize the type and magnitude of instabilities under which the principal components estimator of the factors is consistent and find that these instabilities can be larger than earlier theoretical calculations suggest. We also discuss implications of our results for the robustness of regressions based on the estimated factors and of estimates of the number of factors in the presence of parameter instability. Simulations calibrated to an empirical application indicate that instability in the factor loadings has a limited impact on estimation of the factor space and diffusion index forecasting, whereas estimation of the number of factors is more substantially affected.     

Consistent estimation of a general nonparametric regression function in time series

We propose an estimator of the conditional distribution of X_t|X_t−1,X_t−2,…, and the corresponding regression function , where the conditioning set is of infinite order. We establish consistency of our estimator under stationarity and ergodicity conditions plus a mild smoothness condition.     

Selection of estimation window in the presence of breaks

In situations where a regression model is subject to one or more breaks it is shown that it can be optimal to use pre-break data to estimate the parameters of the model used to compute out-of-sample forecasts. The issue of how best to exploit the trade-off that might exist between bias and forecast error variance is explored and illustrated for the multivariate regression model under the assumption of strictly exogenous regressors. In practice when this assumption cannot be maintained and both the time and size of the breaks are unknown, the optimal choice of the observation window will be subject to further uncertainties that make exploiting the bias–variance trade-off difficult. To that end we propose a new set of cross-validation methods for selection of a single estimation window and weighting or pooling methods for combination of forecasts based on estimation windows of different lengths. Monte Carlo simulations are used to show when these procedures work well compared with methods that ignore the presence of breaks.     

Consistent estimation of equations with composite moving average disturbance terms

The paper examines the circumstances under which an equation with a composite MA disturbance term can be consistently estimated by single-equation non-linear least squares. The composite disturbance may arise as a result of the dependent or an explanatory variable being unobservable either because it is subject to measurement error or because it is a ‘desired’ or ‘expected’ variable. It may also arise as a result of substituting out an explanatory variable from an equation or because a vector MA process is specified for the structural form of a linear simultaneous equation model. The argument of the paper relies on some results relating to the sum of finite MA processes presented in Nelson (1975a) and Darroch and McDonald (1981).     

Consistent estimation when the left-hand variable is exogenous over part of the sample period

This paper proposes a consistent estimation method for regression equations with a left-hand variable that is endogenous for some observations, and exogenous for others. This method is applied to the estimation of a demand-for-money function for Switzerland over a time interval which includes periods of monetary control; that is periods when the quantity of money can best be viewed as exogenous.     

Consistent estimation of correlations between observed interval variables with skewed distributions

Departures from multinormality due to skewness in observed distributions may result in inconsistent estimates of product-moment correlations between interval variables. Therefore, the robustness of the product-moment correlation estimator against skewness in the distributions of sample data on interval variables has been investigated. This estimator is robust against skewness of maximally about 1 in absolute value. If the observed distributions have larger skewnesses, the sample data on interval variables may be redistributed over normally distributed discrete variables with 10 categories each. The estimated polychoric correlations between these discrete variables represent consistent estimates of the product-moment correlations between the original interval variables in the population.     

Instrumental variable estimation in the presence of many moment conditions

This paper develops shrinkage methods for addressing the “many instruments” problem in the context of instrumental variable estimation. It has been observed that instrumental variable estimators may behave poorly if the number of instruments is large. This problem can be addressed by shrinking the influence of a subset of instrumental variables. The procedure can be understood as a two-step process of shrinking some of the OLS coefficient estimates from the regression of the endogenous variables on the instruments, then using the predicted values of the endogenous variables (based on the shrunk coefficient estimates) as the instruments. The shrinkage parameter is chosen to minimize the asymptotic mean square error. The optimal shrinkage parameter has a closed form, which makes it easy to implement. A Monte Carlo study shows that the shrinkage method works well and performs better in many situations than do existing instrument selection procedures.     

Effect of outliers on the estimation of parameters

Summary The estimation of parameter in the type of distributionf(x)=b x ^{–1 /b} exp (–x ^b (/b),x>0, is considered, when several outliers of the type , ,r=1,2, ...,k, are present in the data. The estimates of as well as of 's are put in the closed form. Special cases, Weibull, Gamma and Exponential are considered for the case of single outlier. Actual estimates are calculated from the generated samples of size 2 and 3 for the Weibull and Exponential.     

Efficient estimation of income distribution parameters

The parameters of several families of distributions are estimated by means of minimum χ²; use is made of random samples taken from Dutch income-earning groups in 1973. The numerical search routine used, is the Complex method due to Box. The χ² function is evaluated by standard numerical integration procedures. The lognormal and the Gamma families are rejected because of a poor fit. The log t and the log Pearson IV families are introduced. This results in a considerable improvement of χ² critical levels. The generalized Gamma and the Champernowne function describe the income distribution reasonably well in some cases.     

Simplified estimation of parameters in a cauchy distribution

Simplified estimators of the location and scale parameters of a Cauchy distribution are constructed along the lines developed by D ixon [3, 4]. Symmetrically censored samples are considered. The efficiency of these estimators is shown to be high enough to make them useful in practice.     

Bias in dynamic panel estimation with fixed effects,incidental trends and cross section dependence

Explicit asymptotic bias formulae are given for dynamic panel regression estimators as the cross section sample size N→∞$N\to \infty$. The results extend earlier work by Nickell [1981. Biases in dynamic models with fixed effects. Econometrica 49, 1417–1426] and later authors in several directions that are relevant for practical work, including models with unit roots, deterministic trends, predetermined and exogenous regressors, and errors that may be cross sectionally dependent. The asymptotic bias is found to be so large when incidental linear trends are fitted and the time series sample size is small that it changes the sign of the autoregressive coefficient. Another finding of interest is that, when there is cross section error dependence, the probability limit of the dynamic panel regression estimator is a random variable rather than a constant, which helps to explain the substantial variability observed in dynamic panel estimates when there is cross section dependence even in situations where N is very large. Some proposals for bias correction are suggested and finite sample performance is analyzed in simulations.     

Model selection for hazard rate estimation in presence of censoring

This note presents an estimator of the hazard rate function based on right censored data. A collection of estimators is built from a regression-type contrast, in a general collection of linear models. Then, a penalised model selection procedure provides an estimator which satisfies an oracle inequality. In particular, we can prove that it is adaptive in the minimax sense on Hölder spaces.     

Nonparametric identification and estimation of current status data in the presence of death

We present a nonparametric study of current status data in the presence of death. Such data arise from biomedical investigations in which patients are examined for the onset of a certain disease, for example, tumor progression, but may die before the examination. A key difference between such studies on human subjects and the survival–sacrifice model in animal carcinogenicity experiments is that, due to ethical and perhaps technical reasons, deceased human subjects are not examined, so that the information on their disease status is lost. We show that, for current status data with death, only the overall and disease‐free survival functions can be identified, whereas the cumulative incidence of the disease is not identifiable. We describe a fast and stable algorithm to estimate the disease‐free survival function by maximizing a pseudo‐likelihood with plug‐in estimates for the overall survival rates. It is then proved that the global rate of convergence for the nonparametric maximum pseudo‐likelihood estimator is equal to O_p(n^?1/3) or the convergence rate of the estimated overall survival function, whichever is slower. Simulation studies show that the nonparametric maximum pseudo‐likelihood estimators are fairly accurate in small‐ to medium‐sized samples. Real data from breast cancer studies are analyzed as an illustration.