On the asymptotic normality of the R-estimators of the slope parameters of simple linear regression models with associated errors

ABSTRACT

The purpose of this paper is to prove, under mild conditions, the asymptotic normality of the rank estimator of the slope parameter of a simple linear regression model with stationary associated errors. This result follows from a uniform linearity property for linear rank statistics that we establish under general conditions on the dependence of the errors. We prove also a tightness criterion for weighted empirical process constructed from associated triangular arrays. This criterion is needed for the proofs which are based on that of Koul [Behavior of robust estimators in the regression model with dependent errors. Ann Stat. 1977;5(4):681–699] and of Louhichi [Louhichi S. Weak convergence for empirical processes of associated sequences. Ann Inst Henri Poincaré Probabilités Statist. 2000;36(5):547–567].     

Crossings of a height and related bank order statistics

Former results on BAHADUR efficiency of signed rank tests are carried over to the class of two-sample rank tests. It is shown that the two-sample rank tests are asymptotically optimal at alternatives far away from the hypothesis under fairly general conditions. Surprisingly, the median test appears to be optimal only in case of equal sample sizes.     

Asymptotic Properties of Generalized Multivariate Rank Statistics

This article discusses generalization of the well-known multivariate rank statistics under right-censored data case. Empirical process representation used to get the generalization. The marginal distribution functions are estimated by Kaplan–Meier estimators. Sufficient conditions for asymptotic normality of the generalized multivariate rank statistics under independently right censored data are specified. Several auxiliary results on sup-norm convergence of Kaplan–Meier estimators in randomly exhausting regions are given too.     

Autoregressive moving-average convolution processes on the unit circle

ARMA convolution models for processes in continuous space (in this case the unit circle) and discrete time are derived as a natural extension of the usual Box-Jenkins models. Both weakly time-stationary and nonstationary processes are considered. Sufficient conditions for the existence of weakly time-stationary AR_cMA_c processes are derived, and the covariance functions for some processes are computed. It is demonstrated that the usual scalar and multivariate ARMA processes can be embedded within the larger class of AR_CMA_c models. A possible application of these models to sea-surface temperature prediction is discussed.     

Asymptotic Likelihood Based Inference for Co-integrated Homogenous Gaussian Diffusions

In this paper we consider inference for a multivariate Gaussian homogenous diffusion which is co-integrated, i.e. admits a representation in terms of stable relations (ergodic diffusions) plus Brownian motions. We show that inference on co-integration rank (the number of stable relations) in continuous time can be based on existing asymptotic distributions from discrete time co-integration analysis. Likewise the asymptotic distributions of the co-integration parameters are shown to be mixed Gaussian. Special attention is given to the parametrization of the drift terms. It is shown that the asymptotic distribution of the co-integration rank test statistic does not depend on the level of the process as a result of the chosen parametrization.     

Model averaging based on rank

In this paper, we investigate model selection and model averaging based on rank regression. Under mild conditions, we propose a focused information criterion and a frequentist model averaging estimator for the focused parameters in rank regression model. Compared to the least squares method, the new method is not only highly efficient but also robust. The large sample properties of the proposed procedure are established. The finite sample properties are investigated via extensive Monte Claro simulation study. Finally, we use the Boston Housing Price Dataset to illustrate the use of the proposed rank methods.     

Likelihood Analysis of the I(2) Model

The I (2) model is defined as a submodel of the general vector autoregressive model, by two reduced rank conditions. The model describes stochastic processes with stationary second difference. A parametrization is suggested which makes likelihood inference feasible. Consistency of the maximum likelihood estimator is proved, and the asymptotic distribution of the maximum likelihood estimator is given. It is shown that the asymptotic distribution is either Gaussian, mixed Gaussian or, in some cases, even more complicated.     

The Input Evaluation of Generalized Bernoulli Processes for Salary Lines Construction by Means of Continuous Time Generalized Non-Homogeneous Semi-Markov Processes

Salary line forecasting assumes a relevant role in manpower and in pension funds Previously, the authors presented a generalized Bernoulli process, useful for forecasting the evolution of salary lines, taking into account the salary costs, the number of workers at each rank and the probability transitions between the ranks. The problem with applying this model is constructing the probability of transition between the grades. In this article, we will present a model that allows obtaining these probabilities by means of the solution of the evolution equation of a generalization of continuous time non-homogeneous semi-Markov processes.     

A simple algorithm for the adaptation of scores and power behavior of the corresponding rank test

The purpose of this paper is twofold:On one hand we want to give a very simple algorithm for evaluating a special rank estimator of the type given in Behnen, Neuhaus, and Ruymgaart (1983) for the approximate optimal choice of the scores-generating function of a two-sample linear rank test for the general testing problem H_o:F=G versus H₁:F ≤ G, F ≠ G, in order to demonstrate that the corresponding adaptive rank statistic is simple enough for practical applications. On the other hand we prove the asymptotic normality of the adaptive rank statistic under H (leading to approximate critical values) and demonstrate the adaptive behavior of the corresponding rank test by a Monte Carlo power simulation for sample sizes as low as m=10, n=10.     

On some classifiers based on multivariate ranks

Non parametric approaches to classification have gained significant attention in the last two decades. In this paper, we propose a classification methodology based on the multivariate rank functions and show that it is a Bayes rule for spherically symmetric distributions with a location shift. We show that a rank-based classifier is equivalent to optimal Bayes rule under suitable conditions. We also present an affine invariant version of the classifier. To accommodate different covariance structures, we construct a classifier based on the central rank region. Asymptotic properties of these classification methods are studied. We illustrate the performance of our proposed methods in comparison to some other depth-based classifiers using simulated and real data sets.     

Comparing mean ranks for repeated measures data

Rank tests are considered that compare t treatments in repeated measures designs. A statistic is given that contains as special cases several that have been proposed for this problem, including one that corresponds to the randomized block ANOVA statistic applied to the rank transformed data. Another statistic is proposed, having a null distribution holding under more general conditions, that is the rank transform of the Hotelling statistic for repeated measures. A statistic of this type is also given for data that are ordered categorical rather than fully rankedo Unlike the Friedman statistic, the statistics discussed in this article utilize a single ranking of the entire sample. Power calculations for an underlying normal distribution indicate that the rank transformed ANOVA test can be substantially more powerful than the Friedman test.     

Asymptotic properties of a rank estimate in linear regression with symmetric non-identically distributed errors

In this article, a simple linear regression model with independent and symmetric but non-identically distributed errors is considered. Asymptotic properties of the rank regression estimate defined in Jaeckel [Estimating regression coefficients by minimizing the dispersion of the residuals, Ann. Math. Statist. 43 (1972), pp. 1449–1458] are studied. We show that the studied estimator is consistent and asymptotically normally distributed. The cases of bounded and unbounded score functions are examined separately. The regularity conditions of the article are exemplified for finite mixture distributions.     

A note on the multivariate multisample rank sum and median tests

The muitivariate nonparametric tests analogous to the univar-iate rank sum test and median test are contained in Puri and Sen (1970). These tests provided a practical alternative for the analysis of multivariate data when the assumptions of parametric methods are not satisfied.

In this paper maximum values for L_Nthe asymptotic chi-Square test statistic for both the Multivariate Multisample Rank Sum Test (MMRST) and the Multivariate Multisample Median Test (MMMT) are developed.     

Nonparametric weighted symmetry tests

Weighted symmetry is an extension of the classical notion of symmetry in which the tails of a distribution are similar, up to a scaling factor. The authors develop test statistics of weighted symmetry based on empirical processes. The finite‐dimensional distributions of the proposed statistics are either non‐parametric or conditionally nonparametric, according as the parameters of weighted symmetry are known or estimated. Asymptotically, the distributions of the processes behave like Brownian bridges or motions, leading to familiar distributions for the proposed test statistics. The authors also establish the asymptotic normality of Hodges‐Lehmann type estimators based on a generalization of the Wilcoxon signed rank test. Furthermore, they propose density estimators in mat setting.     

Comparisons of the asymptotic efficiencies of two sample tests for discrete distributions

The asymptotic efficiencies are computed for several popular two sample rank tests when the underlying distributions are Poisson, binomial, discrete uniform, and negative binomial The rank tests examined include the Mann-Whitney test, the van der Waerden test, and the median test. Three methods for handling ties are discussed and compared. The computed asymptotic efficiencies apply also to the k-sample extensions of the above tests, such as the Kruskal-Wallis test, etc.     

Rank procedures in many population forced discrimination problems

In this paper the rank method for forced discrimination in two population problems, introduced by Randies, Broffitt, Ramberg and Hogg (1978), is extended to cover settings involving more than two populations. Several methods of ranking are compared to the normal theory procedure in a Monte Carlo study. Asymptotic theory is included which confirms that the rank method does balance the limiting probabilities of misclassification in a two population setting.     

Stochastic equivalence of ranking methods

The asymptotic behavior of linear rank statistics for comparing the locations of two populations, where the observations are ranked jointly with other populations, is considered. Under certain conditions, the asymptotic behavior of these statistics does not depend on which other populations are included in the ranking. In particular, the difference of a pair of these statistics, with the same score function, but based on two different rankings, converges to zero in probability under Pitman alternatives and Chernoff-Savage conditions on the scores and underlying distributions.     

The Uniform-Laplace Distribution and the D-Test

A new two-sample rank test for location is proposed. This test, called the D-test, is asymptotically efficient for underlying densities which follow a “flat-topped” Laplace distribution. The D-statistic is simple to compute, and the test may be suitable when there is censoring. The D-test includes the median test as a special case.     

Tests for cointegration rank and choice of the alternative

This paper discusses likelihood-ratio (LR) tests on the cointegrating (CI) rank which consider any possible dimension of the CI rank under the alternative. The trace test and lambda-max test are obtained as special cases. Limit quantiles for all the tests in the class are derived. It is found that any of these tests can be used to construct an estimator of the CI rank, with no differences in asymptotic properties when the alternative is fixed. The properties of the class of tests are investigated by local asymptotic analysis, a simulation study and an empirical illustration. It is found that all the tests in the class have comparable power, which deteriorates substantially as the number of random walks increases. Tests constructed for a specific class of alternatives present minor power gains for alternatives in the class, and require the alternative to be far from the null. No test in this class is found to be asymptotically (in-)admissible. Some of the new tests in the class can also be arranged to give a constrained estimator of the CI rank, that restricts the minimum number of common trends. The power gains that these tests can obtain by constraining the minimum number of common trends appears to be limited and outweighted by the risk of inconsistency induced by the constrains. As a consequence, no value of the CI rank should be left untested, unless it can be excluded beyond any reasonable doubt.     

Robust Smoothed Rank Estimation Methods for Accelerated Failure Time Model Allowing Clusters

Smoothed Gehan rank estimation methods are widely used in accelerated failure time (AFT) models with/without clusters. However, most methods are sensitive to outliers in the covariates. In order to solve this problem, we propose robust approaches based on the smoothed Gehan rank estimation methods for the AFT model, allowing for clusters by employing two different weight functions. Simulation studies show that the proposed methods outperform existing smoothed rank estimation methods regarding their biases and standard deviations when there are outliers in the covariates. The proposed methods are also applied to a real dataset from the “Major cardiovascular interventions” study.