Adaptive estimators for parameters of the autoregression function of a Markov chain

Suppose we observe an ergodic Markov chain on the real line, with a parametric model for the autoregression function, i.e. the conditional mean of the transition distribution. If one specifies, in addition, a parametric model for the conditional variance, one can define a simple estimator for the parameter, the maximum quasi-likelihood estimator. It is robust against misspecification of the conditional variance, but not efficient. We construct an estimator which is adaptive in the sense that it is efficient if the conditional variance is misspecified, and asymptotically as good as the maximum quasi-likelihood estimator if the conditional variance is correctly specified. The adaptive estimator is a weighted nonlinear least-squares estimator, with weights given by predictors for the conditional variance.     

Does bias reduction with external estimator of second order parameter work for endpoint?

Bias reduction estimation for tail index has been studied in the literature. One method is to reduce bias with an external estimator of the second order regular variation parameter; see Gomes and Martins [2002. Asymptotically unbiased estimators of the tail index based on external estimation of the second order parameter. Extremes 5(1), 5–31]. It is known that negative extreme value index implies that the underlying distribution has a finite right endpoint. As far as we know, there exists no bias reduction estimator for the endpoint of a distribution. In this paper, we study the bias reduction method with an external estimator of the second order parameter for both the negative extreme value index and endpoint simultaneously. Surprisingly, we find that this bias reduction method for negative extreme value index requires a larger order of sample fraction than that for positive extreme value index. This finding implies that this bias reduction method for endpoint is less attractive than that for positive extreme value index. Nevertheless, our simulation study prefers the proposed bias reduction estimators to the biased estimators in Hall [1982. On estimating the endpoint of a distribution. Ann. Statist. 10, 556–568].     

Estimation of a linear transformation and an associated distributional problem

A multivariate “errors in variables” regression model is proposed which generalizes a model previously considered by Gleser and Watson (1973). Maximum likelihood estimators [MLE's] for the parameters of this model are obtained, and the consistency properties of these estimators are investigated. Distribution of the MLE of the “error” variance is obtained in a simple case while the mean and the variance of the estimator are obtained in this case without appealing to the exact distribution.     

On the simple linear regression model with correlated measurement errors

The simple linear regression model with measurement error has been subject to much research. In this work we will focus on this model when the error in the explanatory variable is correlated with the error in the regression equation. Specifically, we are interested in the comparison between the ordinary errors-in-variables estimator of the regression coefficient β$\beta$ and the estimator that takes account of the correlation between the errors. Based on large sample approximations, we compare the estimators and find that the estimator that takes account of the correlation should be preferred in most situations. We also compare the estimators in small sample situations. This is done by stochastic simulation. The results show that the estimators behave quite similarly in most of the simulated situations, but that the ordinary errors-in-variables estimator performs considerably worse than the estimator that takes account of the correlation for certain parameter combinations. In addition, we look briefly into the bias introduced by ignoring correlated errors when computing sample correlations, and in predictions.     

On multivariate nonlinear regression models with stationary correlated errors

In this paper we consider the statistical analysis of multivariate multiple nonlinear regression models with correlated errors, using Finite Fourier Transforms. Consistency and asymptotic normality of the weighted least squares estimates are established under various conditions on the regressor variables. These conditions involve different types of scalings, and the scaling factors are obtained explicitly for various types of nonlinear regression models including an interesting model which requires the estimation of unknown frequencies. The estimation of frequencies is a classical problem occurring in many areas like signal processing, environmental time series, astronomy and other areas of physical sciences. We illustrate our methodology using two real data sets taken from geophysics and environmental sciences. The data we consider from geophysics are polar motion (which is now widely known as “Chandlers Wobble”), where one has to estimate the drift parameters, the offset parameters and the two periodicities associated with elliptical motion. The data were first analyzed by Arato, Kolmogorov and Sinai who treat it as a bivariate time series satisfying a finite order time series model. They estimate the periodicities using the coefficients of the fitted models. Our analysis shows that the two dominant frequencies are 12 h and 410 days. The second example, we consider is the minimum/maximum monthly temperatures observed at the Antarctic Peninsula (Faraday/Vernadsky station). It is now widely believed that over the past 50 years there is a steady warming in this region, and if this is true, the warming has serious consequences on ecology, marine life, etc. as it can result in melting of ice shelves and glaciers. Our objective here is to estimate any existing temperature trend in the data, and we use the nonlinear regression methodology developed here to achieve that goal.     

On the stability of the jackknife variance estimator in ratio estimation

The stability of a slightly modified version of the usual jackknife variance estimator is evaluated exactly in small samples under a suitable linear regression model and compared with that of two different linearization variance estimators. Depending on the degree of heteroscedasticity of the error variance in the model, the stability of the jackknife variance estimator is found to be somewhat comparable to that of one or the other of the linearization variance estimators under conditions especially favorable to ratio estimation (i.e., regression approximately through the origin with a relatively small coefficient of variation in the x population). When these conditions do not hold, however, the jackknife variance estimator is found to be less stable than either of the linearization variance estimators.     

A Jackknife Method for Estimation of Variance Components

This paper concerns a method of estimation of variance components in a random effect linear model. It is mainly a resampling method and relies on the Jackknife principle. The derived estimators are presented as least squares estimators in an appropriate linear model, and one of them appears as a MINQUE (Minimum Norm Quadratic Unbiased Estimation) estimator. Our resampling method is illustrated by an example given by C. R. Rao [7] and some optimal properties of our estimator are derived for this example. In the last part, this method is used to derive an estimation of variance components in a random effect linear model when one of the components is assumed to be known.     

Statistical inference for a semiparametric measurement error regression model with heteroscedastic errors

Efficient inference for regression models requires that the heteroscedasticity be taken into account. We consider statistical inference under heteroscedasticity in a semiparametric measurement error regression model, in which some covariates are measured with errors. This paper has multiple components. First, we propose a new method for testing the heteroscedasticity. The advantages of the proposed method over the existing ones are that it does not need any nonparametric estimation and does not involve any mismeasured variables. Second, we propose a new two-step estimator for the error variances if there is heteroscedasticity. Finally, we propose a weighted estimating equation-based estimator (WEEBE) for the regression coefficients and establish its asymptotic properties. Compared with existing estimators, the proposed WEEBE is asymptotically more efficient, avoids undersmoothing the regressor functions and requires less restrictions on the observed regressors. Simulation studies show that the proposed test procedure and estimators have nice finite sample performance. A real data set is used to illustrate the utility of our proposed methods.     

An estimator of the common mean of two normal populations

In this paper we consider the estimation of the common mean of two normal populations when the variances are unknown. If it is known that one specified variance is smaller than the other, then it is possible to modify the Graybill-Deal estimator in order to obtain a more efficient estimator. One such estimator is proposed by Mehta and Gurland (1969). We prove that this estimator is more efficient than the Graybill-Deal estimator under the condition that one variance is known to be less than the other.     

Testing Hypotheses for Variance Components in Mixed Linear Models

In the paper the problem of testing hypotheses for variance components in mixed linear models is considered. It is assumed that covariance matrices commute after using the usual invariance procedure with respect to the group of translations. The test for vanishing of single variance component is based on the locally best quadratic unbiased estimator of this component and rejects hypothesis if the ratio of positive and negative part of this estimator is sufficiently large. The power of this test with powers of other four tests for two-way classification models corresponding to block design is compared.     

Nonparametric estimation of the conditional distribution function in a semiparametric censorship model

In this paper we propose a new nonparametric estimator of the conditional distribution function under a semiparametric censorship model. We establish an asymptotic representation of the estimator as a sum of iid random variables, balanced by some kernel weights. This representation is used for obtaining large sample results such as the rate of uniform convergence of the estimator, or its limit distributional law. We prove that the new estimator outperforms the conditional Kaplan–Meier estimator for censored data, in the sense that it exhibits lower asymptotic variance. Illustration through real data analysis is provided.     

Likelihood ratio test against simple stochastic ordering among several multinomial populations

Stochastic ordering between probability distributions has been widely studied in the past 50 years. Because it is often easy to make valuable judgments when such orderings exist, it is desirable to recognize their existence and to model distributional structures under them. Likelihood ratio test is the most commonly used method to test hypotheses involving stochastic orderings. Among the various formally defined notions of stochastic ordering, the least stringent is simple stochastic ordering. In this paper, we consider testing the hypothesis that all multinomial populations are identically distributed against the alternative that they are in simple stochastic ordering. We construct likelihood ratio test statistic for this hypothesis test problem, provide limit form of the objective function corresponding to the test statistic and show that the test statistic is asymptotically distributed as a mixture of chi-squared distributions, i.e., a chi-bar-squared distribution.     

Model Selection,Transformations and Variance Estimation in Nonlinear Regression

The results of analyzing experimental data using a parametric model may heavily depend on the chosen model for regression and variance functions, moreover also on a possibly underlying preliminary transformation of the variables. In this paper we propose and discuss a complex procedure which consists in a simultaneous selection of parametric regression and variance models from a relatively rich model class and of Box-Cox variable transformations by minimization of a cross-validation criterion. For this it is essential to introduce modifications of the standard cross-validation criterion adapted to each of the following objectives: 1. estimation of the unknown regression function, 2. prediction of future values of the response variable, 3. calibration or 4. estimation of some parameter with a certain meaning in the corresponding field of application. Our idea of a criterion oriented combination of procedures (which usually if applied, then in an independent or sequential way) is expected to lead to more accurate results. We show how the accuracy of the parameter estimators can be assessed by a “moment oriented bootstrap procedure", which is an essential modification of the “wild bootstrap” of Härdle and Mammen by use of more accurate variance estimates. This new procedure and its refinement by a bootstrap based pivot (“double bootstrap”) is also used for the construction of confidence, prediction and calibration intervals. Programs written in Splus which realize our strategy for nonlinear regression modelling and parameter estimation are described as well. The performance of the selected model is discussed, and the behaviour of the procedures is illustrated, e.g., by an application in radioimmunological assay.     

Local Influence Analysis for The Ridge Regression Under Stochastic Linear Restrictions

In this article, we assess the local influence for the ridge regression of linear models with stochastic linear restrictions in the spirit of Cook by using the log-likelihood of the stochastic restricted ridge regression estimator. The diagnostics under the perturbations of constant variance, responses and individual explanatory variables are derived. We also assess the local influence of the stochastic restricted ridge regression estimator under the approach suggested by Billor and Loynes. At the end, a numerical example on the Longley data is given to illustrate the theoretic results.     

The Joint Treatment of Exact and Stochastic Restrictions

In regression analysis both exact and stochastic extraneous information may be represented via restrictions on the parameters of a linear model which then may be estimated by applying constrained generalized least squares. It is shown that this estimator can be recast as a computationally simpler estimator that is a combination of the ordinary least squares estimator and the discrepancy between the OLS estimator and both types of restrictions. The variance of the restricted parameters is explicitly shown to depend on the variance of the extraneous information.     

Variance reduction for quantile estimates in simulations via nonlinear controls

Linear controls are a well known simple technique for achieving variance reduction in computer simulation. Unfortunately the effectiveness of a linear control depends upon the correlation between the statistic of interest and the control, which is often low. Since statistics often have a nonlinear relation-ship with the potential control variables, nonlinear controls offer a means for improvement over linear controls. This paper focuses on the use of nonlinear controls for reducing the variance of quantile estimates in simulation. It is shown that one can substantially reduce the analytic effort required to develop a nonlinear control from a quantile estimator by using a strictly monotone transformation to create the nonlinear control. It is also shown that as one increases the sample size for the quantile estimator, the asymptotic multivariate normal distribution of the quantile of interest and the control reduces the effectiveness of the nonlinear control to that of the linear control. However, the data has to be sectioned to obtain an estimate of the variance of the controlled quantile estimate. Graphical methods are suggested for selecting the section size that maximizes the effectiveness of the nonlinear control     

Universal admissibility in second order asymptotics

Some sufficient conditions for an estimator to be universally second order admissible are derived. Those sufficient conditions consist of the elementary integrals with respect to the Fisher information and the limits of some functions characterized by the dealt statistical model, and thus can be checked with comparative ease. In location model and scale model, the sufficient condition for the linear estimator with respect to the maximum likelihood estimator (MLE) to be universally second order admissible is given. Furthermore, a guide for classifying any estimator into either the universal admissibility or the non-universal admissibility is proposed.     

Variance Estimation Based on Invariance Principles

Consistent variance estimators for certain stochastic processes are suggested using the fact that (weak or strong) invariance principles may be available. Convergence rates are also derived, the latter being essentially determined by the approximation rates in the corresponding invariance principles. As an application, a change point test in a simple AMOC renewal model is briefly discussed, where variance estimators possessing good enough convergence rates are required.     

Estimation of variance components in an unbalanced one-way classification

In the unbalanced one-way random effects model the weighted least squares approach with estimated weights is used to develop a relatively simple estimator of variance components. As the number of classes increases, the proposed estimator is seen not only to be best asymptotically normal but also to be asymptotically equivalent to the maximum likelihood estimator.     

Probability density estimation with data missing at random when covariables are present

This paper addresses the problem of the probability density estimation in the presence of covariates when data are missing at random (MAR). The inverse probability weighted method is used to define a nonparametric and a semiparametric weighted probability density estimators. A regression calibration technique is also used to define an imputed estimator. It is shown that all the estimators are asymptotically normal with the same asymptotic variance as that of the inverse probability weighted estimator with known selection probability function and weights. Also, we establish the mean squared error (MSE) bounds and obtain the MSE convergence rates. A simulation is carried out to assess the proposed estimators in terms of the bias and standard error.