Testing for monotone increasing hazard rate

A test of the null hypothesis that a hazard rate is monotone nondecreasing, versus the alternative that it is not, is proposed. Both the test statistic and the means of calibrating it are new. Unlike previous approaches, neither is based on the assumption that the null distribution is exponential. Instead, empirical information is used to effectively identify and eliminate from further consideration parts of the line where the hazard rate is clearly increasing; and to confine subsequent attention only to those parts that remain. This produces a test with greater apparent power, without the excessive conservatism of exponential-based tests. Our approach to calibration borrows from ideas used in certain tests for unimodality of a density, in that a bandwidth is increased until a distribution with the desired properties is obtained. However, the test statistic does not involve any smoothing, and is, in fact, based directly on an assessment of convexity of the distribution function, using the conventional empirical distribution. The test is shown to have optimal power properties in difficult cases, where it is called upon to detect a small departure, in the form of a bump, from monotonicity. More general theoretical properties of the test and its numerical performance are explored.Comment: Published at http://dx.doi.org/10.1214/009053605000000039 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

new test for the parametric form of the variance function in nonparametric regression

In the common nonparametric regression model the problem of testing for the parametric form of the conditional variance is considered. A stochastic process based on the difference between the empirical processes obtained from the standardized nonparametric residuals under the null hypothesis (of a specific parametric form of the variance function) and the alternative is introduced and its weak convergence established. This result is used for the construction of a Cramer von Mises type statistic for testing the parametric form of the conditional variance. The finite sample properties of a bootstrap version of this test are investigated by means of a simulation study. In particular the new procedure is compared with some of the currently available methods for this problem and its performance is illustrated by means of a data example. --Bootstrap ; Kernel estimation ; Nonparametric regression ; Residual distribution ; Testing heteroscedasticity ; Testing homoscedasticity

A goodness-of-fit test for parametric and semi-parametric models in multiresponse regression

We propose an empirical likelihood test that is able to test the goodness of fit of a class of parametric and semi-parametric multiresponse regression models. The class includes as special cases fully parametric models; semi-parametric models, like the multiindex and the partially linear models; and models with shape constraints. Another feature of the test is that it allows both the response variable and the covariate be multivariate, which means that multiple regression curves can be tested simultaneously. The test also allows the presence of infinite-dimensional nuisance functions in the model to be tested. It is shown that the empirical likelihood test statistic is asymptotically normally distributed under certain mild conditions and permits a wild bootstrap calibration. Despite the large size of the class of models to be considered, the empirical likelihood test enjoys good power properties against departures from a hypothesized model within the class.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ208 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Heteroscedastic semiparametric transformation models: estimation and testing for validity

In this paper we consider a heteroscedastic transformation model, where the transformation belongs to a parametric family of monotone transformations, the regression and variance function are modelled nonparametrically and the error is independent of the multidimensional covariates. In this model, we first consider the estimation of the unknown components of the model, namely the transformation parameter, regression and variance function and the distribution of the error. We show the asymptotic normality of the proposed estimators. Second, we propose tests for the validity of the model, and establish the limiting distribution of the test statistics under the null hypothesis. A bootstrap procedure is proposed to approximate the critical values of the tests. Finally, we carry out a simulation study to verify the small sample behavior of the proposed estimators and tests.Comment: 33 pages, 1 figur

Single index regression models in the presence of censoring depending on the covariates

Consider a random vector (X',Y)', where X is d-dimensional and Y is one-dimensional. We assume that Y is subject to random right censoring. The aim of this paper is twofold. First, we propose a new estimator of the joint distribution of (X',Y)'. This estimator overcomes the common curse-of-dimensionality problem, by using a new dimension reduction technique. Second, we assume that the relation between X and Y is given by a mean regression single index model, and propose a new estimator of the parameters in this model. The asymptotic properties of all proposed estimators are obtained.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ464 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Estimation of the Error Density in a Semiparametric Transformation Model

Consider the semiparametric transformation model $\Lambda_{\theta_o}(Y)=m(X)+\epsilon$, where $\theta_o$ is an unknown finite dimensional parameter, the functions $\Lambda_{\theta_o}$ and $m$ are smooth, $\epsilon$ is independent of $X$, and \esp(\epsilon)=0. We propose a kernel-type estimator of the density of the error $\epsilon$, and prove its asymptotic normality. The estimated errors, which lie at the basis of this estimator, are obtained from a profile likelihood estimator of $\theta_o$ and a nonparametric kernel estimator of $m$. The practical performance of the proposed density estimator is evaluated in a simulation study

Estimation of a semiparametric transformation model

This paper proposes consistent estimators for transformation parameters in semiparametric models. The problem is to find the optimal transformation into the space of models with a predetermined regression structure like additive or multiplicative separability. We give results for the estimation of the transformation when the rest of the model is estimated non- or semi-parametrically and fulfills some consistency conditions. We propose two methods for the estimation of the transformation parameter: maximizing a profile likelihood function or minimizing the mean squared distance from independence. First the problem of identification of such models is discussed. We then state asymptotic results for a general class of nonparametric estimators. Finally, we give some particular examples of nonparametric estimators of transformed separable models. The small sample performance is studied in several simulations.Comment: Published in at http://dx.doi.org/10.1214/009053607000000848 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org