Estimation and inference for varying coefficient partially nonlinear models

In this paper, we extend the varying coefficient partially linear model to the varying coefficient partially nonlinear model in which the linear part of the varying coefficient partially linear model is replaced by a nonlinear function of the covariates. A profile nonlinear least squares estimation procedure for the parameter vector and the coefficient function vector of the varying coefficient partially nonlinear model is proposed and the asymptotic properties of the resulting estimators are established. We further propose a generalized likelihood ratio (GLR) test to check whether or not the varying coefficients in the model are constant. The asymptotic null distribution of the GLR statistic is derived and a residual-based bootstrap procedure is also suggested to derive the p-value of the GLR test. Some simulations are conducted to assess the performance of the proposed estimating and testing procedures and the results show that both the procedures perform well in finite samples. Furthermore, a real data example is given to demonstrate the application of the proposed model and its estimating and testing procedures.     

Higher-order asymptotics under model misspecification

Most of the higher-order asymptotic results in statistical inference available in the literature assume model correctness. The aim of this paper is to develop higher-order results under model misspecification. The density functions to O(n^?3/2) of the robust score test statistic and the robust Wald test statistic are derived under the null hypothesis, for the scalar as well as the multiparameter case. Alternate statistics which are robust to O(n^?3/2) are also proposed.     

Local power properties of some asymptotic tests in symmetric linear regression models

In this paper we obtain asymptotic expansions, up to order n^−1/2 and under a sequence of Pitman alternatives, for the nonnull distribution functions of the likelihood ratio, Wald, score and gradient test statistics in the class of symmetric linear regression models. This is a wide class of models which encompasses the t model and several other symmetric distributions with longer-than normal tails. The asymptotic distributions of all four statistics are obtained for testing a subset of regression parameters. Furthermore, in order to compare the finite-sample performance of these tests in this class of models, Monte Carlo simulations are presented. An empirical application to a real data set is considered for illustrative purposes.     

Testing Serial Correlation in Partially Linear Single-Index Errors-in-Variables Models

In this article, we propose two test statistics for testing the underlying serial correlation in a partially linear single-index model Y = η(Z ^τα) + X ^τβ + ? when X is measured with additive error. The proposed test statistics are shown to have asymptotic normal or chi-squared distributions under the null hypothesis of no serial correlation. Monte Carlo experiments are also conducted to illustrate the finite sample performance of the proposed test statistics. The simulation results confirm that these statistics perform satisfactorily in both estimated sizes and powers.     

Goodness-of-fit tests for exponentiality based on a loss-of-memory type functional equation

Three goodness-of-fit tests for exponentiality based on the functional equation characterization 1?F(2x)=[1?F(x)]² for every x?0 are proposed and shown to compare well to several popular tests against common alternative cdf's. Small-sample critical values for α=0.10,0.05 are developed for the two superior test statistics and the asymptotic null-distributions are characterized.     

Testing symmetry based on empirical likelihood

In this paper, we propose a general kth correlation coefficient between the density function and distribution function of a continuous variable as a measure of symmetry and asymmetry. We first propose a root-n moment-based estimator of the kth correlation coefficient and present its asymptotic results. Next, we consider statistical inference of the kth correlation coefficient by using the empirical likelihood (EL) method. The EL statistic is shown to be asymptotically a standard chi-squared distribution. Last, we propose a residual-based estimator of the kth correlation coefficient for a parametric regression model to test whether the density function of the true model error is symmetric or not. We present the asymptotic results of the residual-based kth correlation coefficient estimator and also construct its EL-based confidence intervals. Simulation studies are conducted to examine the performance of the proposed estimators, and we also use our proposed estimators to analyze the air quality dataset.     

Minimum distance lack-of-fit tests for fixed design

This paper discusses a class of tests of lack-of-fit of a parametric regression model when design is non-random and uniform on [0,1]. These tests are based on certain minimized distances between a nonparametric regression function estimator and the parametric model being fitted. We investigate asymptotic null distributions of the proposed tests, their consistency and asymptotic power against a large class of fixed and sequences of local nonparametric alternatives, respectively. The best fitted parameter estimate is seen to be n^1/2-consistent and asymptotically normal. A crucial result needed for proving these results is a central limit lemma for weighted degenerate U statistics where the weights are arrays of some non-random real numbers. This result is of an independent interest and an extension of a result of Hall for non-weighted degenerate U statistics.     

SOME ASYMPTOTIC PROPERTIES OF THE SIMPLE LOGLINEAR MODEL

Consistency and asymptotic normality of the maximum likelihood estimator of β in the loglinear model E(y_i) = e^α+βXi, where y_i are independent Poisson observations, 1 iaan, are proved under conditions which are near necessary and sufficient. The asymptotic distribution of the deviance test for β=β₀ is shown to be chi-squared with 1 degree of freedom under the same conditions, and a second order correction to the deviance is derived. The exponential model for censored survival data is also treated by the same methods.     

Optimal step-stress test under type I progressive group-censoring with random removals

Some traditional life tests result in no or very few failures by the end of test. In such cases, one approach is to do life testing at higher-than-usual stress conditions in order to obtain failures quickly. This paper discusses a k-level step-stress accelerated life test under type I progressive group-censoring with random removals. An exponential failure time distribution with mean life that is a log-linear function of stress and a cumulative exposure model are considered. We derive the maximum likelihood estimators of the model parameters and establish the asymptotic properties of the estimators. We investigate four selection criteria which enable us to obtain the optimum test plans. One is to minimize the asymptotic variance of the maximum likelihood estimator of the logarithm of the mean lifetime at use-condition, and the other three criteria are to maximize the determinant, trace and the smallest eigenvalue of Fisher's information matrix. Some numerical studies are discussed to illustrate the proposed criteria.     

A goodness-of-fit test for the stratified proportional hazards model for survival data

Abstract

Goodness-of-fit testing is addressed in the stratified proportional hazards model for survival data. A test statistic based on within-strata cumulative sums of martingale residuals over covariates is proposed and its asymptotic distribution is derived under the null hypothesis of model adequacy. A Monte Carlo procedure is proposed to approximate the critical value of the test. Simulation studies are conducted to examine finite-sample performance of the proposed statistic.     

Generalized Analysis-of-variance-type Test for the Single-index Quantile Model

In this paper, we are concerned with a test for the index parameter and index function in the single-index model. Based on the estimates obtained by the quantile regression, we extend the generalized analysis-of-variance-type test to the single-index model. We investigate the asymptotic behavior of the proposed test and demonstrate that its limiting null distribution follows an asymptotically χ²-distribution. The simulation studies and real data applications are conducted to illustrate the finite sample performance of the proposed methods.     

Adjusted empirical likelihood inference for additive hazards regression

ABSTRACT

This article develops an adjusted empirical likelihood (EL) method for the additive hazards model. The adjusted EL ratio is shown to have a central chi-squared limiting distribution under the null hypothesis. We also evaluate its asymptotic distribution as a non central chi-squared distribution under the local alternatives of order n^{? 1/2}, deriving the expression for the asymptotic power function. Simulation studies and a real example are conducted to evaluate the finite sample performance of the proposed method. Compared with the normal approximation-based method, the proposed method tends to have more larger empirical power and smaller confidence regions with comparable coverage probabilities.     

Asymptotic power comparison of T2-type test and likelihood ratio test for a mean vector based on two-step monotone missing data

Abstract

In a 2-step monotone missing dataset drawn from a multivariate normal population, T²-type test statistic (similar to Hotelling’s T² test statistic) and likelihood ratio (LR) are often used for the test for a mean vector. In complete data, Hotelling’s T² test and LR test are equivalent, however T²-type test and LR test are not equivalent in the 2-step monotone missing dataset. Then we interest which statistic is reasonable with relation to power. In this paper, we derive asymptotic power function of both statistics under a local alternative and obtain an explicit form for difference in asymptotic power function. Furthermore, under several parameter settings, we compare LR and T²-type test numerically by using difference in empirical power and in asymptotic power function. Summarizing obtained results, we recommend applying LR test for testing a mean vector.     

Jackknifing two-sample statistics

In this paper, a new simple method for jackknifing two-sample statistics is proposed. The method is based on a two-step procedure. In the first step, the point estimator is calculated by leaving one X (or Y) out at a time. At the second step, the point estimator obtained in the first step is further jackknifed, leaving one Y (or X) out at a time, resulting in a simple formula for the proposed point estimator. It is shown that by using the two-step procedure, the bias of the point estimator is reduced in terms of asymptotic order, from O(n⁻¹) up to O(n⁻²), under certain regularity conditions. This conclusion is also confirmed empirically in terms of finite sample numerical examples via a small-scale simulation study. We also discuss the idea of asymptotic bias to obtain parallel results without imposing some conditions that may be difficult to check or too restrictive in practice.     

U-Statistics based on spacings

In this paper, we investigate the asymptotic theory for U-statistics based on sample spacings, i.e. the gaps between successive observations. The usual asymptotic theory for U-statistics does not apply here because spacings are dependent variables. However, under the null hypothesis, the uniform spacings can be expressed as conditionally independent Exponential random variables. We exploit this idea to derive the relevant asymptotic theory both under the null hypothesis and under a sequence of close alternatives.The generalized Gini mean difference of the sample spacings is a prime example of a U-statistic of this type. We show that such a Gini spacings test is analogous to Rao's spacings test. We find the asymptotically locally most powerful test in this class, and it has the same efficacy as the Greenwood statistic.     

Estimation and inference for varying coefficient partially nonlinear errors-in-variables models

In this article, we study the varying coefficient partially nonlinear model with measurement errors in the nonparametric part. A local corrected profile nonlinear least-square estimation procedure is proposed and the asymptotic properties of the resulting estimators are established. Further, a generalized likelihood ratio (GLR) statistic is proposed to test whether the varying coefficients are constant. The asymptotic null distribution of the statistic is obtained and a residual-based bootstrap procedure is employed to compute the p-value of the statistic. Some simulations are conducted to evaluate the performance of the proposed methods. The results show that the estimating and testing procedures work well in finite samples.     

Conditional variance model checking

This paper discusses the problem of fitting a parametric model to the conditional variance function in a class of heteroscedastic regression models. The proposed test is based on the supremum of the Khmaladze type martingale transformation of a certain partial sum process of calibrated squared residuals. Asymptotic null distribution of this transformed process is shown to be the same as that of a time transformed standard Brownian motion. Test is shown to be consistent against a large class of fixed alternatives and to have nontrivial asymptotic power against a class of nonparametric n^-1/2-local$n^{-1/2}\text{-local}$ alternatives, where n is the sample size. Simulation studies are conducted to assess the finite sample performance of the proposed test and to make a finite sample comparison with an existing test.     

Statistical Inference and Applications of Mixture of Varying Coefficient Models

In this paper, we consider a new mixture of varying coefficient models, in which each mixture component follows a varying coefficient model and the mixing proportions and dispersion parameters are also allowed to be unknown smooth functions. We systematically study the identifiability, estimation and inference for the new mixture model. The proposed new mixture model is rather general, encompassing many mixture models as its special cases such as mixtures of linear regression models, mixtures of generalized linear models, mixtures of partially linear models and mixtures of generalized additive models, some of which are new mixture models by themselves and have not been investigated before. The new mixture of varying coefficient model is shown to be identifiable under mild conditions. We develop a local likelihood procedure and a modified expectation–maximization algorithm for the estimation of the unknown non‐parametric functions. Asymptotic normality is established for the proposed estimator. A generalized likelihood ratio test is further developed for testing whether some of the unknown functions are constants. We derive the asymptotic distribution of the proposed generalized likelihood ratio test statistics and prove that the Wilks phenomenon holds. The proposed methodology is illustrated by Monte Carlo simulations and an analysis of a CO₂‐GDP data set.     

Local power of some tests in exponential family nonlinear models

In this paper we obtain asymptotic expansions up to order n^−1/2 for the nonnull distribution functions of the likelihood ratio, Wald, score and gradient test statistics in exponential family nonlinear models (Cordeiro and Paula, 1989), under a sequence of Pitman alternatives. The asymptotic distributions of all four statistics are obtained for testing a subset of regression parameters and for testing the dispersion parameter, thus generalising the results given in Cordeiro et al. (1994) and Ferrari et al. (1997). We also present Monte Carlo simulations in order to compare the finite-sample performance of these tests.     

A Non‐parametric ANOVA‐type Test for Regression Curves Based on Characteristic Functions

This article studies a new procedure to test for the equality of k regression curves in a fully non‐parametric context. The test is based on the comparison of empirical estimators of the characteristic functions of the regression residuals in each population. The asymptotic behaviour of the test statistic is studied in detail. It is shown that under the null hypothesis, the distribution of the test statistic converges to a finite combination of independent chi‐squared random variables with one degree of freedom. The coefficients in this linear combination can be consistently estimated. The proposed test is able to detect contiguous alternatives converging to the null at the rate n ^{? 1 ∕ 2}. The practical performance of the test based on the asymptotic null distribution is investigated by means of simulations.