Bivariate Sign Test for One-Sample Bivariate Location Model Using Ranked Set Sample

In this article, we introduce a bivariate sign test for the one-sample bivariate location model using a bivariate ranked set sample (BVRSS). We show that the proposed test is asymptotically more efficient than its counterpart sign test based on a bivariate simple random sample (BVSRS). The asymptotic null distribution and the non centrality parameter are derived. The asymptotic distribution of the vector of sample median as an estimator of the locations of the bivariate model is introduced. Theoretical and numerical comparisons of the asymptotic efficiency of the BVRSS sign test with respect to the BVSRS sign test are also given.     

Estimation of regression parameters in generalized linear models for cluster correlated data with measurement error

Liang and Zeger (1986) introduced a class of estimating equations that gives consistent estimates of regression parameters and of their asymptotic variances in the class of generalized linear models for cluster correlated data. When the independent variables or covariates in such models are subject to measurement errors, the parameter estimates obtained from these estimating equations are no longer consistent. To correct for the effect of measurement errors, an estimator with smaller asymptotic bias is constructed along the lines of Stefanski (1985), assuming that the measurement error variance is either known or estimable. The asymptotic distribution of the bias-corrected estimator and a consistent estimator of its asymptotic variance are also given. The special case of a binary logistic regression model is studied in detail. For this case, methods based on conditional scores and quasilikelihood are also extended to cluster correlated data. Results of a small simulation study on the performance of the proposed estimators and associated tests of hypotheses are reported.     

Median regression analysis from data with left and right censored observations

Median regression models provide a robust alternative to regression based on the mean. We propose a methodology for fitting a median regression model from data with both left and right censored observations, in which the left censoring variable is always observed. First we set up an adjusted least absolute deviation estimating function using the inverse censoring weighted approach, whose solution specifies the estimator. We derive the consistency and asymptotic normality of the proposed estimator and describe the inference procedure for the regression parameter. Finally, we check the finite sample performance of the proposed procedure through simulation.     

On tests for normality of experimental error in ridge regression

Considered are tests for normality of the errors in ridge regression. If an intercept is included in the model, it is shown that test statistics based on the empirical distribution function of the ridge residuals have the same limiting distribution as in the one-sample test for normality with estimated mean and variance. The result holds with weak assumptions on the behavior of the independent variables; asymptotic normality of the ridge estimator is not required.     

Guaranteed Local Maximum Likelihood Detection of a Change Point in Nonparametric Logistic Regression

We consider nonparametric logistic regression and propose a generalized likelihood test for detecting a threshold effect that indicates a relationship between some risk factor and a defined outcome above the threshold but none below it. One important field of application is occupational medicine and in particular, epidemiological studies. In epidemiological studies, segmented fully parametric logistic regression models are often threshold models, where it is assumed that the exposure has no influence on a response up to a possible unknown threshold, and has an effect beyond that threshold. Finding efficient methods for detection and estimation of a threshold is a very important task in these studies. This article proposes such methods in a context of nonparametric logistic regression. We use a local version of unknown likelihood functions and show that under rather common assumptions the asymptotic power of our test is one. We present a guaranteed non asymptotic upper bound for the significance level of the proposed test. If applying the test yields the acceptance of the conclusion that there was a change point (and hence a threshold limit value), we suggest using the local maximum likelihood estimator of the change point and consider the asymptotic properties of this estimator.     

Multivariate regression models for discrete and continuous repeated measurements

A general class of multivariate regression models is considered for repeated measurements with discrete and continuous outcome variables. The proposed model is based on the seemingly unrelated regression model (Zellner, 1962) and an extension of the model of Park and Woolson(1992). The regression parameters of the model are consistently estimated using the two-stage least squares method. When the out come variables are multivariate normal, the two-stage estimator reduces to Zellner’s two-stage estimator. As a special case, we consider the marginal distribution described by Liang and Zeger (1986). Under this this distributional assumption, we show that the two-stage estimator has similar asymptotic properties and comparable small sample properties to Liang and Zeger's estimator. Since the proposed approach is based on the least squares method, however, any distributional assumption is not required for variables outcome variables. As a result, the proposed estimator is more robust to the marginal distribution of outcomes.     

An Equivalence of Conditional and Unconditional Maximum Likelihood Estimators via Infinite Replication of Observations

Motivated by an application with complex survey data, we show that for logistic regression with a simple matched-pairs design, infinitely replicating observations and maximizing the conditional likelihood results in an estimator exactly identical to the unconditional maximum likelihood estimator based on the original sample, which is inconsistent. Therefore, applying conditional likelihood methods to a pseudosample with observations replicated a large number of times can lead to an inconsistent estimator; this casts doubt on one possible approach to conditional logistic regression with complex survey data. We speculate that for more general designs, an asymptotic equivalence holds.     

An omnibus test of goodness-of-fit for conditional distributions with applications to regression models

We introduce an omnibus goodness-of-fit test for statistical models for the conditional distribution of a random variable. In particular, this test is useful for assessing whether a regression model fits a data set on all its assumptions. The test is based on a generalization of the Cramér–von Mises statistic and involves a local polynomial estimator of the conditional distribution function. First, the uniform almost sure consistency of this estimator is established. Then, the asymptotic distribution of the test statistic is derived under the null hypothesis and under contiguous alternatives. The extension to the case where unknown parameters appear in the model is developed. A simulation study shows that the test has good power against some common departures encountered in regression models. Moreover, its power is comparable to that of other nonparametric tests designed to examine only specific departures.     

Statistical inference for restricted partially linear varying coefficient errors-in-variables models

As a useful extension of partially linear models and varying coefficient models, the partially linear varying coefficient model is useful in statistical modelling. This paper considers statistical inference for the semiparametric model when the covariates in the linear part are measured with additive error and some additional linear restrictions on the parametric component are available. We propose a restricted modified profile least-squares estimator for the parametric component, and prove the asymptotic normality of the proposed estimator. To test hypotheses on the parametric component, we propose a test statistic based on the difference between the corrected residual sums of squares under the null and alterative hypotheses, and show that its limiting distribution is a weighted sum of independent chi-square distributions. We also develop an adjusted test statistic, which has an asymptotically standard chi-squared distribution. Some simulation studies are conducted to illustrate our approaches.     

A heteroscedastic structural errors-in-variables model with equation error

It is not uncommon with astrophysical and epidemiological data sets that the variances of the observations are accessible from an analytical treatment of the data collection process. Moreover, in a regression model, heteroscedastic measurement errors and equation errors are common situations when modelling such data. This article deals with the limiting distribution of the maximum-likelihood and method-of-moments estimators for the line parameters of the regression model. We use the delta method to achieve it, making it possible to build joint confidence regions and hypothesis testing. This technique produces closed expressions for the asymptotic covariance matrix of those estimators. In the moment approach we do not assign any distribution for the unobservable covariate while with the maximum-likelihood approach, we assume a normal distribution. We also conduct simulation studies of rejection rates for Wald-type statistics in order to verify the test size and power. Practical applications are reported for a data set produced by the Chandra observatory and also from the WHO MONICA Project on cardiovascular disease.     

A study on the least squares estimator of multivariate isotonic regression function

Consider the problem of pointwise estimation of f in a multivariate isotonic regression model Z=f(X₁,…,X_d)+ϵ, where Z is the response variable, f is an unknown nonparametric regression function, which is isotonic with respect to each component, and ϵ is the error term. In this article, we investigate the behavior of the least squares estimator of f. We generalize the greatest convex minorant characterization of isotonic regression estimator for the multivariate case and use it to establish the asymptotic distribution of properly normalized version of the estimator. Moreover, we test whether the multivariate isotonic regression function at a fixed point is larger (or smaller) than a specified value or not based on this estimator, and the consistency of the test is established. The practicability of the estimator and the test are shown on simulated and real data as well.     

Preliminary phi-divergence test estimators for linear restrictions in a logistic regression model

The problem of estimation of the parameters in a logistic regression model is considered under multicollinearity situation when it is suspected that the parameter of the logistic regression model may be restricted to a subspace. We study the properties of the preliminary test based on the minimum ϕ -divergence estimator as well as in the ϕ -divergence test statistic. The minimum ϕ -divergence estimator is a natural extension of the maximum likelihood estimator and the ϕ -divergence test statistics is a family of the test statistics for testing the hypothesis that the regression coefficients may be restricted to a subspace.     

Change point problems in the model of logistic regression

The paper considers generalized maximum likelihood asymptotic power one tests which aim to detect a change point in logistic regression when the alternative specifies that a change occurred in parameters of the model. A guaranteed non-asymptotic upper bound for the significance level of each of the tests is presented. For cases in which the test supports the conclusion that there was a change point, we propose a maximum likelihood estimator of that point and present results regarding the asymptotic properties of the estimator. An important field of application of this approach is occupational medicine, where for a lot chemical compounds and other agents, so-called threshold limit values (or TLVs) are specified.We demonstrate applications of the test and the maximum likelihood estimation of the change point using an actual problem that was encountered with real data.     

Bootstrapping quantiles in a fixed design regression model with censored data

We consider the problem of estimating the quantiles of a distribution function in a fixed design regression model in which the observations are subject to random right censoring. The quantile estimator is defined via a conditional Kaplan-Meier type estimator for the distribution at a given design point. We establish an a.s. asymptotic representation for this quantile estimator, from which we obtain its asymptotic normality. Because a complicated estimation procedure is necessary for estimating the asymptotic bias and variance, we use a resampling procedure, which provides us, via an asymptotic representation for the bootstrapped estimator, with an alternative for the normal approximation.     

Ratio detection for mean change in α mixing observations

A ratio test based on the indicators of the data minus the sample median is proposed to detect the change in the mean of α-mixing stochastic sequences. The asymptotic distribution of the test is derived under the null hypothesis. The consistency of the proposed test is also obtained under the hypothesis that μ changes at some unknown time. We also propose a consistent estimator for the change point on the ratio test. Simulations demonstrate that the test and the estimator behaves well for heavy-tailed sequences. At last, an empirical application demonstrate the validity of the test and the estimator.     

On the Breslow–Holubkov estimator

Breslow and Holubkov (J Roy Stat Soc B 59:447–461 1997a) developed semiparametric maximum likelihood estimation for two-phase studies with a case–control first phase under a logistic regression model and noted that, apart for the overall intercept term, it was the same as the semiparametric estimator for two-phase studies with a prospective first phase developed in Scott and Wild (Biometrica 84:57–71 1997). In this paper we extend the Breslow–Holubkov result to general binary regression models and show that it has a very simple relationship with its prospective first-phase counterpart. We also explore why the design of the first phase only affects the intercept of a logistic model, simplify the calculation of standard errors, establish the semiparametric efficiency of the Breslow–Holubkov estimator and derive its asymptotic distribution in the general case.     

Rao's score test with nonparametric density estimators

Traditionally, Rao's score (RS) tests are constructed under a parametric specification of the probability density function. We estimate the density function by a non-parametric estimator and consider a semi-parametric Rao's score (SPRS) test for a set of hypotheses concerning the parametric model. The asymptotic distribution of the SPRS test is analyzed. Further, for the regression model, we carry out a set of Monte Carlo experiments to analyze the size and power of the SPRS test in small samples. The robustness of SPRS test to the choice of the density estimator is also analyzed.     

An improved and efficient biased estimation technique in logistic regression model

Abstract

In this article, we propose a new improved and efficient biased estimation method which is a modified restricted Liu-type estimator satisfying some sub-space linear restrictions in the binary logistic regression model. We study the properties of the new estimator under the mean squared error matrix criterion and our results show that under certain conditions the new estimator is superior to some other estimators. Moreover, a Monte Carlo simulation study is conducted to show the performance of the new estimator in the simulated mean squared error and predictive median squared errors sense. Finally, a real application is considered.     

The asymptotic behaviour of a class of L.-estimators under long-range dependence

This paper obtains asymptotic representations of a class of L-estimators in a linear regression model when the errors are a function of long-range-dependent Gaussian random variables. These representations are then used to address some of the efficiency robustness properties of L-estimators compared to the least-squares estimator. It is observed that under the Gaussian error distribution, each member of the class has the same asymptotic efficiency as that of the least-squares estimator. The results are obtained as a consequence of the asymptotic uniform linearity of some weighted empirical processes based on long-range-dependent random variables.     

Robust M-estimators of scale: Minimax bias versus maximal variance

In the location-scale estimation problem, we study robustness properties of M-estimators of the scale parameter under unknown ?-contamination of a fixed symmetric unimodal error distribution F₀. Within a general class of M-estimators, the estimator with minimax asymptotic bias is shown to lie within the subclass of α-interquantile ranges of the empirical distribution symmetrized about the sample median. Our main result is that as ? → 0, the limiting minimax asymptotic bias estimator is sometimes (e.g., when F_o is Cauchy), but not always, the median absolute deviation about the median. It is also shown that contamination in the neighbourhood of a discontinuity of the influence function of a minimax bias estimator can sometimes inflate the asymptotic variance beyond that achieved by placing all the ?-contamination at infinity. This effect is quantified by a new notion of asymptotic efficiency that takes into account the effect of infinitesimal contamination of the parametric model for the error distribution.