Detections of changes in return by a wavelet smoother with conditional heteroscedastic volatility

In this paper, we propose two estimators, an integral estimator and a discretized estimator, for the wavelet coefficient of regression functions in nonparametric regression models with heteroscedastic variance. These estimators can be used to test the jumps of the regression function. The model allows for lagged-dependent variables and other mixing regressors. The asymptotic distributions of the statistics are established, and the asymptotic critical values are analytically obtained from the asymptotic distribution. We also use the test to determine consistent estimators for the locations of change points. The jump sizes and locations of change points can be consistently estimated using wavelet coefficients, and the convergency rates of these estimators are derived. We perform some Monte Carlo simulations to check the powers and sizes of the test statistics. Finally, we give practical examples in finance and economics to detect changes in stock returns and short-term interest rates using the empirical wavelet method.     

Nonparametric estimators of the bivariate survival function under random censoring

A large number of proposals for estimating the bivariate survival function under random censoring have been made. In this paper we discuss the most prominent estimators, where prominent is meant in the sense that they are best for practical use; Dabrowska's estimator, the Prentice–Cai estimator, Pruitt's modified EM-estimator, and the reduced data NPMLE of van der Laan. We show how these estimators are computed and present their intuitive background. The asymptotic results are summarized. Furthermore, we give a summary of the practical performance of the estimators under different levels of dependence and censoring based on extensive simulation results. This leads also to a practical advise.     

Logarithmic calibration for multiplicative distortion measurement errors regression models

In this article, we propose a new identifiability condition by using the logarithmic calibration for the distortion measurement error models, where neither the response variable nor the covariates can be directly observed but are measured with multiplicative measurement errors. Under the logarithmic calibration, the direct-plug-in estimators of parameters and empirical likelihood based confidence intervals are proposed, and we studied the asymptotic properties of the proposed estimators. For the hypothesis testing of parameter, a restricted estimator under the null hypothesis and a test statistic are proposed. The asymptotic properties for the restricted estimator and test statistic are established. Simulation studies demonstrate the performance of the proposed procedure and a real example is analyzed to illustrate its practical usage.     

Maxbias curves of robust scale estimators based on subranges

A maxbias curve is a powerful tool to describe the robustness of an estimator. It is an asymptotic concept which tells how much an estimator can change due to a given fraction of contamination. In this paper, maxbias curves are computed for some univariate scale estimators based on subranges: trimmed standard deviations, interquantile ranges and the univariate Minimum Volume Ellipsoid (MVE) and Minimum Covariance Determinant (MCD) scale estimators. These estimators are intuitively appealing and easy to calculate. Since the bias behavior of scale estimators may differ depending on the type of contamination (outliers or inliers), expressions for both explosion and implosion maxbias curves are given. On the basis of robustness and efficiency arguments, the MCD scale estimator with 25% breakdown point can be recommended for practical use. Received: February 2000     

Jackknife estimators of a relative risk in 2×2 and 2×2×K contingency tables

Several jackknife estimators of a relative risk in a single 2×2 contingency table and of a common relative risk in a 2×2× K contingency table are presented. The estimators are based on the maximum likelihood estimator in a single table and on an estimator proposed by Tarone (1981) for stratified samples, respectively. For the stratified case, a sampling scheme is assumed where the number of observations within each table tends to infinity but the number of tables remains fixed. The asymptotic properties of the above estimators are derived. Especially, we present two general results which under certain regularity conditions yield consistency and asymptotic normality of every jackknife estimator of a bunch of functions of binomial probabilities.     

Estimation of bias and standard error of an improved estimator of normal mean

This paper considers an improved estimator of normal mean which is obtained by considering a feasible version of minimum mean squared error estimator. The exact expression for the bias and the mean squared error are fairly complicated and do not provide any guidelines as how to estimate the standard error of improved estimator. As is well known that any estimator without a formula for standard error has little practical utility. We therefore derive unbiased estimators for the bias and mean squared error of the improved estimator. Incidently, they turn out to be minimum variance unbiased estimators. Further, this exercise yields a simple formula for estimating the standard error. Based on the criterion of estimated standard error, the efficiency of the improved estimator with respect to the traditional unbiased estimator (i.e., sample mean) is examined numerically. The relationship with asymptotic standard error is also studied.     

Combining kernel estimators in the uniform deconvolution problem

We construct a density estimator and an estimator of the distribution function in the uniform deconvolution model. The estimators are based on inversion formulas and kernel estimators of the density of the observations and its derivative. Initially the inversions yield two different estimators of the density and two estimators of the distribution function. We construct asymptotically optimal convex combinations of these two estimators. We also derive pointwise asymptotic normality of the resulting estimators, the pointwise asymptotic biases and an expansion of the mean integrated squared error of the density estimator. It turns out that the pointwise limit distribution of the density estimator is the same as the pointwise limit distribution of the density estimator introduced by Groeneboom and Jongbloed (Neerlandica, 57, 2003, 136), a kernel smoothed nonparametric maximum likelihood estimator of the distribution function.     

Shrinkage estimation of the proportion in randomized response

This article considers the asymptotic estimation theory for the proportion in randomized response survey usinguncertain prior information (UPI) about the true proportion parameter which is assumed to be available on the basis of some sort of realistic conjecture. Three estimators, namely, the unrestricted estimator, the shrinkage restricted estimator and an estimator based on a preliminary test, are proposed. Their asymptotic mean squared errors are derived and compared. The relative dominance picture of the estimators is presented.     

Large-sample estimation strategies for eigenvalues of a Wishart matrix

The problem of simultaneous asymptotic estimation of eigenvalues of covariance matrix of Wishart matrix is considered under a weighted quadratic loss function. James-Stein type of estimators are obtained which dominate the sample eigenvalues. The relative merits of the proposed estimators are compared to the sample eigenvalues using asymptotic quadratic distributional risk under loal alternatives. It is shown that the proposed estimators are asymptotically superior to the sample eigenvalues. Further, it is demonstrated that the James-Stein type estimator is dominated by its truncated part.     

Semiparametric estimation of logistic regression model with missing covariates and outcome

We consider a semiparametric method to estimate logistic regression models with missing both covariates and an outcome variable, and propose two new estimators. The first, which is based solely on the validation set, is an extension of the validation likelihood estimator of Breslow and Cain (Biometrika 75:11–20, 1988). The second is a joint conditional likelihood estimator based on the validation and non-validation data sets. Both estimators are semiparametric as they do not require any model assumptions regarding the missing data mechanism nor the specification of the conditional distribution of the missing covariates given the observed covariates. The asymptotic distribution theory is developed under the assumption that all covariate variables are categorical. The finite-sample properties of the proposed estimators are investigated through simulation studies showing that the joint conditional likelihood estimator is the most efficient. A cable TV survey data set from Taiwan is used to illustrate the practical use of the proposed methodology.     

Choosing instrumental variables in conditional moment restriction models

Properties of GMM estimators are sensitive to the choice of instrument. Using many instruments leads to high asymptotic asymptotic efficiency but can cause high bias and/or variance in small samples. In this paper we develop and implement asymptotic mean square error (MSE) based criteria for instrument selection in estimation of conditional moment restriction models. The models we consider include various nonlinear simultaneous equations models with unknown heteroskedasticity. We develop moment selection criteria for the familiar two-step optimal GMM estimator (GMM), a bias corrected version, and generalized empirical likelihood estimators (GEL), that include the continuous updating estimator (CUE) as a special case. We also find that the CUE has lower higher-order variance than the bias-corrected GMM estimator, and that the higher-order efficiency of other GEL estimators depends on conditional kurtosis of the moments.     

Statistical Inference in Nonparametric Frontier Models: The State of the Art

Efficiency scores of firms are measured by their distance to an estimated production frontier. The economic literature proposes several nonparametric frontier estimators based on the idea of enveloping the data (FDH and DEA-type estimators). Many have claimed that FDH and DEA techniques are non-statistical, as opposed to econometric approaches where particular parametric expressions are posited to model the frontier. We can now define a statistical model allowing determination of the statistical properties of the nonparametric estimators in the multi-output and multi-input case. New results provide the asymptotic sampling distribution of the FDH estimator in a multivariate setting and of the DEA estimator in the bivariate case. Sampling distributions may also be approximated by bootstrap distributions in very general situations. Consequently, statistical inference based on DEA/FDH-type estimators is now possible. These techniques allow correction for the bias of the efficiency estimators and estimation of confidence intervals for the efficiency measures. This paper summarizes the results which are now available, and provides a brief guide to the existing literature. Emphasizing the role of hypotheses and inference, we show how the results can be used or adapted for practical purposes.     

An application of approximate finite sample results to parameter estimation in a linear errors-in-variables model

In the simple errors-in-variables model the least squares estimator of the slope coefficient is known to be biased towards zero for finite sample size as well as asymptotically. In this paper we suggest a new corrected least squares estimator, where the bias correction is based on approximating the finite sample bias by a lower bound. This estimator is computationally very simple. It is compared with previously proposed corrected least squares estimators, where the correction aims at removing the asymptotic bias or the exact finite sample bias. For each type of corrected least squares estimators we consider the theoretical form, which depends on an unknown parameter, as well as various feasible forms. An analytical comparison of the theoretical estimators is complemented by a Monte Carlo study evaluating the performance of the feasible estimators. The new estimator proposed in this paper proves to be superior with respect to the mean squared error.     

On kernel nonparametric regression designed for complex survey data

In this article, we consider nonparametric regression analysis between two variables when data are sampled through a complex survey. While nonparametric regression analysis has been widely used with data that may be assumed to be generated from independently and identically distributed (iid) random variables, the methods and asymptotic analyses established for iid data need to be extended in the framework of complex survey designs. Local polynomial regression estimators are studied, which include as particular cases design-based versions of the Nadaraya–Watson estimator and of the local linear regression estimator. In this paper, special emphasis is given to the local linear regression estimator. Our estimators incorporate both the sampling weights and the kernel weights. We derive the asymptotic mean squared error (MSE) of the kernel estimators using a combined inference framework, and as a corollary consistency of the estimators is deduced. Selection of a bandwidth is necessary for the resulting estimators; an optimal bandwidth can be determined, according to the MSE criterion in the combined mode of inference. Simulation experiments are conducted to illustrate the proposed methodology and an application with the Canadian survey of labour and income dynamics is presented.     

Effective nonparametric estimation in the case of severely discretized data

Often economic data are discretized or rounded to some extent. This paper proposes a regression and a density estimator that work especially well when discretization causes conventional kernel-based estimators to behave poorly. The estimator proposed here is a weighted average of neighboring frequency estimators, and the weights are composed of cubic B-splines. Interestingly, we show that this estimator can have both a smaller bias and variance than frequency estimators. As a means to obtain asymptotic normality and rates of convergence, we assume that the discreteness becomes finer as the sample size increases.     

Model averaging based on James–Stein estimators

Existing model averaging methods are generally based on ordinary least squares (OLS) estimators. However, it is well known that the James–Stein (JS) estimator dominates the OLS estimator under quadratic loss, provided that the dimension of coefficient is larger than two. Thus, we focus on model averaging based on JS estimators instead of OLS estimators. We develop a weight choice method and prove its asymptotic optimality. A simulation experiment shows promising results for the proposed model average estimator.     

Use of minimum risk approach in the estimation of regression models with missing observations

This article considers a linear regression model with some missing observations on the response variable and presents two estimators of regression coefficients employing the approach of minimum risk estimation. Small disturbance asymptotic properties of these estimators along with the traditional unbiased estimator are analyzed and conditions, that are easy to check in practice, for the superiority of one estimator over the other are derived. Received May 2001     

Kernel-weighted GMM estimators for linear time series models

This paper analyzes the higher-order asymptotic properties of generalized method of moments (GMM) estimators for linear time series models using many lags as instruments. A data-dependent moment selection method based on minimizing the approximate mean squared error is developed. In addition, a new version of the GMM estimator based on kernel-weighted moment conditions is proposed. It is shown that kernel-weighted GMM estimators can reduce the asymptotic bias compared to standard GMM estimators. Kernel weighting also helps to simplify the problem of selecting the optimal number of instruments. A feasible procedure similar to optimal bandwidth selection is proposed for the kernel-weighted GMM estimator.     

Determination of Discrete Spectrum in a Random Field

We consider a two dimensional frequency model in a random field, which can be used to model textures and also has wide applications in Statistical Signal Processing. First we consider the usual least squares estimators and obtain the consistency and the asymptotic distribution of the least squares estimators. Next we consider an estimator, which can be obtained by maximizing the periodogram function. It is observed that the least squares estimators and the estimators obtained by maximizing the periodogram function are asymptotically equivalent. Some numerical experiments are performed to see how the results work for finite samples. We apply our results on simulated textures to observe how the different estimators perform in estimating the true textures from a noisy data.     

Calibration Estimation in Survey Sampling

Calibration estimation, where the sampling weights are adjusted to make certain estimators match known population totals, is commonly used in survey sampling. The generalized regression estimator is an example of a calibration estimator. Given the functional form of the calibration adjustment term, we establish the asymptotic equivalence between the functional-form calibration estimator and an instrumental variable calibration estimator where the instrumental variable is directly determined from the functional form in the calibration equation. Variance estimation based on linearization is discussed and applied to some recently proposed calibration estimators. The results are extended to the estimator that is a solution to the calibrated estimating equation. Results from a limited simulation study are presented.